text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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COTA NT traditionally supports International Volunteers Day with a luncheon for staff and volunteers. In this way, it offers a thank you to its team of committed, hard-working senior volunteers who volunteer their time, skills and resources to COTA and to seniors in the Northern Territory. Without them our program of work would not have the success that it does. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,856 |
Home / Questions / Who was baseball's acting commissioner during the infamous 1994-95 strike?
Who was baseball's acting commissioner during the infamous 1994-95 strike?
Who had a 70s No.1 hit with Kiss You All Over?
Until 1998 by law The QE hotel must do what if you rent a room?
WHO WAS ENGLAND'S MANAGER DURING THE 1970S? | {
"redpajama_set_name": "RedPajamaC4"
} | 7,658 |
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Q: How to be notified only once for new Gmail emails? I've just moved from Android to iPhone and I'm struggling to configure my Gmail notifications to work in the same way.
On Android, I had it configured so that I didn't receive further notifications until I dismissed the previous one. If there was more than one unread email, the icon would be a multiple-envelope symbol (otherwise it was a single envelope).
In case that wasn't clear, here's a little timeline of events for my Android Gmail app:
Email arrives --> Notification!
Email arrives
Email arrives
<clear notifications>
Email arrives --> Notification!
Email arrives
Using the Gmail app on my iPhone 6, I am notified about every email that arrives. Is there any way to replicate the behaviour I enjoyed on my Android phone?
Looking in the settings on the GMail app, I notice the Notifications section only has two options: All New Emails or None. So I'm not holding out too much hope here...
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Q: angular sends an octet-stream to my webservice instead of json I have a spring web service which works fine but it will not add an 'factuur'. If i use a restclient and do a post to add an 'factuur' to my database it will succed but when i do it in my client app which is in angularjs it return a:
org.springframework.web.HttpMediaTypeNotSupportedException: Content type 'application/octet-stream' not supported
my controller in the spring webservice:
@RequestMapping(value = "/",method = RequestMethod.POST)
public @ResponseBody void add(@RequestBody FacturatieView factuur){
facturatieService.addFacturatie(factuur);
}
and in my angular app i have the following:
factory('factuurFactory', function($resource) {
return $resource('http://localhost:8084/publictms/factuur/:id?CALLBACK=JSONP_CALLBACK', {}, {
show: {method: 'GET', params: {id: '@id'}},
delete: {method: 'DELETE', params: {id: '@id'}}
});
}).
// In deze factory kan men alle voertuigen ophalen, wijzigen of een nieuwe toevoegen
factory('facturenFactory', function($resource) {
return $resource('http://localhost:8084/publictms/factuur/?CALLBACK=JSONP_CALLBACK', {}, {
all: {method: 'GET', isArray: true},
create: {method: 'POST', headers: {'Content-Type': 'application/json'}},
update: {method: 'PUT', headers: {'Content-Type': 'application/json'}}
});
}).
// CONTROLLER
// Controller om de facturen te tonen en om een factuur te verwijderen
controller('factuurCtrl', function($scope, $location, factuurFactory, facturenFactory) {
// Haal de lijst van factuur op
$scope.getList = function() {
facturenFactory.all(function(data) {
$scope.facturen = data;
});
};
// Navigeer naar een nieuw factuur aanmaken
$scope.add = function() {
$location.path('admin/facturen/nieuw');
};
// Navigeer naar de details van een factuur
$scope.update = function(factuurId) {
$location.path('admin/factuur/' + factuurId);
};
// Verwijder een factuur
$scope.delete = function(factuurId) {
factuurFactory.delete({id: factuurId}, function() {
$scope.getList();
});
};
$scope.getList();
}).
// Controller om de details van een factuur te tonen en te wijzigen
controller('factuurDetailCtrl', function($scope, $routeParams, factuurFactory, facturenFactory) {
$scope.state = true;
$scope.factuur = factuurFactory.show({id: $routeParams.id});
$scope.edit = function() {
$scope.state = false;
};
$scope.save = function() {
facturenFactory.update($scope.factuur);
$scope.state = true;
};
$scope.cancel = function() {
$scope.state = true;
};
}).
note if i set an mediatype in my controller i will not accept this octet-stream. Anyone knows why angular is sending a octet-stream instead of json? Also the update method works fine.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,147 |
GTU and Partners Stand Against Islamophobia
Challenging Recent Islamophobic Hate Speech, Interfaith Coalition Stands in Solidarity with Muslim Communities PRESS CONTACTS: Safir Ahmed, Zaytuna College, 415.595.0790, sahmed@zaytuna.edu Erin Burns, Pacific School of Religion, 904.742.9139, eburns@psr.edu Doug Davidson, Graduate Theological Union, 510.649.2423, ddavidson@gtu.e...
American Baptist Seminary of the West (ABSW)The Mira and Ajay Shingal Center for Dharma Studies (CDS) Church Divinity School of the Pacific (CDSP)The Center for Islamic Studies (CIS)The Richard S. Dinner Center for Jewish Studies (CJS)Dominican School of Philosophy and Theology (DSPT)Jesuit School of Theology of Santa Clara University (JST-SCU)Pacific Lutheran Theological Seminary of California Lutheran University (PLTS-CLU)Pacific School of Religion (PSR)San Francisco Theological Seminary (SFTS)Starr King School for the Ministry (SKSM)
ABSW | American Baptist Seminary of the WestCIS | Center for Islamic StudiesCDS | The Mira and Ajay Shingal Center for Dharma StudiesCJS | The Richard S. Dinner Center for Jewish StudiesPSR | Pacific School of ReligionJST-SCU | Jesuit School of Theology of Santa Clara Universityzaytuna collegeDSPT | Dominican School of Philosophy & TheologyCDSP | Church Divinity School of the PacificSKSM | Starr King School for the MinistryPLTS | Pacific Lutheran Theological SeminarySFTS | San Francisco Theological Seminary
In Solidarity Against Islamophobia
Thursday, December 17th 2015, 3:00pm
Challenging recent Islamophobic rhetoric, leaders from Jewish, Hindu, Buddhist, Christian, and Muslim traditions unite for illuminating dialogue and action in solidarity with Muslim communities. WHEN: Thursday, December 17th WHERE: "Holy Hill," intersection of Le Conte and Scenic Avenues, Berkeley, CA WHAT: GTU Solidarity Statement and Panel Discussion, 3pm - 5pm: Pacific School of Religion Chapel, 1798 Scenic Ave., Berkeley, CA Scholars and leaders will offer strategies and...
American Baptist Seminary of the West (ABSW)The Mira and Ajay Shingal Center for Dharma Studies (CDS) Church Divinity School of the Pacific (CDSP)The Center for Islamic Studies (CIS)The Richard S. Dinner Center for Jewish Studies (CJS)Dominican School of Philosophy and Theology (DSPT)Jesuit School of Theology of Santa Clara University (JST-SCU)Pacific Lutheran Theological Seminary of California Lutheran University (PLTS-CLU)Pacific School of Religion (PSR)San Francisco Theological Seminary (SFTS)
The Changing Landscape of Theological Education
How Seminaries Are Adapting to New Realities by Stuart J. Moore The handwriting is on the wall. As Americans distance themselves from the label of Christian, preferring "spiritual" or no affiliation, attendance continues to slip across mainline denominations. Schools for ministerial formation are struggling with lower enrollments and less denominational financial support. The composition of the Christian Church is changing and the seminaries must change with it. Why does this shift matter to the Graduate Theological Union? Sometimes we focus so much on the academic programs, M.A. and Ph.D...
Church Divinity School of the Pacific (CDSP)Jesuit School of Theology of Santa Clara University (JST-SCU)Pacific Lutheran Theological Seminary of California Lutheran University (PLTS-CLU)Pacific School of Religion (PSR)Starr King School for the Ministry (SKSM)
CurrentsFall 2013Laurie IsenbergLeAnn Snow FlesherSKSM | Starr King School for the MinistryPLTS | Pacific Lutheran Theological SeminaryPSR | Pacific School of ReligionCDSP | Church Divinity School of the PacificJST-SCU | Jesuit School of Theology of Santa Clara UniversityFST | Franciscan School of Theology
(-) Remove PSR | Pacific School of Religion filter PSR | Pacific School of Religion
Pacific Lutheran Theological Seminary of California Lutheran University (PLTS-CLU) (5) Apply Pacific Lutheran Theological Seminary of California Lutheran University (PLTS-CLU) filter
San Francisco Theological Seminary (SFTS) (4) Apply San Francisco Theological Seminary (SFTS) filter
American Baptist Seminary of the West (ABSW) (3) Apply American Baptist Seminary of the West (ABSW) filter
Dominican School of Philosophy and Theology (DSPT) (2) Apply Dominican School of Philosophy and Theology (DSPT) filter
News & Events (1) Apply News & Events filter
Sarlo Excellence in Teaching Award (1) Apply Sarlo Excellence in Teaching Award filter
The Marriage Plot (1) Apply The Marriage Plot filter | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 592 |
Home › Uncategorized › Jan. 6 was a False Flag
Jan. 6 was a False Flag
By Contributing Writer on August 1, 2021
Pelosi's Jan 6 set-up had two political objectives–both achieved. First, to stop the lawful challenge to the 2020 election certification, and second to label Trump supporters "domestic terrorists." This move refocused attention away from the election malfeasance, and was done to intimidate any citizen opposition to Pelosi's Democrat/RINO/media coup attempt. American citizens are being held as political prisoners without due process. Pelosi and her conspirators belong in jail, not the 468 political prisoners who are languishing there.
Linda Goudsmit August 1, 2021
Rep. Matt Gaetz Implies that Our Own Government Was Behind Jan. 6 False Flag — Demands Release of 14,000 hrs. of Hidden Video (VIDEO)
Rep. Matt Gaetz (R-FL) took to the House floor (without a mask) and implied that our own government was behind the Jan. 6 false flag attacks on the US Capitol.
Rep. Matt Gaetz: "…the 14,000 hours of tape could also show us who was animating that violence. Whether they were connected in any way, directed any way – by federal agencies. It raises great suspicion that we are unable to get access to this information as members of Congress."
Of course, Matt Gaetz is on to something.
Video released by Bobby Powell showed violent hoodlums in all black smashing windows, damaging the US Capitol — but Wray FBI refuses to look for them.
And Black-clad goons were also the first ones up the steps to smash windows at the US Capitol that day.
This was a setup.
Now this…
New video shows these same black-clad military operatives were the first ones up the stairs at the US Capitol on Jan. 6.
So why is the FBI not looking for these guys?
Buy at Barnesand Noble.com
Buy at BarnesandNoble.com
Buy at BarnesandNoble.com or Amazon.com
‹ It's A Matter of Life
Liberty Counsel Battles Covid-19 Mandates › | {
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\section*{Acknowledgments}
This work is supported by DOD-ONR-Office of Naval Research,
DOD-DARPA-Defense Advanced Research Projects Agency Guaranteeing AI Robustness against Deception (GARD), and Adobe, Capital One and JP Morgan faculty fellowships.
\bibliographystyle{plain}
\input{0_neurips_main.bbl}
\newpage
\section{Introduction}
\label{sec:intro}
Deep reinforcement learning (DRL) has achieved impressive results by using deep neural networks (DNN) to learn complex policies in large-scale tasks. However, well-trained DNNs may drastically fail under adversarial perturbations of the input~\citep{akhtar2018threat,chakraborty2018adversarial}. Therefore, before deploying DRL policies to real-life applications, it is crucial to improve the robustness of deep policies against adversarial attacks, especially worst-case attacks that maximally depraves the performance of trained agents~\citep{sun2021strongest}.
\begin{wrapfigure}{r}{0.23\textwidth}
\vspace{-2em}
\centering
\includegraphics[width=0.2\textwidth]{figures/example_5.eps}
\vspace{-0.5em}
\caption{\small{Policies have different vulnerabilities.}}
\label{fig:example}
\vspace{-1.3em}
\end{wrapfigure}
A line of regularization-based robust methods~\cite{zhang2020robust,oikarinen2020robust,shen2020deep} focuses on improving the robustness of the DNN itself and regularizes the policy network to output similar actions under bounded state perturbations.
However, different from supervised learning problems, the vulnerability of a deep policy comes not only from the DNN approximator, but also from the dynamics of the RL environment~\citep{zhang2021robust}.
These regularization-based methods neglect the intrinsic vulnerability of policies under the environment dynamics, and thus may still fail
under strong attacks~\citep{sun2021strongest}.
For example, in the go-home task shown in Figure~\ref{fig:example}, both the green policy and the red policy arrive home without rock collision, when there is no attack.
However, although regularization-based methods may ensure a minor action change under a state perturbation, the red policy may still be susceptible to a low reward under attacks, as a very small divergence can lead it to the bomb.
On the contrary, the green policy is more robust to adversarial attacks since it stays away from the bomb.
Therefore, besides promoting the robustness of DNN approximators (such as the policy network), it is also important to learn a policy with stronger intrinsic robustness.
There is another line of work considering the long-term robustness of a deep policy under strong adversarial attacks.
In particular, it is theoretically proved~\citep{zhang2020robust,sun2021strongest} that the strongest (worst-case) attacker against a policy can be learned as an RL problem, and training the agent under such a learned attacker can result in a robust policy. Zhang et al.~\cite{zhang2021robust} propose the \emph{Alternating Training with Learned Adversaries (ATLA)} framework, which alternately trains an RL agent and an RL attacker. Sun et al.~\citep{sun2021strongest} further propose PA-ATLA, which alternately trains an agent and the proposed more efficient PA-AD RL attacker, obtaining state-of-the-art robustness in many MuJoCo environments.
However, training an RL attacker requires extra samples from the environment, and the attacker's RL problem may even be more difficult and sample expensive to solve than the agent's original RL problem~\citep{zhang2021robust,sun2021strongest}, especially in large-scale environments such as Atari games with pixel observations. Therefore, although ATLA and PA-ATLA are able to achieve high long-term reward under attacks, they double the computational burden and sample complexity to train the robust agent.
The above analysis of existing literature suggests two main challenges in improving the adversarial robustness of DRL agents:
(1) correctly characterizing the long-term reward vulnerability of an RL policy, and
(2) efficiently training a robust agent without requiring much more effort than vanilla training.
To tackle these challenges, in this paper, we propose a generic and efficient robust training framework named \textit{{Worst-case-aware Robust RL}\xspace ({WocaR-RL}\xspace)} that estimates and improves the long-term robustness of an RL agent.
{WocaR-RL}\xspace has 3 key mechanisms.
\underline{First}, {WocaR-RL}\xspace introduces a novel \emph{{worst-attack Bellman operator}\xspace} which uses existing off-policy samples to estimate the lower bound of the policy value under the worst-case attack. Compared to prior works~\citep{zhang2021robust,sun2021strongest} which attempt to learn the worst-case attack by RL methods, {WocaR-RL}\xspace does not require any extra interaction with the environment.
\underline{Second}, using the estimated worst-case policy value, {WocaR-RL}\xspace optimizes the policy to select actions that not only achieve high natural future reward, but also achieve high worst-case reward when there are adversarial attacks. Therefore, {WocaR-RL}\xspace learns a policy with less intrinsic vulnerability.
\underline{Third}, {WocaR-RL}\xspace regularizes the policy network with a carefully designed state importance weight. As a result, the DNN approximator tolerates state perturbations, especially for more important states where decisions are crucial for future reward.
The above 3 mechanisms can also be interpreted from a geometric perspective of adversarial policy learning, as detailed in Appendix~\ref{app:understand}.
Our \textbf{contributions} can be summarized as below.
\textbf{(1)} We provide an approach to estimate the worst-case value of any policy under any bounded $\ell_p$ adversarial attacks. This helps evaluate the robustness of a policy without learning an attacker which requires extra samples and exploration.
\textbf{(2)} We propose a novel and principled robust training framework for RL, named \textit{{Worst-case-aware Robust RL}\xspace ({WocaR-RL}\xspace)}, which characterizes and improves the worst-case robustness of an agent. {WocaR-RL}\xspace can be used to robustify existing DRL algorithms (e.g. PPO~\citep{schulman2017proximal}, DQN~\citep{mnih2013playing}).
\textbf{(3)} We show by experiments that {WocaR-RL}\xspace achieve \textbf{improved robustness} against various adversarial attacks as well as \textbf{higher efficiency}, compared with state-of-the-art (SOTA) robust RL methods in many MuJoCo and Atari games.
For example, compared to the SOTA algorithm PA-ATLA-PPO~\citep{sun2021strongest} in the Walker environment, we obtain 20\% more worst-case reward (under the strongest attack algorithm), with only about 50\% training samples and 50\% running time.
Moreover, {WocaR-RL}\xspace learns \textbf{more interpretable ``robust behaviors''} than PA-ATLA-PPO in Walker as shown in Figure~\ref{fig:walker}.
\begin{figure}[!hb]
\vspace{-0.5em}
\centering
\includegraphics[width=0.85\columnwidth]{figures/compare_walker.eps}
\vspace{-0.8em}
\caption{\small{The robust Walker agents trained with \textbf{(top)} the state-of-the-art method PA-ATLA-PPO~\citep{sun2021strongest} and \textbf{(bottom)} our {WocaR-RL}\xspace.
Although PA-ATLA-PPO agent also achieves high reward under attacks, it learns to jump with one leg, which is counter-intuitive and may indicate some level of overfitting to a specific attacker.
In contrast, our {WocaR-RL}\xspace agent learns to lower down its body, which is more intuitive and interpretable.
The full agent trajectories in Walker and other environments are provided in supplementary materials as GIF figures.
}}
\label{fig:walker}
\vspace{-1em}
\end{figure}
\section{Related Work}
\label{sec:related}
\vspace{-0.5em}
\textbf{Defending against Adversarial Perturbations on State Observations.}\quad
\textbf{(1)}
\textit{Regularization-based methods}~\citep{zhang2020robust, shen2020deep,oikarinen2020robust} enforce the policy to have similar outputs under similar inputs,
which achieves certifiable performance for DQN in some Atari games. But in continuous control tasks, these methods may not reliably improve the worst-case performance. A recent work by Korkmaz~\citep{korkmaz2021investigating} points out that these adversarially trained models may still be sensible to new perturbations.
\textbf{(2)} \textit{Attack-driven methods} train DRL agents with adversarial examples. Some early works~\citep{kos2017delving, behzadan2017whatever, mandlekar2017adversarially, pattanaik2017robust} apply weak or strong gradient-based attacks on state observations to train RL agents against adversarial perturbations.
Zhang et al.~\citep{zhang2021robust} propose Alternating Training with Learned Adversaries (ATLA), which alternately trains an RL agent and an RL adversary and significantly improves the policy robustness in continuous control games.
Sun et al.~\citep{sun2021strongest} further extend this framework to PA-ATLA with their proposed more advanced RL attacker PA-AD. Although ATLA and PA-ATLA achieve strong empirical robustness, they require training an extra RL adversary that can be computationally and sample expensive.
\textbf{(3)} There is another line of work studying \textit{certifiable robustness} of RL policies.
Several works~\citep{lutjens2020certified, oikarinen2020robust, fischer2019online} computed lower bounds of the action value network $Q^\pi$ to certify robustness of action selection at every step. However, these bounds do not consider the distribution shifts caused by attacks, so some actions that appear safe for now can lead to extremely vulnerable future states and low long-term reward under future attacks. Moreover, these methods cannot apply to continuous action spaces.
Kumar et al. and Wu et al.\citep{kumar2021policy, wu2021crop} both extend randomized smoothing~\citep{cohen2019certified} to derive robustness certificates for trained policies. But these works mostly focus on theoretical analysis, and effective robust training approaches rather than robust training.
\textbf{Adversarial Defenses against Other Adversarial Attacks.}\quad
Besides observation perturbations, attacks can happen in many other scenarios. For example, the agent's executed actions can be perturbed~\citep{xiao2019characterizing,tan2020robustifying, tessler2019action,lee2020query}. Moreover, in a multi-agent game, an agent's behavior can create adversarial perturbations to a victim agent~\citep{gleave2019adversarial}.
Pinto et al.~\citep{pinto2017robust} model the competition between the agent and the attacker as a zero-sum two-player game, and train the agent under a learned attacker to tolerate both environment shifts and adversarial disturbances.
We point out that although we mainly consider state adversaries, our {WocaR-RL}\xspace can be extended to action attacks as formulated in Appendix~\ref{app:extension}.
Note that we focus on robustness against test-time attacks, different from poisoning attacks which alter the RL training process~\citep{behzadan2017vulnerability,huang2019deceptive,sun2020vulnerability,zhang2020adaptive,rakhsha2020policy}.
\textbf{Safe RL and Risk-sensitive RL.}\quad
There are several lines of work that study RL under safety/risk constraints~\citep{Heger1994ConsiderationOR,gaskett2003reinforcement,garcia2015comprehensive,bechtle2020curious,thomas2021safe} or under intrinsic uncertainty of environment dynamics~\citep{lim2013reinforcement,Mankowitz2020Robust}. However, these works do not deal with adversarial attacks, which can be adaptive to the learned policy. More comparison between these methods and our proposed method is discussed in Section~\ref{sec:alg}.
\section{Preliminaries and Background}
\label{sec:prelim}
\setlength\abovedisplayskip{2pt}
\setlength\belowdisplayskip{2pt}
\vspace{-0.5em}
\textbf{Reinforcement Learning (RL).}\quad
An RL environment is modeled by a Markov Decision Process (MDP), denoted by a tuple $\mathcal{M}=\langle\mathcal{S}, \mathcal{A}, P, R, \gamma\rangle$,
where $\mathcal{S}$ is a state space, $\mathcal{A}$ is an action space, $P: \mathcal{S} \times \mathcal{A} \rightarrow \Delta(\mathcal{S})$ is a stochastic dynamics model\footnote{$\Delta(\mathcal{X})$ denotes the space of probability distributions over $\mathcal{X}$.}, $R: \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$ is a reward function and $\gamma \in[0,1)$ is a discount factor.
An agent takes actions based on a policy $\pi: \mathcal{S} \rightarrow \Delta(\mathcal{A})$. For any policy, its \emph{natural performance} can be measured by the value function
$V^\pi(s) := \mathbb{E}_{P,\pi}[\sum_{t=0}^{\infty} \gamma^{t} R\left(s_t, a_t\right) \mid s_0=s]$,
and the action value function
$Q^\pi(s,a) := \mathbb{E}_{P,\pi}[\sum_{t=0}^{\infty} \gamma^{t} R\left(s_t, a_t\right) \mid s_0=s, a_0=a]$.
We call $V^\pi$ the \textit{natural value} and $Q^\pi$ the \textit{natural action value} in contrast to the values under attacks, as will be introduced in Section~\ref{sec:alg}.
\textbf{Deep Reinforcement Learning (DRL).}\quad
In large-scale problems, a policy can be parameterized by a neural network. For example, value-based RL methods (e.g. DQN~\citep{mnih2013playing}) usually fit a Q network and take the greedy policy $\pi(s)=\mathrm{argmax}_a Q(s,a)$. In actor-critic methods (e.g. PPO~\citep{schulman2017proximal}), the learner directly learns a policy network and a critic network.
In practice, an agent usually follows a stochastic policy during training that enables exploration, and executes a trained policy deterministically in test-time, e.g. the greedy policy learned with DQN.
Throughout this paper, we use $\pi_\theta$ to denote the training-time stochastic policy parameterized by $\theta$, while $\pi$ denotes the trained deterministic policy that maps a state to an action.
\textbf{Test-time Adversarial Attacks.}\quad
After training, the agent is deployed into the environment and executes a pre-trained fixed policy $\pi$.
An attacker/adversary, during the deployment of the agent, may perturb the state observation of the agent/victim at every time step with a certain attack budget $\epsilon$.
Note that the attacker only perturbs the inputs to the policy, and the underlying state in the environment does not change. This is a realistic setting because real-world observations can come from noisy sensors or be manipulated by malicious attacks. For example, an auto-driving car receives sensory observations; an attacker may add imperceptible noise to the camera, or perturb the GPS signal, although the underlying environment (the road) remains unchanged.
In this paper, we consider the $\ell_p$ \textit{thread model} which is widely used in adversarial learning literature: at step $t$, the attacker alters the observation $s_t$ into $\tilde{s}_t \in \mathcal{B}_\epsilon(s_t)$, where $\mathcal{B}_\epsilon(s_t)$ is a $\ell_p$ norm ball centered at $s_t$ with radius $\epsilon$.
The above setting ($\ell_p$ constrained observation attack) is the same with many prior works~\citep{huang2017adversarial, pattanaik2017robust, zhang2020robust, zhang2021robust, sun2021strongest}.
\subsection{A Closer Look at Robust RL}
\begin{figure}[!htbp]
\centering
\begin{subfigure}[t]{0.22\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/training_vanilla.eps}
\vspace{-2em}
\caption{\small{Vanilla Training}}
\label{sfig:train_vanilla}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.22\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/training_consistent.eps}
\vspace{-2em}
\caption{\small{Lipschitz-driven}}
\label{sfig:train_consistent}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.22\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/training_worst.eps}
\vspace{-2em}
\caption{\small{Attack-driven}}
\label{sfig:train_worst}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.22\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/training_ours.eps}
\vspace{-2em}
\caption{\small{Our {WocaR-RL}\xspace}}
\label{sfig:train_ours}
\end{subfigure}
\caption{Geometric understanding of different training methods following the polytope theory by~\cite{dadashi2019value} and~\cite{sun2021strongest}.
$x,y$ axes represent the policy value for $s_1\in\mathcal{S}$ and $s_2\in\mathcal{S}$.
The grey polytope depicts the value space of all policies, while the pink polytope (referred to as value perturbation polytope) contains the values of policy $\pi$ under all attacks with given constraint ($\epsilon$-radius $\ell_p$ perturbations on the state input to the policy).
$V^\pi$ denotes the value of a learned policy, and $\underline{V}^\pi$ stands for the {worst-attack value}\xspace of this policy $\pi$ (located at the bottom leftmost vertex of the value perturbation polytope). \\
\textbf{Two relations between the value perturbation polytope and policy robustness:}
The more distant the pink value perturbation polytope's bottom leftmost vertex is from the origin, the higher {worst-attack value}\xspace $\pi$ has.
The smaller the pink value perturbation polytope is, the less vulnerable the policy is (i.e., an $\epsilon$-bounded state perturbation can not lead to a drastic change of the policy value). \\
\textbf{Our method:} {WocaR-RL}\xspace makes a policy more robust via {worst-attack value estimation}\xspace, {worst-case-aware policy optimization}\xspace and {value-enhanced state regularization}\xspace, which shrink the value perturbation polytope and move the value perturbation polytope's bottom leftmost vertex away from the origin.}
\label{fig:training}
\end{figure}
In real-world applications where observations may be noisy or perturbed, it is important to ensure that the agent not only makes good decisions, but also makes safe decisions.
\textbf{Existing Robust RL Approaches.}\quad
There are many existing robust training methods for RL, and we summarize the common ideas as the following two categories.\\
\textit{\textbf{(1)} Lipschitz-driven methods:}
encourage the policy to output similar actions for any pair of clean state and perturbed state, i.e., $\min_{\theta} \max_{s\in\mathcal{S}, \tilde{s}\in\mathcal{B}_\epsilon(s)} \mathsf{Dist} (\pi_\theta(s), \pi_\theta(\tilde{s}))$, where $\mathsf{Dist}$ can be any distance metric. Therefore, the policy function (network) has small local Lipschitz constant at each clean state.
Note that this idea is similar to many certifiable robust training methods~\citep{gowal2018effectiveness} in supervised learning.
For example, Fischer et al.~\cite{fischer2019online} achieve provable robustness for DQN by applying the DiffAI~\citep{mirman2018differentiable} approach, so that the DQN agent selects the same action for any element inside $\mathcal{B}_\epsilon(s)$.
Zhang et al.~\cite{zhang2020robust} propose to minimize the total variance between $\pi(s)$ and $\pi(\tilde{s})$ using convex
relaxations of NNs.
Although Lipschitz-driven methods are relatively efficient in training,
they usually treat all states equally, and do not explicitly consider long-term rewards.
Therefore, it is hard to obtain a non-vacuous reward certification, especially in continuous-action environments.
\\
\textit{\textbf{(2)} Attack-driven methods:}
train the agent under adversarial attacks, which is analogous to Adversarial Training (AT)~\citep{madry2018towards}.
However, different from AT, a PGD attacker may not induce a robust policy in an RL problem due to the uncertainty and complexity of the environment.
Zhang et al.~\cite{zhang2021robust} propose to alternately train an agent and an RL-based ``optimal'' adversary, so that the agent can adapt to the worst-case input perturbation. Therefore, attack-driven method can be formulated as $\max_\theta \underline{V}^{\pi_\theta}$. Zhang et al.~\cite{zhang2021robust} and a follow-up work by Sun et al.~\cite{sun2021strongest} apply the alternate training approach and obtain state-of-the-art robust performance.
However, learning the optimal attacker using RL algorithms doubles the learning complexity and the required samples, making it hard to apply these methods to large-scale problems. Moreover, although these attack-driven methods improve the worst-case performance of an agent, the natural reward can be sacrificed.\\
Note that we discuss methods that improve the robustness of deep policies during training. Therefore, the focus is different from some important works~\citep{lutjens2020certified,wu2021crop,kumar2021policy} that directly use non-robust policies and execute them in a robust way.
\textbf{Our Motivation: Geometric Understanding of Robust RL.}\quad
The robustness of a learned RL policy can be understood from a geometric perspective.
Dadashi et al.~\cite{dadashi2019value} point out that the value functions of all policies in a finite MDP form a polytope, as shown by the grey area in Figure~\ref{fig:training}. Sun et al.~\cite{sun2021strongest} further find that $V^{\tilde{\pi}}$, possible values of a policy $\pi$ under all $\epsilon$-constrained $\ell_p$ perturbations, also form a polytope (pink area in Figure~\ref{fig:training}), which we refer to as the \textit{value perturbation polytope}.
Recall that in robust RL, we pursue a high natural value $V^\pi$, and a high worst-case value $\underline{V}^\pi$ which is the lower leftmost vertex of the value perturbation polytope. A vulnerable policy that outputs a different action for a perturbed state as a larger value perturbation polytope.
Lipschitz-driven methods, as Figure~\ref{fig:training}(a) shows, attempts to shrink the size of the value perturbation polytope, but does not necessarily result in a high $\underline{V}^\pi$. Attack-driven methods, as Figure~\ref{fig:training} shows, improves $\underline{V}^\pi$, but have no control over the size of the value perturbation polytope, and may not obtain a high natural value $V^\pi$.
\textbf{Our Proposed Robust RL Principle.}\quad
In contrast to prior Lipschitz-driven methods and Attack-driven methods, we propose to both ``lift the position'' and ``shrink the size'' of the value perturbation polytope.
To achieve the above principle in an efficient way, we propose to
(1) directly estimate and optimize the worst-case value of a policy without training the optimal attacker ({worst-attack value estimation}\xspace and {worst-case-aware policy optimization}\xspace mechanisms of {WocaR-RL}\xspace), and
(2) regularize the local Lipschitz constants of the policy with value-enhanced weights ({value-enhanced state regularization}\xspace mechanism of {WocaR-RL}\xspace).
See Section~\ref{sec:alg} for more details of the proposed algorithm.
\section{{Worst-case-aware Robust RL}\xspace}
\label{sec:alg}
\vspace{-0.5em}
In this section, we present \textit{{Worst-case-aware Robust RL}\xspace({WocaR-RL}\xspace)}, a generic framework that can be fused with any DRL approach to improve the adversarial robustness of an agent.
We will introduce the three key mechanisms in {WocaR-RL}\xspace: {worst-attack value estimation}\xspace, {worst-case-aware policy optimization}\xspace, and {value-enhanced state regularization}\xspace, respectively.
Then, we will illustrate how to incorporate these mechanisms into existing DRL algorithms to improve their robustness.
\textbf{Mechanism 1: Worst-attack Value Estimation}
Traditional RL aims to learn a policy with the maximal value $V^\pi$.
However, in a real-world problem where observations can be noisy or even adversarially perturbed, it is not enough to only consider the natural value $V^\pi$ and $Q^\pi$. As motivated in Figure~\ref{fig:example}, two policies with similar natural rewards can get totally different rewards under attacks.
To comprehensively evaluate how good a policy is in an adversarial scenario and to improve its robustness, we should be aware of the lowest possible long-term reward of the policy when its observation is adversarially perturbed with a certain attack budget $\epsilon$ at every step (with an $\ell_p$ attack model introduced in Section~\ref{sec:prelim}).
The worst-case value of a policy is, by definition, the cumulative reward obtained under the optimal attacker.
As justified by prior works~\citep{zhang2020robust,sun2021strongest}, for any given victim policy $\pi$ and attack budget $\epsilon>0$, there exists an optimal attacker, and finding the optimal attacker is equivalent to learning the optimal policy in another MDP.
We denote the optimal (deterministic) attacker's policy as $h^*$.
However, learning such an optimal attacker by RL algorithms requires extra interaction samples from the environment, due to the unknown dynamics. Moreover, learning the attacker by RL can be hard and expensive, especially when the state observation space is high-dimensional.
Instead of explicitly learning the optimal attacker with a large amount of samples, we propose to directly estimate the worst-case cumulative reward of the policy by characterizing the vulnerability of the given policy.
We first define the \textit{{worst-attack action value}\xspace} of policy $\pi$ as $\underline{Q}^\pi(s,a) := \mathbb{E}_{P} [\sum\nolimits_{t=0}^{\infty} \gamma^{t} R\left(s_t, \pi(h^*(s_t))\right) \mid s_0=s, a_0=a ].$
The \textit{{worst-attack value}\xspace} $\underline{V}^\pi$ can be defined using $h^*$ in the same way, as shown in Definition~\ref{def:worstv} in Appendix~\ref{app:theory}.
Then, we introduce a novel operator $\worstbell^\pi$, namely the \textit{{worst-attack Bellman operator}\xspace}, defined as below.
\begin{definition}[{Worst-attack Bellman Operator}\xspace]
\label{def:bellman}
For MDP $\mathcal{M}$, given a fixed policy $\pi$ and attack radius $\epsilon$, define the {worst-attack Bellman operator}\xspace
$\worstbell^\pi$ as
\begin{equation}
\label{eq:bellman}
\left(\worstbell^\pi Q\right)(s, a):=\mathbb{E}_{s^{\prime} \sim P(s, a)} [R(s, a)+\gamma \min_{a^{\prime} \in \mathcal{A}_{\mathrm{adv}}(s^\prime, \pi)} Q\left(s^{\prime}, a^{\prime}\right) ],
\end{equation}
where $\forall s\in\mathcal{S}$, $\mathcal{A}_{\mathrm{adv}}(s, \pi)$ is defined as
\begin{equation}
\label{eq:adv_action}
\mathcal{A}_{\mathrm{adv}}(s, \pi) := \{a\in\mathcal{A}: \exists \tilde{s}\in\mathcal{B}_\epsilon(s) \text{ s.t. } \pi(\tilde{s}) = a \}.
\end{equation}
\end{definition}
\vspace{-0.5em}
Here $\mathcal{A}_{\mathrm{adv}}(s^\prime, \pi)$ denotes the set of actions an adversary can mislead the victim $\pi$ into selecting by perturbing the state $s^\prime$ into a neighboring state $\tilde{s}\in\mathcal{B}_\epsilon(s^\prime)$. This hypothetical perturbation to the \textit{future} state $s^\prime$ is the key for characterizing the worst-case long-term reward under attack. The following theorem associates the {worst-attack Bellman operator}\xspace and the {worst-attack action value}\xspace.
\begin{theorem}[Worst-attack Bellman Operator and Worst-attack Action Value]
\label{thm:main}
For any given policy $\pi$, $\underline{\mathcal{T}}^\pi$ is a contraction whose fixed point is $\underline{Q}^\pi$, the {worst-attack action value}\xspace of $\pi$ under any $\ell_p$ observation attacks with radius $\epsilon$.
\end{theorem}
\vspace{-0.5em}
Theorem~\ref{thm:main} proved in Appendix~\ref{app:theory} suggests that the lowest possible cumulative reward of a policy under bounded observation attacks can be computed by {worst-attack Bellman operator}\xspace. The corresponding {worst-attack value}\xspace $\underline{V}^\pi$ can be obtained by $\underline{V}^\pi(s)=\min_{a\in\mathcal{A}_{\mathrm{adv}}(s, \pi)} \underline{Q}^\pi(s,a)$.
\textbf{How to Compute $\mathcal{A}_{\mathrm{adv}}$.}\quad
To obtain $\mathcal{A}_{\mathrm{adv}}(s,\pi)$, we need to identify the actions that can be the outputs of the policy $\pi$ when the input state $s$ is perturbed within $\mathcal{B}_\epsilon(s)$. This can be solved by commonly-used convex relaxation of neural networks~\citep{gowal2019scalable,zhang2018finding,wong2018provable,zhang2020towards,gowal2018effectiveness}, where layer-wise lower and upper bounds of the neural
network are derived.
That is, we calculate $\overline{\pi}$ and $\underline{\pi}$ such that $\overline{\pi}(s)\geq \pi(\hat{s}) \geq \underline{\pi}(s), \forall \hat{s}\in\mathcal{B}_\epsilon(s)$.
With such a relaxation, we can obtain a superset of $\mathcal{A}_{\mathrm{adv}}$, namely $\hat{\mathcal{A}}_{\mathrm{adv}}$. Then, the fixed point of Equation~\eqref{eq:bellman} with $\mathcal{A}_{\mathrm{adv}}$ being replaced by $\hat{\mathcal{A}}_{\mathrm{adv}}$ becomes a lower bound of the {worst-attack action value}\xspace.
For a continuous action space, $\hat{\mathcal{A}}_{\mathrm{adv}}(s,\pi)$ contains actions bounded by $\overline{\pi}(s)$ and $\underline{\pi}(s)$.
For a discrete action space, we can first compute the maximal and minimal probabilities of taking each action, and derive the set of actions that are likely to be selected.
The computation of $\hat{\mathcal{A}}_{\mathrm{adv}}$ is not expensive, as
there are many efficient convex relaxation methods~\citep{mirman2018differentiable,zhang2020towards} which compute $\overline{\pi}$ and $\underline{\pi}$ with only constant-factor more computations than directly computing $\pi(s)$.
Experiment in Section~\ref{sec:exp} verifies the efficiency of our approach, where we use a well-developed toolbox $\mathrm{auto\_LiRPA}$~\citep{xu2020automatic} to calculate the convex relaxation.
More implementation details and explanations are provided in Appendix~\ref{app:ibp}.
\textbf{Estimating Worst-attack Value.}\quad
Note that the {worst-attack Bellman operator}\xspace $\worstbell^\pi$ is similar to the optimal Bellman operator $\mathcal{T}^*$, although it uses $\min_{a\in\mathcal{A}_{\mathrm{adv}}}$ instead of $\max_{a\in\mathcal{A}}$. Therefore, once we identify $\mathcal{A}_{\mathrm{adv}}$ as introduced above, it is straightforward to compute the {worst-attack action value}\xspace using Bellman backups.
To model the {worst-attack action value}\xspace, we train a network named \textit{{worst-attack critic}\xspace}, denoted by {\small{$\underline{Q}^\pi_\phi$}}, where $\phi$ is the parameterization.
Concretely, for any mini-batch $\{s_t,a_t,r_t,s_{t+1}\}_{t=1}^N$, {\small{$\underline{Q}^\pi_\phi$}} is optimized by minimizing the following estimation loss:
\setlength\abovedisplayskip{2pt}
\setlength\belowdisplayskip{2pt}
\begin{align}
\mathcal{L}_{\mathrm{est}}(\underline{Q}^\pi_\phi) \! :=\! \frac{1}{N} \sum_{t=1}^N(\underline{y}_t \!-\underline{Q}^\pi_\phi(s_t,a_t))^{2},
\text{where }\underline{y}_t \! = r_t + \gamma \min_{\hat{a}\in\mathcal{A}_{\mathrm{adv}}(s_{t+1},\pi)} \underline{Q}^\pi_\phi(s_{t+1},\hat{a}). \label{loss:est}
\end{align}
For a discrete action space, $\mathcal{A}_{\mathrm{adv}}$ is a discrete set and solving {\small{$\underline{y}_t$}} is straightforward.
For a continuous action space, we use gradient descent to approximately find the minimizer $\hat{a}$.
Since $\mathcal{A}_{\mathrm{adv}}$ is in general small, this minimization is usually easy to solve. In MuJoCo, we find that 50-step gradient descent already converges to a good solution with little computational cost, as detailed in Appendix~\ref{app:exp:eff}.
\textbf{Differences with Worst-case Value Estimation in Related Work.}
Our proposed {worst-attack Bellman operator}\xspace is different from the worst-case Bellman operator in the literature of risk-sensitive RL~\citep{Heger1994ConsiderationOR,gaskett2003reinforcement,tamar2013scaling,garcia2015comprehensive,bechtle2020curious,thomas2021safe}, whose goal is to avoid unsafe trajectories under the intrinsic uncertainties of the MDP.
These inherent uncertainties of the environment are independent of the learned policy.
In contrast, our focus is to defend against adversarial perturbations created by malicious attackers that can be \emph{adaptive} to the policy.
The GWC reward proposed by \cite{oikarinen2020robust} also estimates the worst-case reward of a policy under state perturbations. But their evaluation is based on a greedy strategy and requires interactions with the environment, which is different from our estimation.
\textbf{Mechanism 2: Worst-case-aware Policy Optimization}
So far we have introduced how to evaluate the {worst-attack value}\xspace of a policy by learning a {worst-attack critic}\xspace.
Inspired by the actor-critic framework, where the actor policy network $\pi_\theta$ is optimized towards a direction that the critic value increases the most, we can regard {worst-attack critic}\xspace as a special critic that directs the actor to increase the {worst-attack value}\xspace. That is, we encourage the agent to select an action with a higher {worst-attack action value}\xspace, by minimizing the worst-attack policy loss below:
\setlength\abovedisplayskip{2pt}
\setlength\belowdisplayskip{2pt}
\begin{equation}
\label{loss:worst}
\mathcal{L}_{\mathrm{wst}}(\pi_\theta;\underline{Q}^\pi_\phi) := - \frac{1}{N} \sum_{t=1}^N \sum_{a\in\mathcal{A}} \pi_\theta(a|s_t) \underline{Q}^\pi_\phi(s_t, a),
\end{equation}
where $\underline{Q}^\pi_\phi$ is the {worst-attack critic}\xspace learned via $\mathcal{L}_{\mathrm{est}}$ introduced in Equation~\eqref{loss:est}.
Note that $\mathcal{L}_{\mathrm{wst}}$ is a general form, while the detailed implementation of the worst-attack policy optimization can vary depending on the architecture of $\pi_\theta$ in the base RL algorithm (e.g. PPO has a policy network, while DQN acts using the greedy policy induced by a Q network). In Appendix~\ref{app:ppo} and Appendix~\ref{app:dqn}, we illustrate how to implement $\mathcal{L}_{\mathrm{wst}}$ for PPO and DQN as two examples.
The proposed {worst-case-aware policy optimization}\xspace has several \textbf{merits} compared to prior ATLA~\citep{zhang2021robust} and PA-ATLA~\citep{sun2021strongest} methods which alternately train the agent and an RL attacker.
\textbf{(1)} Learning the optimal attacker $h^*$ requires collecting extra samples using the current policy (on-policy estimation). In contrast, {\small{$\underline{Q}^\pi_\phi$}} can be learned using off-policy samples, e.g., historical samples in the replay buffer, and thus is more suitable for training where the policy changes over time. ({\small{$\underline{Q}^\pi_\phi$}} depends on the current policy via the computation of {\small{$\mathcal{A}_{\mathrm{adv}}$}}.)
\textbf{(2)} We properly exploit the policy function that is being trained by computing the set of possibly selected actions {\small{$\hat{\mathcal{A}}_{\mathrm{adv}}$}} for any state. In contrast, ATLA~\citep{zhang2021robust} learns an attacker by treating the current policy as a black box, ignoring the intrinsic properties of the policy. PA-ATLA~\citep{sun2021strongest}, although assumes white-box access to the victim policy, also needs to explore and learn from extra on-policy interactions.
\textbf{(3)} The attacker trained with DRL methods, namely $\hat{h}^*$, is not guaranteed to converge to an optimal solution, such that the performance of $\pi$ estimated under $\hat{h}^*$ can be overly optimistic.
Our estimation, as mentioned in Mechanism 1, computes a lower bound of $\underline{Q}^\pi$ and thus can better indicate the robustness of a policy.
\textbf{Mechanism 3: Value-enhanced State Regularization}
As discussed in Section~\ref{sec:intro}, the vulnerability of a deep policy comes from both the policy's intrinsic vulnerability with the RL dynamics and the DNN approximator. The first two mechanisms of {WocaR-RL}\xspace mainly focus on the policy's intrinsic vulnerability, i.e., let the policy select actions that are less vulnerable to possible attacks in all future steps. However, if a bounded state perturbation can cause the network to output a very different action, then the $\mathcal{A}_{\mathrm{adv}}$ set will be large and $\underline{Q}^\pi$ can thus be low. Therefore, it is also important to encourage the trained policy to output similar actions for the clean state $s$ and any $\tilde{s}\in\mathcal{B}_\epsilon(s)$, as is done in prior work~\citep{zhang2020robust,shen2020deep,fischer2019online}.
But different from these prior methods, we note that different states should be treated differently. Some states are ``critical'' where selecting a bad action will result in catastrophic consequences. For example, when the agent gets close to the bomb in Figure~\ref{fig:example}, we should make the network more resistant to adversarial state perturbations.
To differentiate states based on their impacts on future reward, we propose to measure the importance of states with Definition~\ref{def:weight} below.
\begin{definition}[State Importance Weight]
\label{def:weight}
\!\!Define state importance weight of $s\in\mathcal{S}$ for policy $\pi$ as
\begin{equation}
\setlength\abovedisplayskip{2pt}
\setlength\belowdisplayskip{2pt}
w(s) = \max_{a_1 \in \mathcal{A}} Q^\pi(s, a_1) - \min_{a_2 \in \mathcal{A}} Q^\pi(s, a_2).\label{eq:weight}
\end{equation}
\end{definition}
\vspace{-1em}
\begin{wrapfigure}{r}{0.27\textwidth}
\centering
\vspace{-1em}
\begin{subfigure}[t]{0.13\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/maxQ.png}
\vspace{-0.5em}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.13\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/minQ.png}
\vspace{-0.5em}
\end{subfigure}
\vspace{-1em}
\caption{\small{States in Pong with \\
\textbf{(left)} high weight $w(s)$ and \\
\textbf{(right)} low weight $w(s)$.}}
\label{fig:pong}
\vspace{-1em}
\end{wrapfigure}
To justify whether Definition~\ref{def:weight} can characterize state importance, we train a DQN network in an Atari game Pong, and show the states with the highest weight and the lowest weight in Figure~\ref{fig:pong}, among many state samples.
We can see that the state with higher weight in Figure~\ref{fig:pong}(left) is indeed crucial for the game, as the green agent paddle is close to the ball. Conversely, a less-important state in Figure~\ref{fig:pong}(right) does not have significantly different future rewards under different actions.
Computing $w(s)$ is easy in a discrete action space, while in a continuous action space, one can use gradient descent to approximately find the maximal and the minimal Q values for a state.
Similar to the computation of Equation~\eqref{loss:est} with a continuous action space, we find that a 50-step gradient descent works well in experiments.
By incorporating the state importance weight $w(s)$, we regularize the policy network and let it pay more attention to more crucial states, by minimizing the following loss:
\setlength\abovedisplayskip{2pt}
\setlength\belowdisplayskip{2pt}
\begin{equation}
\label{loss:reg}
\mathcal{L}_{\mathrm{reg}}(\pi_\theta) = \frac{1}{N} \sum_{t=1}^N w(s_t) \max_{\tilde{s}_t\in\mathcal{B}_\epsilon(s_t)} \mathsf{Dist} (\pi_\theta(s_t), \pi_\theta(\tilde{s}_t)),
\end{equation}
where $\mathsf{Dist}$ can be any distance measure between two distributions (e.g., KL-divergence). Minimizing $\mathcal{L}_{\mathrm{reg}}$ can result in a smaller $\mathcal{A}_{\mathrm{adv}}$, and thus the {worst-attack value}\xspace will be closer to the natural value.
\textbf{{WocaR-RL}\xspace: A Generic Robust Training Framework}
\begin{wrapfigure}{r}{0.55\textwidth}
\vspace{-2em}
\centering
\includegraphics[width=0.54\columnwidth]{figures/diagram_new.eps}
\vspace{-0.5em}
\caption{Training architecture of {WocaR-RL}\xspace. (Components proposed in this paper are colored as red.)
}
\label{fig:diagram_loss}
\vspace{-1em}
\end{wrapfigure}
So far we have introduced three key mechanisms and their loss functions, $\mathcal{L}_{\mathrm{est}}$ in Equation~\eqref{loss:est}, $\mathcal{L}_{\mathrm{wst}}$ in Equation~\eqref{loss:worst} and $\mathcal{L}_{\mathrm{reg}}$ in Equation~\eqref{loss:reg}.
Then, our robust training framework {WocaR-RL}\xspace combines these losses with any base RL algorithm.
To be more specific, as shown in Figure~\ref{fig:diagram_loss}, for any base RL algorithm that trains policy $\pi_\theta$ using loss $\mathcal{L}_{\mathrm{RL}}$, we learn an extra {worst-attack critic}\xspace network $\underline{Q}^\pi_\phi$ by minimizing
\setlength\abovedisplayskip{2pt}
\setlength\belowdisplayskip{2pt}
\begin{equation}
\label{loss:critic}
\mathcal{L}_{\underline{Q}^\pi_\phi} := \mathcal{L}_{\mathrm{est}}(\underline{Q}^\pi_\phi),
\end{equation}
and combine $\mathcal{L}_{\mathrm{wst}}$ and $\mathcal{L}_{\mathrm{reg}}$ with $\mathcal{L}_{\mathrm{RL}}$ to optimize $\pi_\theta$ by minimizing
\setlength\abovedisplayskip{2pt}
\setlength\belowdisplayskip{2pt}
\begin{equation}
\label{loss:policy}
\mathcal{L}_{\pi_\theta} := \mathcal{L}_{\mathrm{RL}}(\pi_\theta) + \kappa_{\mathrm{wst}} \mathcal{L}_{\mathrm{wst}}(\pi_\theta;\underline{Q}^\pi_\phi) + \kappa_{\mathrm{reg}} \mathcal{L}_{\mathrm{reg}}(\pi_\theta),
\end{equation}
where $\kappa_{\mathrm{wst}}$ and $\kappa_{\mathrm{reg}}$ are hyperparameters balancing between natural performance and robustness.
Note that $\underline{Q}^\pi_\phi$ is trained together but independently with $\pi_\theta$ using historical transition samples, so {WocaR-RL}\xspace does not require extra samples from the environment.
{WocaR-RL}\xspace can also be interpreted from a geometric perspective based on prior RL polytope theory~\citep{dadashi2019value,sun2021strongest} as detailed in Appendix~\ref{app:understand}.
Our {WocaR-RL}\xspace is a generic robust training framework that can be used to robustify existing DRL algorithms.
We provide two case studies:
\textbf{(1)} combining {WocaR-RL}\xspace with a policy-based algorithm PPO~\citep{schulman2017proximal}, namely \textit{{WocaR-PPO}\xspace}, and
\textbf{(2)} combining {WocaR-RL}\xspace with a value-based algorithm DQN~\citep{mnih2013playing}, namely \textit{{WocaR-DQN}\xspace}.
The pseudocodes of {WocaR-PPO}\xspace and {WocaR-DQN}\xspace are illustrated in Appendix~\ref{app:ppo} and Appendix~\ref{app:dqn}.
The application of {WocaR-RL}\xspace to other DRL methods is then straightforward, since most DRL methods are either policy-based or value-based.
Next, we show by experiments that {WocaR-PPO}\xspace and {WocaR-DQN}\xspace achieve state-of-the-art robustness with superior efficiency, in various continuous control tasks and video game environments.
We also empirically verify the effectiveness of each of the 3 mechanisms of {WocaR-RL}\xspace and their weights by ablation study in Section~\ref{sec:result_effect}.
\section{Experiments and Discussion}
\label{sec:exp}
\vspace{-0.5em}
In this section, our experimental evaluations on various MuJoCo and Atari environments aim to study the following questions:
\textbf{(1)} Can {WocaR-RL}\xspace learn policies with better \textbf{robustness} under existing strong adversarial attacks?
\textbf{(2)} Can {WocaR-RL}\xspace maintain \textbf{natural performance} when improving robustness?
\textbf{(3)} Can {WocaR-RL}\xspace learn more \textbf{efficiently} during robust training?
\textbf{(4)} Is each mechanism in {WocaR-RL}\xspace \textbf{effective}?
Problem (1), (2) and (3) are answered in Section~\ref{sec:result_main} with detailed empirical results, and problem (4) is studied in Section~\ref{sec:result_effect} via ablation experiments.
\subsection{Experiments and Evaluations}
\label{sec:result_main}
\textbf{Environments.}\quad
Following most prior works~\citep{zhang2020robust,zhang2021robust,oikarinen2020robust} and the released implementation, we apply our {WocaR-RL}\xspace to PPO~\citep{schulman2017proximal} on 4 MuJoCo tasks with continuous action spaces, including Hopper, Walker2d, Halfcheetah and Ant, and to DQN~\citep{mnih2013playing} agents on 4 Atari games including Pong, Freeway, BankHeist and RoadRunner, which have high dimensional pixel inputs and discrete action spaces.
\textbf{Baselines and Implementation.}\quad
We compare our algorithm with several state-of-the-art robust training methods, including
(1) \emph{SA-PPO/SA-DQN}~\cite{zhang2020robust}: regularizing policy networks by convex relaxation.
(2) \emph{ATLA-PPO}~\cite{zhang2021robust}: alternately training an agent and an RL attacker.
(3) \emph{PA-ATLA-PPO}~\cite{sun2021strongest}: alternately training an agent and a more advanced RL attacker PA-AD.
(4) \emph{RADIAL-PPO/RADIAL-DQN}~\cite{oikarinen2020robust}: optimizing policy network by designed adversarial loss functions based on robustness bounds.
SA and RADIAL have both PPO and DQN versions, which are compared with our {WocaR-PPO}\xspace and {WocaR-DQN}\xspace.
But ATLA and PA-ATLA do not provide DQN versions, since alternately training on DQN can be expensive as explained in the original papers~\citep{sun2021strongest}. (PA-ATLA has an A2C version, which we compare in Appendix~\ref{app:exp:res}.)
Therefore, we reproduce their ATLA-PPO and PA-ATLA-PPO results and compare them with our {WocaR-PPO}\xspace.
More implementation and hyperparameter details are provided in Appendix~\ref{app:exp:imp}.
\textbf{Case I: Robust PPO for MuJoCo Continuous Control}
\textbf{Evaluation Metrics.}\quad
To reflect both the natural performance and robustness of trained agents, we report the average episodic rewards under no attack and against various attacks. For a comprehensive robustness evaluation, we attack the trained robust models with multiple existing attack methods, including:
(1) \textit{MaxDiff} \cite{zhang2020robust} (maximal action difference),
(2) \textit{Robust Sarsa (RS)} \cite{zhang2020robust} (attacking with a robust action-value function),
(3) \textit{SA-RL} \citep{zhang2020robust} (finding the optimal state adversary) and
(4) \textit{PA-AD} \citep{sun2021strongest} (the \textit{existing strongest attack} by learning the optimal policy adversary with RL).
For a clear comparison, we use the same attack radius $\epsilon$ as in most baselines~\citep{zhang2020robust,zhang2021robust,sun2021strongest}.
\textbf{Performance and Robustness of {WocaR-PPO}\xspace}\quad
Figure~\ref{fig:mujoco_curve} (left four columns) shows performance curves during training under four different adversarial attacks.
Among all four attack algorithms, {WocaR-PPO}\xspace converges much faster than baselines, and often achieves the best asymptotic robust performance, especially under the strongest PA-AD attack. It is worth emphasizing that since we train a robust agent without explicitly learning an RL attacker, our method not only obtains stronger robustness and much higher efficiency, but also a more general defense: {WocaR-PPO}\xspace obtains comprehensively superior performance against a variety of attacks compared against existing SOTA algorithms based on learned attackers (ATLA-PPO, PA-ATLA-PPO).
Additionally, in our experiments, {WocaR-PPO}\xspace learns relatively more universal defensive behaviors as shown in Figure~\ref{fig:walker}, which can physically explain why our algorithm can defend against diverse attacks. We provide policy demonstrations in multiple tasks in our supplementary materials.\\
The comparison of natural performance and the worst-case performance appears in Figure~\ref{fig:mujoco_curve} (right). We see that {WocaR-PPO}\xspace maintains competitive natural rewards under no attack compared with other baselines, which demonstrates that our algorithm gains more robustness without losing too much natural performance.
The full results of baselines and our algorithm under different attack evaluations are provided by Table~\ref{tab:mujoco_app} in Appendix~\ref{app:exp:res} (including performance under random attacks).
\input{fig_curve_add}
\textbf{Efficiency of Training {WocaR-PPO}\xspace.}\quad
The learning curves in Figure~\ref{fig:mujoco_curve} (left) directly show the sample efficiency of {WocaR-PPO}\xspace. Following the optimal settings provided in \citep{zhang2020robust, zhang2021robust, oikarinen2020robust}, our method takes \emph{50\% training steps} required by RADIAL-PPO and ATLA methods on Hopper, Walker2d, and Halfcheetah because RADIAL-PPO needs more steps to ensure convergence and ATLA methods require additional adversary training steps. When solving high dimensional environments like Ant, {WocaR-PPO}\xspace only requires \emph{75\% steps} compared with all other baselines to converge. We also provide additional results of baselines using the same training steps as {WocaR-PPO}\xspace in Appendix~\ref{app:exp:add}.\\
In terms of time efficiency, {WocaR-PPO}\xspace saves \emph{50\% training time} for convergence on Hopper, Walker2d, and Halfcheetah, and \emph{32\% time} on Ant compared with the SOTA method. Therefore, \textit{{WocaR-PPO}\xspace achieves both higher computational efficiency and higher sample efficiency than SOTA baselines.} Detailed costs in time and sampling are in Appendix~\ref{app:exp:eff}.
\textbf{Case II: Robust DQN for Atari Video Games}\quad
\textbf{Evaluation Metrics.}\quad
Since Atari games have pixel state spaces and discrete action spaces, the applicable attacking algorithms also differ from those in MuJoCo tasks.
We include the following common attacks: (1) 10-step untargeted \textit{PGD} (projected gradient descent) attack, (2) \textit{MinBest}~\citep{huang2017adversarial}, which minimizes the probability of choosing the "best" action, (3) \textit{PA-AD}~\citep{sun2021strongest}, as the state-of-the-art RL-based adversarial attack algorithm.
\textbf{Performance and Robustness of {WocaR-DQN}\xspace.}\quad
Table~\ref{tab:atari} presents the results on four Atari games under attack radius $\epsilon=3/255$, while results and analysis under smaller attack radius $1/255$ are in Appendix~\ref{app:exp:res}.
We can see that \textit{our {WocaR-DQN}\xspace consistently outperforms baselines under MinBest and PA-AD attacks in all environments, with a significant advance under the strongest (worst-case) PA-AD attacks compared with other robust agents.} Under PGD attacks, {WocaR-DQN}\xspace performs comparably with the state-of-the-art in Freeway and Pong (which are simpler games) and gains higher rewards than other agents in BankHeist and Roadrunner. Since SA-DQN and RADIAL-DQN focus on bounding and smoothing the policy network and do not consider the policy's intrinsic vulnerability, they are robust under the PGD attack but still vulnerable against the stronger PA-AD attack.
\textbf{Efficiency of Training {WocaR-DQN}\xspace.}\quad The total training time for SA-DQN, RADIAL-DQN, and our {WocaR-DQN}\xspace are roughly 35, 17, and 18 hours, respectively. All baselines are trained for 6 million frames on the same hardware. Therefore, {WocaR-DQN}\xspace is 49\% faster (and is more robust) than SA-DQN. Compared to the more advanced baseline RADIAL-DQN, although {WocaR-DQN}\xspace is 5\% slower, it achieves better robustness (539\% higher reward than RADIAL-DQN in RoadRunner).
\input{table_atari_full}
\input{fig_curve_less}
\input{fig_ablation}
\subsection{Verifying Effectiveness of {WocaR-RL}\xspace}
\label{sec:result_effect}
Now we dive deeper into the algorithmic design and verify the effectiveness of {WocaR-RL}\xspace by ablation studies on {WocaR-PPO}\xspace.
\textbf{(1) Worst-attack value estimation.}
We show the learned worst-attack value estimation, {\small{$\underline{Q}^\pi_\phi$}}, during the training process in Figure~\ref{sfig:curves_walker_value_less} and \ref{sfig:curves_ant_value_less}, in comparison with the actual reward under the strongest attack (PA-AD~\citep{sun2021strongest}) in Figure~\ref{sfig:curves_walker_reward_less} and \ref{sfig:curves_ant_reward_less}. The pink curves in both plots suggest that \emph{our worst-attack value estimation matches the trend of actual worst-case reward under attacks,} although the network estimated value and the real reward have different scales due to the commonly-used reward normalization for learning stability. Therefore, the effectiveness of our proposed worst-attack value estimation ($\mathcal{L}_{\mathrm{est}}$) is verified.
\textbf{(2) Worst-case-aware policy optimization.}
Compared to vanilla PPO and SA-PPO, we can see that \emph{{WocaR-PPO}\xspace improves the worst-attack value and the worst-case reward during training}, suggesting the effectiveness of our worst-attack value improvement ($\mathcal{L}_{\mathrm{wst}}$).
The comparison of natural rewards, as well as curves in other environments, are provided in Appendix~\ref{app:exp:curve}.
Moreover, the adjustable weight $\kappa_{\mathrm{wst}}$ in Equation~\eqref{loss:policy} controls the trade-off between natural value and worst-attack value in policy optimization. When $\kappa_{\mathrm{wst}}$ is high, the policy pays more attention to its worst-attack value. Appendix~\ref{app:exp:prefer} verifies that \emph{{WocaR-RL}\xspace, with different values of weight $\kappa_{\mathrm{wst}}$, produces different robustness and natural performance while consistently dominating other robust agents.}
\textbf{(3) Value-enhanced state regularization.}
We conduct ablation experiments to analyze the effect of two techniques: our proposed state importance weight $w(s)$ and the state regularization loss $\mathcal{L}_{\mathrm{reg}}$ \citep{zhang2020robust}. In Figure~\ref{sfig:abl_h1}, we compare the performance of the original {WocaR-PPO}\xspace to a variant of {WocaR-PPO}\xspace without the state importance weight $w(s)$ on Halfcheetah, which visually indicates that $w(s)$ can help agents boost the robustness. Since {SA-PPO}\xspace~\citep{zhang2020robust} also uses a state regularization technique, the improvement of {SA-PPO}\xspace added with $w(s)$ also show the universal effectiveness of our state importance. Without $w(s)$, our algorithm also achieves similar or better performance than baselines, but including this inexpensive technique $w(s)$ gives {WocaR-RL}\xspace a greater advantage, especially under learned strong attacks SA-RL and PA-AD.
Figure~\ref{sfig:reg_h} presents the performance of ATLA methods and our algorithm without $\mathcal{L}_{\mathrm{reg}}$ on Hopper, which verifies that {WocaR-PPO}\xspace also yields the superior performance when removing the regularization technique. And the comparison between {WocaR-PPO}\xspace and {WocaR-PPO}\xspace without $\mathcal{L}_{\mathrm{reg}}$ demonstrates that the weighted state regularization is beneficial to enhancing the robustness in our algorithm. Detailed ablation studies for $w(s)$ and $\mathcal{L}_{\mathrm{reg}}$ on four MuJoCo environments are shown in Appendix~\ref{app:exp:reg}.
\section{Conclusion and Discussion}
\label{sec:conclusion}
This paper proposes a robust RL training framework, {WocaR-RL}\xspace, that evaluates and improves the long-term robustness of a policy via {worst-attack value estimation}\xspace, {worst-case-aware policy optimization}\xspace, and {value-enhanced state regularization}\xspace.
Different from recent state-of-the-art adversarial training methods~\citep{sun2021strongest,zhang2021robust} which train an extra adversary to improve the robustness of an agent, we directly estimate and improve the lower bound of the agent's cumulative reward. As a result, {WocaR-RL}\xspace not only achieves better robustness than state-of-the-art robust RL approaches, but also halves the total sample complexity and computation complexity, in a wide range of Atari and MuJoCo tasks.
There are several aspects to improve or extend the current approach.
First, the proposed {worst-attack Bellman operator}\xspace in theory gives the exact worst-case value of a policy under $\ell_p$ bounded attacks. But in practice, it is hard to compute the set $\mathcal{A}_{\mathrm{adv}}$ directly, so we use convex relaxation to obtain a superset of it, $\hat{\mathcal{A}}_{\mathrm{adv}}$. As a result, the fixed point of {worst-attack Bellman operator}\xspace with $\mathcal{A}_{\mathrm{adv}}$ being replaced by $\hat{\mathcal{A}}_{\mathrm{adv}}$ is a lower bound of the worst-case value. Then, our algorithm increases the worst-case value by improving its lower bound, as visualized and explained in Figure~\ref{fig:training} in Appendix~\ref{app:understand}. Therefore, one potential way of further improving the robustness is using a tighter relaxation.
In addition, this paper only considers the $\ell_p$ threat model as is common in most related works. But in real-world applications, other attack models could exist (e.g. patch attacks~\citep{brown2017adversarial}), and improving the robustness of RL agents in these scenarios is another important research direction.
\section{Theoretical Analysis}
\label{app:theory}
Similar to the {worst-attack action value}\xspace, we can define the {worst-attack value}\xspace as below:
\begin{definition}[Worst-attack Value]
\label{def:worstv}
For a given policy $\pi$, define the {worst-attack value}\xspace of $\pi$ as
\begin{equation}
\underline{V}^\pi(s) := \mathbb{E}_{P} [\sum_{t=0}^{\infty} \gamma^{t} R\left(s_t, \pi(h^*(s_t))\right) \mid s_0=s],
\end{equation}
where $h^*$ is the optimal attacker which minimizes the victim's cumulative reward under the $\epsilon$ constraint.
\end{definition}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
First, we show that $\underline{\mathcal{T}}^\pi$ is a contraction.
For any two Q functions $Q_1: \mathcal{S}\times\mathcal{A} \to \mathbb{R}$ and $Q_2: \mathcal{S}\times\mathcal{A} \to \mathbb{R}$, we have
\begin{small}
\begin{equation*}
\begin{split}
&\left\|\underline{\mathcal{T}}^\pi Q_1 - \underline{\mathcal{T}}^\pi Q_2\right\|_{\infty} \\
&=\max _{s,a} \left|\sum_{s^{\prime} \in \mathcal{S}} P\left(s^{\prime} \mid s, a\right)\left[R(s, a)+\gamma \min _{a^{\prime} \in \mathcal{A}_{\mathrm{adv}}(s^{\prime}, \pi)} Q_1\left(s^{\prime}, a^{\prime}\right) - R(s, a)+\gamma \min _{a^{\prime} \in \mathcal{A}_{\mathrm{adv}}(s^{\prime}, \pi)} Q_2\left(s^{\prime}, a^{\prime}\right)\right] \right|\\
&=\gamma \max _{s,a}\left|\sum_{s^{\prime} \in \mathcal{S}} P\left(s^{\prime} \mid s, a\right) \left[\min _{a^{\prime} \in \mathcal{A}_{\mathrm{adv}}(s^{\prime}, \pi)} Q_1\left(s^{\prime}, a^{\prime}\right) - \min _{a^{\prime} \in \mathcal{A}_{\mathrm{adv}}(s^{\prime}, \pi)} Q_2\left(s^{\prime}, a^{\prime}\right)\right]\right|\\
&\leq \gamma \max _{s,a}\sum_{s^{\prime} \in \mathcal{S}} P\left(s^{\prime} \mid s, a\right) \left|\min _{a^{\prime} \in \mathcal{A}_{\mathrm{adv}}(s^{\prime}, \pi)} Q_1\left(s^{\prime}, a^{\prime}\right) - \min _{a^{\prime} \in \mathcal{A}_{\mathrm{adv}}(s^{\prime}, \pi)} Q_2\left(s^{\prime}, a^{\prime}\right)\right|\\
&\leq \gamma \max _{s,a}\sum_{s^{\prime} \in \mathcal{S}} P\left(s^{\prime} \mid s, a\right) \max _{a^{\prime} \in \mathcal{A}_{\mathrm{adv}}(s^{\prime}, \pi)}\left|Q_1\left(s^{\prime}, a^{\prime}\right) - Q_2\left(s^{\prime}, a^{\prime}\right)\right|\\
&=\gamma \max _{s,a}\sum_{s^{\prime} \in \mathcal{S}} P\left(s^{\prime} \mid s, a\right)\left\|Q_1 - Q_2\right\|_{\infty}\\
&=\gamma\left\|Q_1 - Q_2\right\|_{\infty}\\
\end{split}
\end{equation*}
\end{small}
The second inequality comes from the fact that,
\begin{equation*}
\left|\min _{x_1}f(x_1) - \min _{x_2}g(x_2)\right| \leq \max _{x}\left|f(x)-g(x)\right|
\end{equation*}
The operator $\underline{\mathcal{T}}^\pi$ satisfies,
\begin{equation*}
\left\|\underline{\mathcal{T}}^\pi Q_1 - \underline{\mathcal{T}}^\pi Q_2\right\|_{\infty} \leq \gamma\left\|Q_1 - Q_2\right\|_{\infty}
\end{equation*}
so it is a contraction in the sup-norm.
Recall the definition of {worst-attack action value}\xspace:
\begin{equation}
\underline{Q}^\pi(s,a) := \mathbb{E}_{P} [\sum_{t=0}^{\infty} \gamma^{t} R\left(s_t, \pi(h^*(s_t))\right) \mid s_0=s, a_0=a ],
\end{equation}
where $h^*$ is the optimal attacker which minimizes the victim's cumulative reward under the $\epsilon$ constraint.
That is, the optimal attacker $h^*$ lets the agent select the worst possible action among all achievable actions in $\mathcal{A}_{\mathrm{adv}}$.
Hence, we have $\underline{Q}^{\pi}(s, a)= \underline{\mathcal{T}}^\pi \underline{Q}^{\pi}(s, a)$. Therefore, $\underline{Q}^{\pi}(s, a)$ is the fixed point of the Bellman operator $\underline{\mathcal{T}}^\pi$.
\end{proof}
\section{Algorithm Details}
\label{app:algo}
\subsection{Computing $\mathcal{A}_{\mathrm{adv}}$ by Network Bounding Techniques}
\label{app:ibp}
Recall that $\mathcal{A}_{\mathrm{adv}}(s, \pi) = \{a\in\mathcal{A}: \exists \tilde{s}\in\mathcal{B}_\epsilon(s) \text{ s.t. } \pi(\tilde{s}) = a \}$ is the set of actions that $\pi$ may be misled to select in state $s$. Computing the exact $\mathcal{A}_{\mathrm{adv}}$ is difficult due to the complexity of neural networks, so we use relaxations of network such as Interval Bound Propagation (IBP)~\cite{wong2018provable, gowal2019scalable} to approximately calculate $\mathcal{A}_{\mathrm{adv}}$.
\textbf{A Brief Introduction to Convex Relaxation Methods.}\quad
Convex relaxation methods are techniques to bound a neural network that provide the upper and lower bound of the neural network output given a bounded $l_p$ perturbation to the input. In particular, we take $l_\infty$ as an example, which has been studied extensively in prior works. Formally, let $f_\theta$ be a real-valued function parameterized by a neural network $\theta$, and let $f_\theta(s)$ denote the output of the neural network with the input $s$. Given an $l_\infty$ perturbation budget $\epsilon$, convex relaxation method outputs $(\underline{f_\theta(s)}, \overline{f_\theta(s)})$ such that
\begin{equation*}
\underline{f_\theta(s)}\leq \min_{\|s^\prime-s\|_\infty\leq\epsilon}f_\theta(s')\leq
\max_{\|s^\prime-s\|_\infty\leq\epsilon}f_\theta(s')
\leq\overline{f_\theta(s)}
\end{equation*}
Recall that we use $\pi_\theta$ to denote the parameterized policy being trained that maps a state observation to a distribution over the action space, and $\pi$ denotes the deterministic policy refined from $\pi_\theta$ with $\pi(s)=\mathrm{argmax}_{a\in\mathcal{A}} \pi_\theta(a|s)$.
$\mathcal{A}_{\mathrm{adv}}(s,\pi)$ contains actions that could be selected by $\pi$ (with the highest probability in $\pi_\theta$'s output) when $s$ is perturbed within a $\epsilon$-radius ball.
Our goal is to approximately identify a superset of $\mathcal{A}_{\mathrm{adv}}(s,\pi)$, i.e., $\hat{\mathcal{A}}_{\mathrm{adv}}(s,\pi)$, via the convex relaxation of networks introduced above.\\
\textbf{Computing $\mathcal{A}_{\mathrm{adv}}$ in Continuous Action Space.}\quad
The most common policy parameterization in a continuous action space is through a Gaussian distribution. Let $\mu_\theta(s)$ be the mean of Gaussian computed by $\pi_\theta(s)$, then $\pi=\mu(s)$. Therefore, we can use network relaxation to compute an upper bound and a lower bound of $\mu_\theta$ with input $\mathcal{B}_\epsilon(s)$. Then, $\hat{\mathcal{A}}_{\mathrm{adv}}(s,\pi)=[\underline{\mu_\theta(s)}, \overline{\mu_\theta(s)}]$, i.e., a set of actions that are coordinate-wise bounded by $\underline{\mu_\theta(s)}$ and $\overline{\mu_\theta(s)}$.
For other continuous distributions, e.g., Beta distribution, the computation is similar, as we only need to find the largest and smallest actions.
In summary, we can compute $\hat{\mathcal{A}}_{\mathrm{adv}}(s,\pi)=[\underline{\pi_\theta(s)}, \overline{\pi_\theta(s)}]$.
\textbf{Computing $\mathcal{A}_{\mathrm{adv}}$ in Discrete Action Space.}\quad
For a discrete action space, the output of $\pi_\theta$ is a categorical distribution, and $\pi$ selects the action with the highest probability. Or equivalently, in value-based algorithms like DQN, the Q network (can be regarded as $\pi_\theta$) outputs the Q estimates for each action, and $\pi$ selects the action with the highest Q value.
In this case, we can compute the upper and lower bound of $\pi_\theta$ in every dimension (corresponding to an action), denoted as $\overline{a}_i,\underline{a}_i$, $\forall 1 \leq i \leq |\mathcal{A}|$.
Then, an action $a_i\in\mathcal{A}$ is in $\hat{\mathcal{A}}_{\mathrm{adv}}$ if for all $1 \leq j \leq |\mathcal{A}|, j\neq i$, we have $\overline{a}_i > \underline{a}_j$.
\textbf{Implementation details of $\mathcal{A}_{\mathrm{adv}}$}
For a continuous action space, interval bound propagation (IBP) is the cheapest method to implement convex relaxation. We use IBP+Backward relaxation provided by \textit{auto\_LiRPA} library, following \citep{zhang2020robust} to efficiently produce tighter bounds $\mathcal{A}_{\mathrm{adv}}$ for the policy networks $\pi_{theta}$. For a discrete action space, we compute the layer-wise output bounds for the Q-network by applying robustness verification algorithms from \citep{oikarinen2020robust}.
\subsection{Worst-case-aware Robust PPO ({WocaR-PPO}\xspace)}
\label{app:ppo}
In policy-based DRL methods~\citep{schulman2015trust,lillicrap2015continuous,schulman2017proximal} such as PPO, the actor policy $\pi_\theta$ is optimized so that it increases the probability of selecting actions with higher critic values. Therefore, we combine our {worst-attack critic}\xspace and the original critic function, and optimize $\pi_\theta$ such that both the natural value ($\mathcal{L}_{\mathrm{RL}}$) and the {worst-attack action value}\xspace $\underline{Q}^\pi_\phi$ ($\mathcal{L}_{\mathrm{wst}}$) can be increased ($\mathcal{L}_{\mathrm{RL}}$ and $\mathcal{L}_{\mathrm{wst}}$). At the same time, $\pi_\theta$ is also regularized by $\mathcal{L}_{\mathrm{reg}}$.
We provide the full algorithm of {WocaR-PPO}\xspace in Algorithm \ref{alg:ppo} and highlight the differences with the prior method SA-PPO. {WocaR-PPO}\xspace needs to train an additional {worst-attack critic}\xspace $\underline{Q}^\pi_\phi$ to provide the robust-PPO-clip objective. The perturbation budget $\epsilon_t$ increases slowly during training.
The implementation of $\mathcal{L}_{\mathrm{reg}}$ is the same as the SA-regularizer~\cite{zhang2020robust}. For computing the state importance weight $w_{s_t}$, because there is no Q-value network in PPO, we provide a different formula to measure the state importance without extra calculation (Line 11 in Algorithm~\ref{alg:ppo}).
\input{algo_ppo.tex}
\subsection{Worst-case-aware Robust DQN ({WocaR-DQN}\xspace)}
\label{app:dqn}
For value-based DRL methods~\citep{mnih2013playing,guez2015deep,wang2016dueling} such as DQN, a Q network is learned to evaluate the natural action value. Although the policy is not directly modeled by a network, the Q network induces a greedy policy by $\pi(s)=\mathrm{argmax}_a Q(s,a)$.
To distinguish the acting policy and the natural action value, we keep the original Q network, and learn a new Q network that serves as a robust policy. This new Q network is called a \textit{robust Q network}, denoted by $Q_r$, which is used to take greedy actions $a=\pi(s):=\mathrm{argmax}_a Q_r(s,a)$
In addition to the original vanilla Q network $Q_v$ and the robust Q network $Q_r$, we learn the {worst-attack critic}\xspace network $\underline{Q}^\pi_\phi$, which evaluates the {worst-attack action value}\xspace of the greedy policy induced by $Q_r$. Then, we update $Q_r$ by assigning higher values for actions with both high natural Q value and high {worst-attack action value}\xspace ($\mathcal{L}_{\mathrm{RL}}$ and $\mathcal{L}_{\mathrm{wst}}$), while enforcing the network to output the same action under bounded state perturbations ($\mathcal{L}_{\mathrm{reg}}$).
{WocaR-DQN}\xspace is presented in Algorithm \ref{alg:dqn}. {WocaR-DQN}\xspace trains three Q-value functions including a vanilla Q network, a worst-case Q network, and a robust Q network. The worst-case Q $\underline{Q}^\pi_\phi$ is learned to estimate the worst-case performance and the robust Q is updated using the vanilla value and worst-case value together. Moreover, a target Q network is used as the original DQN implementation, to compute the target value when updating the vanilla Q network (Line 8 to 10 in Algorithm~\ref{alg:dqn}). To learn the worst-case critic $\underline{Q}^\pi_\phi$, we select the worst-attack action from the estimated possible perturbed action set $\hat{\mathcal{A}}_{\mathrm{adv}}$ to compute the worst-case TD loss $\mathcal{L}_{\mathrm{est}}$ (Line 11 to 15). The implementation of $\mathcal{L}_{\mathrm{reg}}$ is the same as the SA-regularizer~\cite{zhang2020robust}, where the robust Q network is regularized. To update the robust Q, we use a special $y^r_i$ which combines the target Q $Q_{v^{\prime}}$ and $Q_r$ for the next state to compute the TD loss, and minimize the $\mathcal{L}_{\mathrm{reg}}$ weighted by the state importance $w(s_i)$ (Line 16 to 17). In {WocaR-DQN}\xspace, we use an increasing $\epsilon_t$ schedule and a more slowly increasing worst-case schedule $\kappa_{wst}(t)$ for robust Q training.
\input{algo_dqn.tex}
\subsection{Worst-case-aware Robust A2C ({WocaR-A2C}\xspace)}
\label{app:a2c}
We also provide WocaR-A2C based on A2C implementation in Algorithm~\ref{alg:a2c}. Differ from the original A2C, {WocaR-A2C}\xspace needs to learn an additional $\underline{Q}^\pi_\phi$ similar to {WocaR-PPO}\xspace. To learn $\underline{Q}^\pi_\phi$, we compute the output bounds for the policy network $\pi_{\theta_{\pi}}$ under $\epsilon$-bounded perturbations and then select the worst action $\hat{a}_{t+1}$ to calculate the TD-loss $\mathcal{L}_{\mathrm{est}}$ (Line 6 to 9). The solutions for state importance weight $w(s_t)$ and regularization $\mathcal{L}_{\mathrm{reg}}$ are same as {WocaR-PPO}\xspace (Line 10-11). To learn the policy network $\pi_{\theta_{\pi}}$, we minimize the $\underline{Q}^\pi_\phi$ value together with the original actor loss (Line 12).
\input{algo_a2c.tex}
\subsection{Extension to Action Attacks}
\label{app:extension}
Although our paper mainly focuses on state attack, our proposed techniques and algorithms based on the worst-attack Bellman operator can be easily extended to action attack, which is another threat model studied in previous works ~\citep{pinto2017robust,tan2020robustifying,tessler2019action}. In fact, for action attack, we even do not need to apply IBP for the worst-attack Bellman backup. We could just simply replace $\mathcal{A}_{\mathrm{adv}}$ with the set of actions that the agent could take under attack, then the rest of the algorithms will follow the exact same as the ones presented here.
\section{Experiment Details and Additional Results}
\label{app:exp}
\subsection{Implementation Details}
\label{app:exp:imp}
For reproducibility, the reported results are selected from 30 agents for different training methods with medium performance due to the high variance in RL training.
\subsubsection{PPO in MuJoCo}
\textbf{(a) PPO Baselines}\quad
\textbf{Vanilla PPO}\quad We use the optimal hyperparameters from \cite{zhang2020robust} with the original fully connected (MLP) structure as the policy network for vanilla PPO training on all environments. On Hopper, Walker2d and Halfcheetah, we train for 2 million steps (976 iterations) , and 10 million steps (4882 iterations) on Ant to ensure convergence, which are consistent with other baselines (except ATLA methods).\\
\textbf{SA-PPO}\quad We use the hyperparameters using a grid search and solve the regularizer using convex relaxation with the IBP+Backward scheme to solve the regularizer. The regularization parameter $kappa$ is chosen in $\{0.01, 0.03, 0.1, 0.3, 1.0\}$. \\
\textbf{ATLA-PPO}\quad The hyperparameters for both policy and adversary are tuned for vanilla PPO with LSTM models. A larger entropy bonus coefficient is set to allow sufficient exploration. We set $N_v = N_{\pi} = 1$ for all experiments. We train 2441 iterations for Hopper, Walker2d, and Halfcheetah as well as 4882 iterations for Ant.\\
\textbf{PA-ATLA-PPO}\quad We use the hyperparameters similar to ATLA-PPO and conduct a grid search for a part of adversary hyperparameters including the learning rate and the entropy bonus coefficient.\\
\textbf{RADIAL-PPO}\quad RADIAL-PPO applies the same value of hyperparameters from \citep{oikarinen2020robust}. We train agents with the same iterations aligning vanilla PPO for fair comparison.
\textbf{(b) PPO Attackers}\quad
For \textbf{Random} and \textbf{MaxDiff} attack, we directly use the implementation from \cite{zhang2021robust}. The reported rewards under RS attack are from 30 trained robust value function, which is used to attack agents.\\
For \textbf{SA-RL} attack, a grid search of the optimal hyperparameters for each robust agents is conducted to find the strongest attacker. The strength of the regularization $\kappa$ is set as $1 \times 10^{-6}$ to 1.\\
For \textbf{PA-AD} attack, the adversaries are trained by PPO with a grid search of hyperparameters to obtain the strongest adversary.\\
For different types of RL-based attacks, we respectively train 100 adversaries and report the worst rewards among all trained adversaries.
\textbf{(c) {WocaR-PPO}\xspace} \quad We use the same LSTM structure (single layer with 64 hidden neurons as in vanilla PPO agents. With a grid search experiment, we find the optimal hyperparameters for {WocaR-PPO}\xspace. Specially, we use PGD to compute bounds for the policy network and convex relaxation to solve the state regularization. The number of {WocaR-PPO}\xspace training steps in all environments are the same as those in vanilla PPO. We tune the adjustable weight $\kappa_{wst}$ and increase $\kappa_{wst}$ from $0$ to the target value. For Hopper, Walker2d and Halfcheetah, $\kappa_{wst}$ is linearly increasing and we set the target value as 0.8. For Ant, we choose the exponential increase and the target value as 0.5.
\subsubsection{DQN in Atari}
\textbf{(a) DQN Baselines} \quad
\textbf{Vanilla DQN}\quad We follow \cite{zhang2020robust} and \citep{oikarinen2020robust} in hyperparameters and network structures for vanilla DQN training. The implementation of all our baselines applies Double DQN~\citep{van2016deep} and Prioritized Experience Replay~\citep{schaul2015prioritized}. For each Atari environment without framestack, we normalize the pixel values to $[0, 1]$ and clip rewards to $[-1, +1]$. For reliably convergence, we run $6\times 10^6$ steps for all baselines on all environments. Additionally, we use a replay buffer with a capacity of $5\times 10$. During testing, we evaluate agents without epsilon greedy exploration for 1000 episodes.\\
\textbf{SA-DQN} \quad
SA-DQN use the same settings of network structures and hyperparameters as in vanilla DQN. The regularization parameter $\kappa$ is chosen from ${0.005, 0.01, 0.02}$ and the schedule of $\epsilon$ during training also follows \cite{zhang2020robust}. \\
\textbf{RADIAL-DQN}\quad
Following the original implementation from \cite{oikarinen2020robust}, we reproduce the results of RADIAL-DQN with our environment settings.
\textbf{(b) DQN Attackers} \quad
For \textbf{PGD} attacks, we apply 10-step untargeted PGD attacks. We also try 50-step PGD attacks, but we find that the rewards of robust agents do not further reduce.\\
For \textbf{MinBest} attacks, we use FGSM to compute state perturbations following \cite{huang2017adversarial}.\\
For \textbf{PA-AD} attacks, the PA-AD attackers are learned with the ACKTR algorithm. We use a learning rate $0.0001$ and train the attackers for 5 million frames.
\textbf{(c) {WocaR-DQN}\xspace} \quad
For {WocaR-DQN}\xspace, we keep the same network architectures and hyperparameters as in vanilla DQN agents. During training, we set the adjustable weight $\kappa_{wst}$ as 0 for the first $2 \times 10^6$ steps, and then exponentially increase it from 0 to 0.5 for $4 \times 10^6$ steps.
\subsection{Additional Experiment Results on Robustness Performance}
\label{app:exp:res}
\input{table_mujoco_app}
\textbf{MuJoCo Experiments}\quad We reported all results in Table \ref{tab:mujoco_app} including episode rewards of well-trained robust models under various adversarial attacks. Under this full adversarial evaluation, we provide a robustness comparison between baselines and our algorithm from a comprehensive angle. We report the attack performance under a common chosen perturbation budget $\epsilon$ following \cite{zhang2020robust, zhang2021robust}. Results in all four MuJoCo environments show that our {WocaR-PPO}\xspace is the most robust method.
We emphasize that Table~\ref{tab:mujoco_app} reports the final performance of all robust training baselines after convergence, but some baselines takes much more steps than our {WocaR-PPO}\xspace. Table~\ref{tab:mujoco_app_less} in Appendix~\ref{app:exp:add} compares all methods under the same number of training steps, where {WocaR-PPO}\xspace outperforms baselines more significantly.
\textbf{Atari Experiments}\quad In Table \ref{tab:atari_app}, we present performance based on DQN on four Atari environments under 1/255 and 3/255 $\epsilon$ attack. Under $\epsilon$ of 1/255, our {WocaR-DQN}\xspace achieves competitive performance under PGD attacks and outperforms all baselines under MinBest and PA-AD attacks, which shows better robustness of {WocaR-DQN}\xspace under weaker attacks.\\
Based on vanilla A2C, we implement SA-A2C\citep{zhang2020robust} and PA-ATLA-A2C\citep{sun2021strongest} as robust baselines. We implement {WocaR-A2C}\xspace to compare with ATLA methods on Atari. In Table \ref{tab:a2c_app}, under any $\epsilon$ value, our WocaR-A2C outperforms other robust baselines across different attacks. We can conclude that our method considerably enhance more robustness than ATLA methods on Atari.
\input{table_atari_app}
\input{table_a2c}
\subsection{Additional Evaluation and Ablation Studies}
\label{app:exp:abl}
\subsubsection{Robustness Evaluation Using Multiple $\epsilon$}
\label{app:exp:eps}
To study how {WocaR-PPO}\xspace performs under attacks with different value of $\epsilon$, Figure \ref{fig:eps} shows the evaluation of our algorithms under different $\epsilon$ attacks compared with the baselines in Hopper and Walker2d. We can conclude that our robustly trained model universally and significantly outperforms other robust agents considering various attack budget $\epsilon$.
\input{fig_eps}
\subsubsection{Additional Evaluation on Sample Efficiency}
\label{app:exp:add}
\input{table_mujoco_less}
In Table~\ref{tab:mujoco_app_less}, we report the performance of {WocaR-PPO}\xspace and all robust PPO baselines using the same training steps.
We find that under limited training steps, ATLA-PPO, PA-ATLA-PPO and RADIAL-PPO obtain sub-optimal robustness, which suggests that these methods are more sample-hungry. In contrast, {WocaR-PPO}\xspace converges under fewer steps and achieves best performance with a large advantage, which shows the higher efficiency of {WocaR-PPO}\xspace.
\subsubsection{Additional Results of Time Efficiency}
\label{app:exp:eff}
\input{table_efficient}
We show the training efficiency of {WocaR-PPO}\xspace from three aspects including time, training iterations, and sampling in MuJoCo environments by comparing with SA-PPO and state-of-the-art methods ATLA-PPO, PA-ATLA-PPO, and RADIAL-PPO in Table \ref{efficiency}. For a fair comparison, we use the same {\it GeForce RTX 1080 Ti GPUs} to train all the robust agents. \\
It needs to mention that in continuous action spaces when estimating the worst-case value, we solve $\min_{\hat{a}\in\hat{\mathcal{A}}_{\mathrm{adv}}} \underline{Q}^\pi_\phi(s_{t+1},\hat{a})$ using 50-step gradient descent. The running time of this 50-step gradient descent is about \textbf{1.68 seconds} per batch with batch size 128. In total, this gradient descent computation takes 18\% of the total training time, thus it is not the computation bottleneck.
Without training with an adversary, \textit{our algorithm requires much less (only 50\% or 75\%) steps to reliably converge.} {WocaR-PPO}\xspace only takes less than half of time for low-dimensional environments to converge compared to ATLA methods and RADIAL-PPO. In high-dimensional environments like Ant, we only need 4 hours for training, while ATLA methods require at least 7 hours. When solving harder tasks, the efficiency advantage of {WocaR-PPO}\xspace is more obvious.
\subsubsection{Effectiveness of Worst-attack Policy Optimization}
\label{app:exp:curve}
\input{fig_curve}
In addition to Figure~\ref{fig:curves_less}, we show the learning curves in Walker2d and Ant in Figure \ref{fig:curve} to verify the effectiveness of {worst-attack value estimation}\xspace and {worst-case-aware policy optimization}\xspace. Figure \ref{fig:curve}(a) and (d) show the natural rewards of agents during training without attacks.
The actual worst-attack rewards in Figure \ref{fig:curve}(b) and (e) refer to the the reward obtained by the agents under PA-AD attack~\citep{sun2021strongest} which is the existing strongest attacking algorithm. To study the worst-case performance during training, We evaluate PPO, SA-PPO and {WocaR-PPO}\xspace agents after every 20 iterations using all types of attacks and report the worst-case rewards for each checkpoint.
We also present the trend of the estimated worst-case values during training in Figure \ref{fig:curve}(c) and (f), which are tested by the trained worst-attack value functions $\underline{Q}^\pi_\phi$. \\
We observe from the curves that our {worst-attack critic}\xspace estimation matches the trend of actual worst-attack rewards. Also, the increases of estimated worst-attack values and actual worst-attack rewards of {WocaR-PPO}\xspace show that our {WocaR-RL}\xspace significantly improves the robustness of agents by enhancing worst-attack values.
\subsubsection{Trade-off between Natural Performance and Robustness}
\label{app:exp:prefer}
As mentioned in Section~\ref{sec:result_effect}, the adjustable weight $\kappa_{\mathrm{wst}}$ controls the trade-off between natural performance and robustness.
To discuss the effect of $\kappa_{\mathrm{wst}}$, we train agents using {WocaR-PPO}\xspace in Hopper, Walker2d, and Halfcheetah with uniformly sampled 40 different values of weight $\kappa_{\mathrm{wst}}$ in range $(0,1]$. \\
Figure~\ref{app:fig:weight} plots the worst-case performance and natural performance of robust training baselines and 10 agents trained by {WocaR-PPO}\xspace with various values of $\kappa_{wst}$. We can see that when reward under worst-case perturbations increases, it leads to a reduction of the natural reward.
The choice of the worst-case value's weight $\kappa_{wst}$ is to control the trade-off between the final natural performance and robustness. It does not affect the convergence of the algorithm. When we increase the weight of worst-case values $\kappa_{wst}$, the reward under worst-case perturbations increases, but it leads to a reduction of the natural reward. Equally, when $\kappa_{wst}$ is set close to 0, the algorithm is similar to standard training, where the policy achieves high reward under no attack, but extremely low reward under attacks. Hence, $\kappa_{wst}$ is necessary for our algorithm to balance these two kinds of performance. In practice, one can adjust $\kappa_{wst}$ according to their preferences to robustness and natural performance.
We report the results in Table \ref{tab:mujoco_app} with significant better worst-case robustness and comparable natural performance compared with baselines. {WocaR-PPO}\xspace can always find policies which dominate other robust agents.
\input{fig_weight}
\subsubsection{Additional Ablation Studies}
\label{app:exp:reg}
\input{fig_abl}
\input{fig_reg}
We provide full ablation experimental results for the state importance weight $w(s)$ and the regularization loss $\mathcal{L}_{\mathrm{reg}}$ \citep{zhang2020robust} on four MuJoCo environments.
For the state importance weight $w(s)$, we compare the performance between the original {WocaR-PPO}\xspace and {WocaR-PPO}\xspace without $w(s)$ in Figure~\ref{app:fig:abl}. Additionally, we also equip {SA-PPO}\xspace with $w(s)$ to show the universal applicability of this design.
In all four MuJoCo environments, we can see that with $w(s)$, both {WocaR-PPO}\xspace and SA-PPO get boosted robustness, verifying the effectiveness of the state importance weight.
For the state regularization loss $\mathcal{L}_{\mathrm{reg}}$, Figure~\ref{app:fig:reg} verifies that $\mathcal{L}_{\mathrm{reg}}$ enhances the robustness of {WocaR-PPO}\xspace, since the performance of {WocaR-PPO}\xspace drops without $\mathcal{L}_{\mathrm{reg}}$.
On the other hand, Figure~\ref{app:fig:reg} also compares the performance of ATLA methods and our algorithm without $\mathcal{L}_{\mathrm{reg}}$ (note that ATLA methods also regularizes the PPO policies during training). The results indicate that \textit{the decisive contribution of {WocaR-PPO}\xspace to robustness improving comes from the worst-attack-aware policy optimization. }
These ablation studies demonstrate that all the techniques are beneficial for robustness improvement and further show that our worst-case-aware training performs better than training with attackers.
\section{Geometric Understanding of {WocaR-RL}\xspace}
\label{app:understand}
\input{4_idea}
\section{Potential Societal Impacts}
\label{app:limit}
This work focuses on improving the robustness of deep RL agents, which can make RL models more reliable in high-stakes applications. Although it is generally positive for the community to build more robust agents, such robust agents may also bring some potentially negative impacts, including the possibility of robust robots replacing some occupations and causing mass unemployment.
\section*{Checklist}
\begin{enumerate}
\item For all authors...
\begin{enumerate}
\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
\answerYes{See our abstract and Section~\ref{sec:intro}.}
\item Did you describe the limitations of your work?
\answerYes{We discuss the limitations in Appendix~\ref{app:limit}.}
\item Did you discuss any potential negative societal impacts of your work?
\answerYes{We discuss the potential negative societal impacts in Appendix~\ref{app:limit}.}
\item Have you read the ethics review guidelines and ensured that your paper conforms to them?
\answerYes{Our paper conforms to the guidelines.}
\end{enumerate}
\item If you are including theoretical results...
\begin{enumerate}
\item Did you state the full set of assumptions of all theoretical results?
\answerYes{See Appendix~\ref{app:theory}.}
\item Did you include complete proofs of all theoretical results?
\answerYes{See Appendix~\ref{app:theory}.}
\end{enumerate}
\item If you ran experiments...
\begin{enumerate}
\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
\answerYes{}
\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
\answerYes{See Appendix~\ref{app:exp:imp}}
\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
\answerYes{For each empirical result, we report error bars.}
\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
\answerYes{See efficiency analysis in Appendix~\ref{app:exp:eff}.}
\end{enumerate}
\item If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
\begin{enumerate}
\item If your work uses existing assets, did you cite the creators?
\answerYes{}
\item Did you mention the license of the assets?
\answerYes{}
\item Did you include any new assets either in the supplemental material or as a URL?
\answerYes{}
\item Did you discuss whether and how consent was obtained from people whose data you're using/curating?
\answerYes{}
\item Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
\answerYes{}
\end{enumerate}
\item If you used crowdsourcing or conducted research with human subjects...
\begin{enumerate}
\item Did you include the full text of instructions given to participants and screenshots, if applicable?
\answerNA{}
\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
\answerNA{}
\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
\answerNA{}
\end{enumerate}
\end{enumerate}
\section{1.Intro}
RL is vulnerable to adversarial perturbations.
I: value matters instead of individual points (why we have state weights) (the Pong example)
SA-DQN / RADIAL-DQN: do not consider the worst case value (consider individual states separately, similarly to supervised learning)
II: efficiency matters (why we propose the bellman operator)
Existing methods achieve good performance and robustness by doing xxx. Not efficient enough
inefficiency mainly due to the evaluating of the worst case value
ATLA(sa, pa): learn an RL attacker against the current policy which requires extra samples
Ours: use the past samples to directly estimate the worst case performance of the current policy
What we propose and contributions
\section{2.}
\input{related work}
\section{3.}
\subsection{3.1 }
How to weight states
\subsection{3.2}
Worst case Bellman Operator
\section{4. Algorithm Description}
\section{5. }
\subsection{5.1}
main results
\subsection{5.2 discussion on the trade-off between two values}
Theory example and the empirical evidence
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,373 |
package gov.nih.nci.cagrid.wsenum;
import gov.nih.nci.cagrid.common.Utils;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.StringReader;
import java.io.StringWriter;
import java.lang.reflect.Field;
import java.lang.reflect.Method;
import java.util.ArrayList;
import java.util.List;
import java.util.NoSuchElementException;
import javax.xml.namespace.QName;
import javax.xml.soap.SOAPElement;
import junit.framework.TestCase;
import org.apache.axis.AxisEngine;
import org.apache.axis.EngineConfiguration;
import org.apache.axis.MessageContext;
import org.apache.axis.configuration.FileProvider;
import org.apache.axis.server.AxisServer;
import org.apache.axis.types.Duration;
import org.globus.ws.enumeration.EnumIterator;
import org.globus.ws.enumeration.IterationConstraints;
import org.globus.ws.enumeration.IterationResult;
import org.globus.ws.enumeration.TimeoutException;
import org.projectmobius.bookstore.Book;
/**
* CompleteEnumIteratorBaseTest
* Test case for complete enumeration iterator implementations
*
* @author David Ervin
*
* @created Apr 10, 2007 12:16:58 PM
* @version $Id: CompleteEnumIteratorBaseTest.java,v 1.4 2008-11-04 15:27:15 dervin Exp $
*/
public abstract class CompleteEnumIteratorBaseTest extends TestCase {
private List<Object> objectList;
private EnumIterator enumIterator;
private String iteratorClassName;
public CompleteEnumIteratorBaseTest(String iteratorClassName) {
super("EnumIter testing for " + iteratorClassName);
this.iteratorClassName = iteratorClassName;
}
protected List<Object> getObjectList() {
return objectList;
}
protected EnumIterator getEnumIterator() {
return enumIterator;
}
public void setUp() {
// need a list of data objects
objectList = new ArrayList<Object>();
for (int i = 0; i < 10; i++) {
Book b = new Book();
b.setAuthor("caGrid Testing Book " + i);
b.setISBN(String.valueOf(i));
objectList.add(b);
}
// set up the enum iterator
try {
Class iterClass = Class.forName(iteratorClassName);
Method createIteratorMethod = iterClass.getDeclaredMethod(
"createIterator", new Class[] {List.class, QName.class});
Object[] args = {objectList, TestingConstants.BOOK_QNAME};
enumIterator = (EnumIterator) createIteratorMethod.invoke(null, args);
} catch (Exception ex) {
ex.printStackTrace();
fail("Error initializing the Concurrent Iterator: " + ex.getMessage());
}
}
public void tearDown() {
enumIterator.release();
try {
enumIterator.next(new IterationConstraints());
fail("Enumeration released, but did not throw exception on next() call");
} catch (Exception ex) {
assertEquals("Unexpected exception thrown",
NoSuchElementException.class.getName(), ex.getClass().getName());
enumIterator = null;
}
}
public void testRetrieveSingleResult() {
IterationConstraints cons = new IterationConstraints(1, -1, null);
IterationResult result = enumIterator.next(cons);
SOAPElement[] rawElements = result.getItems();
assertTrue("No elements were returned", rawElements != null);
assertEquals("Unexpected number of results returned", 1, rawElements.length);
// deserialize the result
try {
String xml = rawElements[0].toString();
Book b = (Book) deserializeDocumentString(xml, Book.class);
boolean found = bookInOriginalList(b);
assertTrue("Returned book was not found in original object list", found);
} catch (Exception ex) {
ex.printStackTrace();
fail("Error deserializing result: " + ex.getMessage());
}
}
public void testRetrieveMultipleResults() {
IterationConstraints cons = new IterationConstraints(3, -1, null);
IterationResult result = enumIterator.next(cons);
SOAPElement[] rawElements = result.getItems();
assertTrue("No elements were returned", rawElements != null);
assertEquals("Unexpected number of results returned", 3, rawElements.length);
for (int i = 0; i < rawElements.length; i++) {
// deserialize the result
try {
Book b = (Book) deserializeDocumentString(
rawElements[i].toString(), Book.class);
boolean found = bookInOriginalList(b);
assertTrue("Returned book not found in original object list", found);
} catch (Exception ex) {
ex.printStackTrace();
fail("Error deserializing result: " + ex.getMessage());
}
}
}
public void testRetrieveAllResults() {
// ask for more results than we actually have
IterationConstraints cons = new IterationConstraints(objectList.size() + 1, -1, null);
IterationResult result = enumIterator.next(cons);
SOAPElement[] rawElements = result.getItems();
assertTrue("No elements were returned", rawElements != null);
assertEquals("Unexpected number of results returned",
objectList.size(), rawElements.length);
assertTrue("End of was not sequence reached", result.isEndOfSequence());
for (int i = 0; i < rawElements.length; i++) {
// deserialize the result
try {
Book b = (Book) deserializeDocumentString(
rawElements[i].toString(), Book.class);
boolean found = bookInOriginalList(b);
assertTrue("Returned book not found in original object list", found);
} catch (Exception ex) {
ex.printStackTrace();
fail("Error deserializing result: " + ex.getMessage());
}
}
}
public void testResultsTimeout() {
slowDownIterator();
// this duration (1 sec) should time out
Duration maxWait = new Duration(false, 0, 0, 0, 0, 0, 1);
// ask for all the results
IterationConstraints cons = new IterationConstraints(
objectList.size(), -1, maxWait);
try {
// this should timeout
enumIterator.next(cons);
fail("Query did not time out");
} catch (TimeoutException ex) {
// expected
} catch (Exception ex) {
ex.printStackTrace();
fail("Unexpected exception of type "
+ ex.getClass().getName() + " occured: " + ex.getMessage());
}
}
public void testCharLimitExceded() {
// ask for all the results, but only enough chars for the first element
int charCount = -1;
StringWriter writer = new StringWriter();
try {
Utils.serializeObject(objectList.get(0),
TestingConstants.BOOK_QNAME, writer);
// trim() because serializeObject() appends a newline to the writer
String text = writer.getBuffer().toString().trim();
charCount = (text.length() * 2) - 1;
} catch (Exception ex) {
ex.printStackTrace();
fail("Error determining object char count: " + ex.getMessage());
}
IterationConstraints cons = new IterationConstraints(
objectList.size(), charCount, null);
IterationResult result = enumIterator.next(cons);
SOAPElement[] rawResults = result.getItems();
assertTrue("Enumeration did not return results", rawResults != null);
assertFalse("Enumeration mistakenly returned all results",
rawResults.length == objectList.size());
assertEquals("Unexpected number of results returned", 1, rawResults.length);
// verify content
Book original = null;
Book returned = null;
try {
original = (Book) deserializeDocumentString(
writer.getBuffer().toString(), Book.class);
returned = (Book) deserializeDocumentString(
rawResults[0].toString(), Book.class);
} catch (Exception ex) {
fail("Error deserializing objects: " + ex.getMessage());
}
boolean equal = original.getAuthor().equals(returned.getAuthor())
&& original.getISBN().equals(returned.getISBN());
assertTrue("Expected book and returned book do not match", equal);
}
protected boolean bookInOriginalList(Book g) {
// verify the book is part of the original object list
for (int i = 0; i < objectList.size(); i++) {
Book current = (Book) objectList.get(i);
if (current.getAuthor().equals(g.getAuthor())
&& current.getISBN().equals(g.getISBN())) {
return true;
}
}
return false;
}
protected Object deserializeDocumentString(String xmlDocument, Class objectClass) throws Exception {
return Utils.deserializeObject(new StringReader(xmlDocument), objectClass);
}
protected MessageContext createMessageContext(InputStream configStream) {
EngineConfiguration engineConfig = new FileProvider(configStream);
AxisEngine engine = new AxisServer(engineConfig);
MessageContext context = new MessageContext(engine);
context.setEncodingStyle("");
context.setProperty(AxisEngine.PROP_DOMULTIREFS, Boolean.FALSE);
// the following two properties prevent xsd types from appearing in
// every single element in the serialized XML
context.setProperty(AxisEngine.PROP_EMIT_ALL_TYPES, Boolean.FALSE);
context.setProperty(AxisEngine.PROP_SEND_XSI, Boolean.FALSE);
return context;
}
/**
* "Fixes" the reader for xml data to wait 500ms every time it reads a line
* from disk. This effectively slows down the calls to next() inside
* the iterator to the point that timeouts are a real possibility
*/
protected void slowDownIterator() {
// parent class has the file reader
Class iterClass = enumIterator.getClass().getSuperclass();
try {
Field readerField = iterClass.getDeclaredField("fileReader");
readerField.setAccessible(true);
BufferedReader originalReader = (BufferedReader) readerField.get(enumIterator);
BufferedReader slowReader = new BufferedReader(originalReader) {
public String readLine() throws IOException {
try {
Thread.sleep(500);
} catch (Exception ex) {
// whatever
}
return super.readLine();
}
};
readerField.set(enumIterator, slowReader);
} catch (Exception ex) {
ex.printStackTrace();
fail("Error slowing down the iterator: " + ex.getMessage());
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,056 |
<!doctype html>
<html>
<head>
<title>Twitter Tweet Button</title>
<script type="text/javascript" src="js/jquery-1.7.2.min.js"></script>
<script type="text/javascript" src="js/twitter.widgets.js"></script>
<script type="text/javascript" src="js/libjs.O2.min.js"></script>
<style type="text/css">
body{
font-size: 2.5em;
font-family: monaco;
color:#BFB;
background-color: #000;
text-shadow: #FFF 0px 0px 20px;
}
</style>
</head>
<body>
<h1>Twitter Demo </h1>
<h2>modified by PG</h2>
<h1 id="PG_go">GO</h1>
<div id="twitcontainer">
<a id="twitbuttdest" href="https://twitter.com/share" data-url="http://jterrace.github.com/js.js/twitter/" data-text="I just clicked a Twitter button created by running Twitter's script inside #js.js" class="twitter-share-button" data-via="jterrace" data-size="large">Tweet</a>
<pre id="PG_pre">
</pre>
</div>
<script type="text/javascript">
function reportMessage(m) {
console.log(m);
$("#PG_pre").append(m+"\n");
}
//$(document).ready(function(){alert("OK");});
$("#PG_go").click(function() {
$("#PG_go").html("Running....");
PG_start = new Date().getTime();
alert("running");
var srcContents = WIDGETS_JS;
function globalResolveProperty(propName) {
reportMessage("global resolve " + propName);
return 1;
}
function globalGetProperty(propName) {
reportMessage('getting global property ' + propName);
return undefined;
}
var jsObjs = JSJS.Init({'global_getter': JSJS.wrapGetter(globalGetProperty, JSJS.Types.bool),
'global_resolver': JSJS.wrapResolver(globalResolveProperty)});
function locationGetProperty(propName) {
reportMessage("location get property " + propName);
switch(propName) {
case "hash":
return {'type': JSJS.Types.charPtr, 'val': window.location.hash};
break;
case "host":
return {'type': JSJS.Types.charPtr, 'val': window.location.host};
break;
case "hostname":
return {'type': JSJS.Types.charPtr, 'val': window.location.hostname};
break;
case "href":
return {'type': JSJS.Types.charPtr, 'val': window.location.href};
break;
case "pathname":
return {'type': JSJS.Types.charPtr, 'val': window.location.pathname};
break;
case "port":
return {'type': JSJS.Types.double, 'val': window.location.port};
break;
case "protocol":
return {'type': JSJS.Types.charPtr, 'val': window.location.protocol};
break;
case "search":
return {'type': JSJS.Types.charPtr, 'val': window.location.search};
break;
}
throw "Not Implemented location prop - " + propName;
}
var jsLocationClass = JSJS.CreateClass(JSJS['JSCLASS_GLOBAL_FLAGS'],
JSJS['PropertyStub'],
JSJS['PropertyStub'],
JSJS.wrapGetter(locationGetProperty, JSJS.Types.bool),
JSJS['StrictPropertyStub'],
JSJS['EnumerateStub'],
JSJS['ResolveStub'],
JSJS['ConvertStub'],
JSJS['FinalizeStub']);
var jsLocation = JSJS.DefineObject(jsObjs.cx, jsObjs.glob, "location", jsLocationClass, 0, 0);
function screenGetProperty(propName) {
reportMessage("screen get property " + propName);
switch(propName) {
case "width":
return {'type': JSJS.Types.double, 'val': screen.width};
break;
case "height":
return {'type': JSJS.Types.double, 'val': screen.height};
break;
}
throw "Not Implemented screen prop - " + propName;
}
var jsScreenClass = JSJS.CreateClass(JSJS['JSCLASS_GLOBAL_FLAGS'],
JSJS['PropertyStub'],
JSJS['PropertyStub'],
JSJS.wrapGetter(screenGetProperty, JSJS.Types.bool),
JSJS['StrictPropertyStub'],
JSJS['EnumerateStub'],
JSJS['ResolveStub'],
JSJS['ConvertStub'],
JSJS['FinalizeStub']);
var jsScreen = JSJS.DefineObject(jsObjs.cx, jsObjs.glob, "screen", jsScreenClass, 0, 0);
function navigatorGetProperty(propName) {
reportMessage("navigator get property " + propName);
switch(propName) {
case "doNotTrack":
return {'type': JSJS.Types.double, 'val': 1};
break;
case "userAgent":
return {'type': JSJS.Types.charPtr, 'val': navigator.userAgent};
break;
}
throw "Not Implemented navigator prop - " + propName;
}
var jsNavigatorClass = JSJS.CreateClass(JSJS['JSCLASS_GLOBAL_FLAGS'],
JSJS['PropertyStub'],
JSJS['PropertyStub'],
JSJS.wrapGetter(navigatorGetProperty, JSJS.Types.bool),
JSJS['StrictPropertyStub'],
JSJS['EnumerateStub'],
JSJS['ResolveStub'],
JSJS['ConvertStub'],
JSJS['FinalizeStub']);
var jsNavigator = JSJS.DefineObject(jsObjs.cx, jsObjs.glob, "navigator", jsNavigatorClass, 0, 0);
function wrapEventObject(evOrig, srcPtr) {
function eventGetProperty(propName) {
reportMessage("event get property " + propName);
switch(propName) {
case "source":
return {'type': JSJS.Types.objPtr, 'val': srcPtr};
break;
case "data":
return {'type': JSJS.Types.charPtr, 'val': evOrig.data};
break;
}
throw "Not Implemented event prop - " + propName;
}
var jsEventClass = JSJS.CreateClass(JSJS['JSCLASS_GLOBAL_FLAGS'],
JSJS['PropertyStub'],
JSJS['PropertyStub'],
JSJS.wrapGetter(eventGetProperty, JSJS.Types.bool),
JSJS['StrictPropertyStub'],
JSJS['EnumerateStub'],
JSJS['ResolveStub'],
JSJS['ConvertStub'],
JSJS['FinalizeStub']);
var jsEvent = JSJS.NewObject(jsObjs.cx, jsEventClass, 0, 0);
return jsEvent;
}
var wrappedMap = {};
function makeWrappedElement(elem, getter, setter, resolver) {
function _elementBoxGetProperty(propName) {
reportMessage('element box get ' + propName);
return true;
}
function _elementBoxSetProperty(cx, obj, idStrReal, strict, val, jsval) {
reportMessage('setting element box property ' + idStrReal + " = " + val + " / " + jsval);
return undefined;
}
function _elementBoxResolveProperty(idStrReal) {
reportMessage("element box resolve property " + idStrReal);
return 1;
}
var elementBoxGetProperty = getter || _elementBoxGetProperty;
var elementBoxSetProperty = setter || _elementBoxSetProperty
var elementBoxResolveProperty = resolver || _elementBoxResolveProperty;
var jsElementBox = JSJS.CreateClass(JSJS['JSCLASS_GLOBAL_FLAGS'],
JSJS['PropertyStub'],
JSJS['PropertyStub'],
JSJS.wrapGetter(elementBoxGetProperty, JSJS.Types.bool),
JSJS.wrapSetter(elementBoxSetProperty),
JSJS['EnumerateStub'],
JSJS.wrapResolver(elementBoxResolveProperty),
JSJS['ConvertStub'],
JSJS['FinalizeStub']);
var objPtr = JSJS.NewObject(jsObjs.cx, jsElementBox, 0, 0);
function elementCompareDocumentPosition(otherNode) {
reportMessage("elementCompareDocumentPosition");
throw "Not Implemented";
return 0;
}
var wrappedCompareDocumentPosition = JSJS.wrapFunction({
func: elementCompareDocumentPosition,
args: [JSJS.Types.objPtr],
returns: JSJS.Types.double});
var ptrCompareDocumentPosition = JSJS.NewFunction(jsObjs.cx, wrappedCompareDocumentPosition, 1, "compareDocumentPosition");
JSJS.SetProperty(jsObjs.cx, objPtr, "compareDocumentPosition", ptrCompareDocumentPosition);
wrappedMap[objPtr] = elem;
return objPtr;
}
function jsGetElementsByTagName(something) {
reportMessage("getelementsbytagname " + something);
return [];
}
var wrappedGetElementsByTagName = JSJS.wrapFunction({
func: jsGetElementsByTagName,
args: [JSJS.Types.charPtr],
returns: JSJS.Types.arrayPtr});
var ptrGetElementsByTagName = JSJS.NewFunction(jsObjs.cx, wrappedGetElementsByTagName, 1, "getElementsByTagName");
function jsGetElementById(elemId) {
reportMessage("getelementbyid " + elemId);
var elem = document.getElementById(elemId);
if (elem == null) {
return 0;
} else {
throw "Not implemented getelementbyid";
}
}
var wrappedGetElementById = JSJS.wrapFunction({
func: jsGetElementById,
args: [JSJS.Types.charPtr],
returns: JSJS.Types.objPtr});
var ptrGetElementById = JSJS.NewFunction(jsObjs.cx, wrappedGetElementById, 1, "getElementById");
function jsIframeCloneNode(deep) {
reportMessage("cloneIframeNode ");
{
var wrappedElem = document.createElement("iframe");
function jsIframeSetAttribute(attrName, attrValue) {
reportMessage("wrapped Iframe setAttribute " + attrName);
var attrConverted = JSJS.identifyConvertValue(jsObjs.cx, attrValue);
switch (attrName) {
case "allowTransparency":
case "frameBorder":
case "scrolling":
case "tabIndex":
case "name":
reportMessage("setting iframe " + attrName + "=" + attrConverted);
wrappedElem.setAttribute(attrName, attrConverted);
return null;
}
throw "Not Allowed";
}
var wrappedIframeSetAttribute = JSJS.wrapFunction({
func: jsIframeSetAttribute,
args: [JSJS.Types.charPtr, JSJS.Types.dynamicPtr],
returns: null});
var ptrIframeSetAttribute = JSJS.NewFunction(jsObjs.cx, wrappedIframeSetAttribute, 2, "setAttribute");
function jsIframeGetAttribute(attrName) {
reportMessage("wrapped Iframe getAttribute " + attrName);
switch (attrName) {
case "style":
return {'type': JSJS.Types.charPtr, 'val': wrappedElem.style};
break;
}
throw "Not Implemented";
}
var wrappedIframeGetAttribute = JSJS.wrapFunction({
func: jsIframeGetAttribute,
args: [JSJS.Types.charPtr],
returns: null});
var ptrIframeGetAttribute = JSJS.NewFunction(jsObjs.cx, wrappedIframeGetAttribute, 1, "getAttribute");
function jsIframeAddEventListener(type, listener, useCapture) {
reportMessage("wrapped Iframe addEventListener " + type);
if (type == "load") {
wrappedElem.addEventListener("load", function() {
var jsvalout = JSJS.CreateJSVal(jsObjs.cx, listener, JSJS.Types.objPtr);
reportMessage("about to call iframe callback");
JSJS.CallFunctionValue(jsObjs.cx, jsObjs.glob, jsvalout);
reportMessage("finished calling iframe load");
});
return undefined;
}
throw "Not Implemented";
}
var wrappedIframeAddEventListener = JSJS.wrapFunction({
func: jsIframeAddEventListener,
args: [JSJS.Types.charPtr, JSJS.Types.objPtr, JSJS.Types.bool],
numRequired: 2,
returns: null});
var ptrIframeAddEventListener = JSJS.NewFunction(jsObjs.cx, wrappedIframeAddEventListener, 1, "addEventListener");
return makeWrappedElement(wrappedElem,
function getter(propName) {
reportMessage("iframe getter " + propName);
switch (propName) {
case "id":
return {'type': JSJS.Types.charPtr, 'val': wrappedElem.id};
case "setAttribute":
return {'type': JSJS.Types.funcPtr, 'val': ptrIframeSetAttribute};
case "getAttribute":
return {'type': JSJS.Types.funcPtr, 'val': ptrIframeGetAttribute};
case "style":
return {'type': JSJS.Types.charPtr, 'val': wrappedElem.style};
case "addEventListener":
return {'type': JSJS.Types.funcPtr, 'val': ptrIframeAddEventListener};
case "contentWindow":
return {'type': JSJS.Types.objPtr, 'val': makeWrappedElement(wrappedElem.contentWindow)};
}
},
function setter(cx, obj, propName, strict, val, jsval) {
reportMessage("iframe setter " + propName);
if (propName == "id") {
wrappedElem.id = val;
} else if (propName == "src") {
var l = document.createElement("a");
l.href = val;
if (l.hostname == "platform.twitter.com") {
reportMessage("setting iframe source to " + val);
wrappedElem.src = val;
}
}
},
function resolver(propName) {
reportMessage("iframe resolve " + propName);
return 1;
}
);
}
}
var wrappedIframeCloneNode = JSJS.wrapFunction({
func : jsIframeCloneNode,
args : [ JSJS.Types.bool ],
returns : JSJS.Types.objPtr
});
var ptrIframeClone = JSJS.NewFunction(jsObjs.cx,
wrappedIframeCloneNode, 1, "cloneNode");
var twitButContainer = document.getElementById('twitcontainer');
var twitButDestA = document.getElementById('twitbuttdest');
function jsCreateElement(elemName) {
reportMessage("create element " + elemName);
if (elemName == "a") {
return JSJS.NewObject(jsObjs.cx, 0, 0, jsDocument);
} else if (elemName == "iframe") {
return makeWrappedElement(document.createElement("iframe"),
function(propName) {
reportMessage("wrapped iframe getter " + propName);
switch (propName) {
case "cloneNode":
return {
'type' : JSJS.Types.funcPtr,
'val' : ptrIframeClone
};
break;
}
});
} else if (elemName == "span") {
var wrapperSpan = document.createElement("span");
return makeWrappedElement(wrapperSpan,
function(propName) {
reportMessage("wrapped span getter " + propName);
function spanSetAttribute(attrName, attrValue) {
reportMessage("span setAttribute " + attrName);
var attrConverted = JSJS.identifyConvertValue(jsObjs.cx, attrValue);
switch(attrName) {
case "style":
wrapperSpan.setAttribute(attrName, attrConverted);
break;
default:
throw "not implemented span setattribute";
}
}
var wrappedSpanSetAttribute = JSJS.wrapFunction({
func: spanSetAttribute,
args: [JSJS.Types.charPtr, JSJS.Types.dynamicPtr],
returns: null});
var ptrSpanSetAttribute = JSJS.NewFunction(jsObjs.cx, wrappedSpanSetAttribute, 1, "setAttribute");
switch (propName) {
case "setAttribute":
return {type: JSJS.Types.funcPtr, val: ptrSpanSetAttribute};
break;
case "clientWidth":
return {type: JSJS.Types.double, val: 0};
break;
case "clientHeight":
return {type: JSJS.Types.double, val: 0};
break;
case "offsetHeight":
return {type: JSJS.Types.double, val: 19};
break;
case "offsetWidth":
return {type: JSJS.Types.double, val: 21};
break;
}
throw "not implemented span";
});
} else if (elemName == "div") {
var wrapperDiv = document.createElement("div");
var wrappedIframeChild;
return makeWrappedElement(wrapperDiv,
function(propName) {
reportMessage("wrapped div getter " + propName);
switch (propName) {
case "firstChild":
function divIFSetAttribute(attrName, attrValue) {
reportMessage("div iframe setAttribute " + attrName);
var attrConverted = JSJS.identifyConvertValue(jsObjs.cx, attrValue);
switch(attrName) {
case "title":
wrapperDiv.firstChild.setAttribute(attrName, attrValue);
break;
default:
throw "not implemented";
}
}
var wrappedDivIFSetAttribute = JSJS.wrapFunction({
func: divIFSetAttribute,
args: [JSJS.Types.charPtr, JSJS.Types.dynamicPtr],
returns: null});
var ptrDivIFSetAttribut = JSJS.NewFunction(jsObjs.cx, wrappedDivIFSetAttribute, 1, "setAttribute");
wrappedIframeChild = makeWrappedElement(wrapperDiv.firstChild,
function(propName) {
reportMessage("wrapped iframe get " + propName);
switch (propName) {
case "style":
return {type: JSJS.Types.charPtr, val: wrapperDiv.firstChild.style};
case "setAttribute":
return {type: JSJS.Types.funcPtr, val: ptrDivIFSetAttribut};
default:
throw "wrappedno";
}
},
function(cx, obj, propName, strict, val, jsval) {
reportMessage("wrapped iframe set " + propName);
switch (propName) {
case "compareDocumentPosition":
return;
case "src":
var l = document.createElement("a");
l.href = val;
if (l.hostname == "platform.twitter.com") {
reportMessage("setting iframe source to " + val);
wrapperDiv.firstChild.src = val;
return;
}
case "className":
wrapperDiv.firstChild.className = val;
return;
}
throw "wrappedno2";
});
return {type: JSJS.Types.objPtr, val: wrappedIframeChild};
default:
throw "nope";
}
}, function(cx, obj, propName, strict, val, jsval) {
reportMessage("trying to set wrapped div attr " + propName);
switch (propName) {
case "compareDocumentPosition":
return;
case "innerHTML":
if (val == "<iframe allowtransparency='true' frameborder='0' scrolling='no'></iframe>") {
wrapperDiv.innerHTML = val;
}
return;
default:
throw "not allowed in wrapped div setter";
}
});
}
return null;
}
var wrappedCreateElement = JSJS.wrapFunction({
func : jsCreateElement,
args : [ JSJS.Types.charPtr ],
returns : JSJS.Types.objPtr
});
var ptrCreateElement = JSJS.NewFunction(jsObjs.cx, wrappedCreateElement, 1, "createElement");
function jsAddEventListener(type, listener, useCapture) {
reportMessage("document add event listener " + type);
if (type == "DOMContentLoaded") {
var jsvalout = JSJS.CreateJSVal(jsObjs.cx, listener, JSJS.Types.objPtr);
reportMessage("about to call DOMContentLoaded callback " + jsvalout);
JSJS.CallFunctionValue(jsObjs.cx, jsObjs.glob, jsvalout);
reportMessage("finished calling DOMContentLoaded");
return undefined;
}
throw "Not Implemented";
}
var wrappedAddEventListener = JSJS.wrapFunction({
func: jsAddEventListener,
args: [JSJS.Types.charPtr, JSJS.Types.objPtr, JSJS.Types.bool],
numRequired: 2,
returns: null});
var ptrAddEventListener = JSJS.NewFunction(jsObjs.cx, wrappedAddEventListener, 3, "addEventListener");
function jsRemoveEventListener(type, listener, useCapture) {
reportMessage("document remove event listener " + type);
if (type == "DOMContentLoaded") {
return undefined;
}
throw "Not Implemented";
}
var wrappedRemoveEventListener = JSJS.wrapFunction({
func: jsRemoveEventListener,
args: [JSJS.Types.charPtr, JSJS.Types.objPtr, JSJS.Types.bool],
numRequired: 2,
returns: null});
var ptrRemoveEventListener = JSJS.NewFunction(jsObjs.cx, wrappedRemoveEventListener, 3, "removeEventListener");
function aGetAttribute(attr) {
reportMessage("wrapped A getAttribute " + attr);
switch (attr) {
case "data-rendering":
case "data-count":
case "data-size":
case "data-lang":
case "data-text":
case "data-align":
case "data-via":
case "data-related":
case "data-counturl":
case "data-searchlink":
case "data-placeid":
case "data-hashtags":
case "data-button-screen-name":
case "data-button-hashtag":
case "data-url":
case "data-dnt":
case "data-twttr-rendered":
case "data-enable-new-sizing":
var attrValue = twitButDestA.getAttribute(attr);
reportMessage("=== GETTING === " + attr + " === " + attrValue);
var attrType = typeof(twitButDestA.getAttribute(attr));
if (attrType == 'object' && attrValue == null) {
return {'type': JSJS.Types.objPtr, 'val': null};
} else if (attrType == 'string') {
return {'type': JSJS.Types.charPtr, 'val': attrValue};
}
throw "wrong value!";
default:
throw "unimplemented";
}
}
var wrappedAGetAttribute = JSJS.wrapFunction({
func: aGetAttribute,
args: [JSJS.Types.charPtr],
returns: null});
var ptrAGetAttribute = JSJS.NewFunction(jsObjs.cx, wrappedAGetAttribute, 1, "getAttribute");
function aSetAttribute(attrName, attrValue) {
reportMessage("wrapped A setAttribute " + attrName);
var attrConverted = JSJS.identifyConvertValue(jsObjs.cx, attrValue);
switch(attrName) {
case "data-rendering":
case "data-twttr-rendered":
reportMessage("=== SETTING === " + attrName + " === " + attrValue);
twitButDestA.setAttribute(attrName, attrValue);
break;
default:
throw "not implemented";
}
}
var wrappedASetAttribute = JSJS.wrapFunction({
func: aSetAttribute,
args: [JSJS.Types.charPtr, JSJS.Types.dynamicPtr],
returns: null});
var ptrASetAttribute = JSJS.NewFunction(jsObjs.cx, wrappedASetAttribute, 1, "setAttribute");
function divReplaceChild(newChild, oldChild) {
reportMessage("replaceChild " + oldChild + " with " + newChild);
if (!(newChild in wrappedMap && oldChild in wrappedMap)) {
throw "Wrong div replacement!";
}
twitButContainer.replaceChild(wrappedMap[newChild], wrappedMap[oldChild]);
}
var wrappedDivReplaceChild = JSJS.wrapFunction({
func: divReplaceChild,
args: [JSJS.Types.objPtr, JSJS.Types.objPtr],
returns: null});
var ptrDivReplaceChild = JSJS.NewFunction(jsObjs.cx, wrappedDivReplaceChild, 2, "replaceChild");
var wrappedTwitContainer = makeWrappedElement(twitButContainer,
function containergetter(propName) {
reportMessage("container getter " + propName);
switch (propName) {
case "lang":
return {'type': JSJS.Types.charPtr, 'val': twitButContainer.lang};
case "parentNode":
return {'type': JSJS.Types.objPtr, 'val': null};
case "replaceChild":
return {'type': JSJS.Types.funcPtr, 'val': ptrDivReplaceChild};
}
throw "not implcontainergetter";
});
function jsQuerySelectorAll(selectors) {
reportMessage("querySelectorAll " + selectors);
switch (selectors) {
case "a.twitter-share-button":
return [[JSJS.Types.objPtr, makeWrappedElement(twitButDestA,
function wrappedAgetter(propName) {
reportMessage("wrapped A getter propname " + propName);
switch (propName) {
case "getAttribute":
return {'type': JSJS.Types.funcPtr, 'val': ptrAGetAttribute};
case "setAttribute":
return {'type': JSJS.Types.funcPtr, 'val': ptrASetAttribute};
case "id":
return {'type': JSJS.Types.charPtr, 'val': twitButDestA.id};
case "href":
return {'type': JSJS.Types.charPtr, 'val': twitButDestA.href};
case "lang":
return {'type': JSJS.Types.charPtr, 'val': twitButDestA.lang};
case "className":
return {'type': JSJS.Types.charPtr, 'val': twitButDestA.className};
case "parentNode":
return {'type': JSJS.Types.objPtr, 'val': wrappedTwitContainer};
default:
throw "not implemeneted";
}
}, function wrappedAsetter(cx, obj, propName, strict, val, vp) {
reportMessage("wrapped A setter propname " + propName);
switch (propName) {
case "className":
twitButDestA.className = val;
return;
case "compareDocumentPosition":
return;
default:
throw "wrapped a setter oh no!";
}
})]];
default:
return [];
}
throw "Not implemented";
}
var wrappedQuerySelectorAll = JSJS.wrapFunction({
func : jsQuerySelectorAll,
args : [ JSJS.Types.charPtr ],
returns : JSJS.Types.arrayPtr
});
var ptrQuerySelectorAll = JSJS.NewFunction(jsObjs.cx, wrappedQuerySelectorAll, 1, "querySelectorAll");
function jsQuerySelector(selectors) {
reportMessage("querySelector " + selectors);
throw "Not implemented"
}
var wrappedQuerySelector = JSJS.wrapFunction({
func : jsQuerySelector,
args : [ JSJS.Types.charPtr ],
returns : null
});
var ptrQuerySelector = JSJS.NewFunction(jsObjs.cx, wrappedQuerySelector, 1, "querySelector");
function documentResolveProperty(propName) {
reportMessage("document resolve " + propName);
return 1;
}
function documentGetProperty(propName) {
reportMessage('getting document property ' + propName);
switch (propName) {
case "getElementsByTagName":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrGetElementsByTagName};
break;
case "getElementById":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrGetElementById};
break;
case "createElement":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrCreateElement};
break;
case "readyState":
return {'type' : JSJS.Types.charPtr, 'val' : "complete"};
break;
case "addEventListener":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrAddEventListener};
break;
case "removeEventListener":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrRemoveEventListener};
break;
case "title":
return {'type' : JSJS.Types.charPtr, 'val' : document.title};
break;
case "documentElement":
return {'type' : JSJS.Types.objPtr, 'val' : makeWrappedElement(document.documentElement)};
break;
case "querySelectorAll":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrQuerySelectorAll};
break;
case "querySelector":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrQuerySelector};
break;
case "location":
return {'type' : JSJS.Types.objPtr, 'val' : jsLocation};
break;
case "body":
{
function jsBodyInsertBefore(newElement, referenceElement) {
reportMessage("body insertBefore");
if (newElement in wrappedMap && wrappedMap[newElement].tagName == "IFRAME") {
//document.body.insertBefore(wrappedMap[newElement], document.body.firstChild);
return null;
}
reportMessage(wrappedMap);
throw "Not Implemented (" + newElement + ")";
}
var wrappedBodyInsertBefore = JSJS.wrapFunction({
func: jsBodyInsertBefore,
args: [JSJS.Types.objPtr, JSJS.Types.objPtr],
returns: null});
var ptrBodyInsertBefore = JSJS.NewFunction(jsObjs.cx, wrappedBodyInsertBefore, 2, "insertBefore");
function jsBodyAppendChild(newElement) {
reportMessage("IGNORING body appendchild");
}
var wrappedBodyAppendChild = JSJS.wrapFunction({
func: jsBodyAppendChild,
args: [JSJS.Types.objPtr],
returns: null});
var ptrBodyAppendChild = JSJS.NewFunction(jsObjs.cx, wrappedBodyAppendChild, 1, "appendChild");
function jsBodyRemoveChild(newElement) {
reportMessage("IGNORING body removechild");
}
var wrappedBodyRemoveChild = JSJS.wrapFunction({
func: jsBodyRemoveChild,
args: [JSJS.Types.objPtr],
returns: null});
var ptrBodyRemoveChild = JSJS.NewFunction(jsObjs.cx, wrappedBodyRemoveChild, 1, "removeChild");
return {'type': JSJS.Types.objPtr, 'val': makeWrappedElement(document.body,
function getter(propName) {
reportMessage("document.body getter " + propName);
switch (propName) {
case "insertBefore":
return {'type': JSJS.Types.funcPtr, 'val': ptrBodyInsertBefore};
case "appendChild":
return {'type': JSJS.Types.funcPtr, 'val': ptrBodyAppendChild};
case "removeChild":
return {'type': JSJS.Types.funcPtr, 'val': ptrBodyRemoveChild};
}
}, function setter(cx, obj, propName, strict, val, jsval) {
reportMessage("document.body setter " + propName);
})}
}
}
reportMessage("returning nothing for document." + propName);
return true;
}
var jsDocumentClass = JSJS.CreateClass(JSJS['JSCLASS_GLOBAL_FLAGS'],
JSJS['PropertyStub'], JSJS['PropertyStub'], JSJS.wrapGetter(
documentGetProperty, JSJS.Types.bool),
JSJS['StrictPropertyStub'], JSJS['EnumerateStub'], JSJS
.wrapResolver(documentResolveProperty),
JSJS['ConvertStub'], JSJS['FinalizeStub']);
var jsDocument = JSJS.DefineObject(jsObjs.cx, jsObjs.glob, "document",
jsDocumentClass, 0, 0);
JSJS.SetProperty(jsObjs.cx, jsDocument, "querySelectorAll",
ptrQuerySelectorAll);
JSJS.SetProperty(jsObjs.cx, jsDocument, "querySelector",
ptrQuerySelector);
function jsPostMessage(message, origin) {
reportMessage("postMessage " + origin);
throw "Not implemented";
}
var wrappedPostMessage = JSJS.wrapFunction({
func : jsPostMessage,
args : [ JSJS.Types.objPtr, JSJS.Types.charPtr ],
returns : null
});
var ptrPostMessage = JSJS.NewFunction(jsObjs.cx, wrappedPostMessage, 2, "postMessage");
function jsWindowAddEventListener(type, listener, useCapture) {
reportMessage("Window addEventListener " + type);
var jsvalout = JSJS.CreateJSVal(jsObjs.cx, listener, JSJS.Types.objPtr);
if (type == "message") {
window.addEventListener("message", function(event) {
reportMessage("window message received");
for (wrappedPtr in wrappedMap) {
if (wrappedMap[wrappedPtr] == event.source) {
reportMessage("about to call message callback");
JSJS.CallFunctionValue(jsObjs.cx, jsObjs.glob, jsvalout,
[JSJS.Types.objPtr], [wrapEventObject(event, wrappedPtr)]);
reportMessage("done calling message callback");
return;
}
}
reportMessage("returning because no valid message handler found");
});
return undefined;
}
throw "Not Implemented";
}
var wrappedWindowAddEventListener = JSJS.wrapFunction({
func: jsWindowAddEventListener,
args: [JSJS.Types.charPtr, JSJS.Types.objPtr, JSJS.Types.bool],
numRequired: 2,
returns: null});
var ptrWindowAddEventListener = JSJS.NewFunction(jsObjs.cx, wrappedWindowAddEventListener, 1, "addEventListener");
function windowGetProperty(propName) {
reportMessage('getting window property ' + propName);
switch (propName) {
case "postMessage":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrPostMessage};
break;
case "addEventListener":
return {'type' : JSJS.Types.funcPtr, 'val' : ptrWindowAddEventListener};
break;
default:
reportMessage("returning global value for prop " + propName);
return {'type' : null, 'val' : JSJS.GetProperty(jsObjs.cx, jsObjs.glob, propName)};
break;
}
return true;
}
function windowSetProperty(cx, obj, idStrReal, strict, val, jsval) {
reportMessage('setting window property ' + idStrReal + " = " + val
+ " / " + jsval);
// Copy window property sets to the global object
JSJS.SetProperty(jsObjs.cx, jsObjs.glob, idStrReal, jsval);
return true;
}
function windowResolveProperty(propName) {
reportMessage("window resolve prop " + propName);
return 1;
}
var jsWindowClass = JSJS.CreateClass(JSJS['JSCLASS_GLOBAL_FLAGS'],
JSJS['PropertyStub'], JSJS['PropertyStub'], JSJS.wrapGetter(
windowGetProperty, JSJS.Types.bool), JSJS
.wrapSetter(windowSetProperty), JSJS['EnumerateStub'],
JSJS.wrapResolver(windowResolveProperty), JSJS['ConvertStub'],
JSJS['FinalizeStub']);
var jsWindow = JSJS.DefineObject(jsObjs.cx, jsObjs.glob, "window",
jsWindowClass, 0, 0);
reportMessage("executing script");
var rval = JSJS.EvaluateScript(jsObjs.cx, jsObjs.glob, srcContents);
PG_end = new Date().getTime();
PG_time = "Time: " + (PG_end - PG_start)/1000 + " s";
alert(PG_time);
$("#PG_go").html(PG_time);
});
</script>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,670 |
{"url":"https:\/\/discuss.codechef.com\/questions\/67870\/bwgame-editorial","text":"\u00d7\n\n# BWGAME - Editorial\n\nAuthor: Sergey Kulik\nTester: Sergey Kulik\n\nHard\n\n# PREREQUISITES:\n\nMatrix Algebra, Binary merging, heaps.\n\n# PROBLEM:\n\nYou are given a $N$ pairs of integers, ($L[i], R[i]$) $i$ ranging from 1 to $N$. Two players play a game. Alex can choose a permutation of integers from 1 to $N$, P[1...$N$], if and only if $L[i] <= P[i] <= R[i]$ for all $1<=i<=N$ and the number of inversions in this permutation is even. Similarly, the number of inversions should be odd for Fedya. You have to find, who among Alex and Fedya has more number of permutations. If both have same, then the answer is a Draw.\n\n# EXPLANATION:\n\nLet $Pp$ = the number of permutations that are suitable for Alex, $Np$ = the number of permutations that are suitable for Fedya.\n\nBy the definition, if we replace black cells with 1, and white with 0, (and obtain the matrix $A$), then determinant of matrix $A = Pp - Np$. So the problem is about the calculation of the $det A$.\n\nThe first point is that the determinant will be either -1, 0, or +1. The proof can be seen in the later part of the editorial.\n\nIn the matrix $A$, all the elements in the columns from $L[i]$ to $R[i]$, in the $i^{th}$ row will be equal to 1. Let us maintain an array of Min-Heaps, where heap[i] is a Min-heap that stores all the segments which start in the $i^{th}$ column.\n\nIn order to calculate the determinant, you can follow the below strategy:\n\n1. Pick a segment of 1's that starts in the first column, if there are a many of them, pick the shortest one, just pop out the minimum element in the heap[1].\n2. Let its length be $X$, Now you can subtract this segment from all the segments that start at the first column as well, but since they weren't chosen, they're longer than this one (if there is a segment with the equal length, the determinant is zero). Subtracting all the segments will be same as shifting all the segments from heap[1] to heap[$X$ + 1].\n3. Now we've obtained the situation, where there is only one row that has non-zero element in the first column. Here, according to Laplace's theorem, we can just eliminate the first column and this row and find the det of new matrix and multiply it by some (-1)^(row_number + column_number) and go on, but here we will have only the matrix of the size that is one unit less than the size of the original matrix.\n\nSo, doing this process N times, we'll get the solution to the problem. If we see our new matrix we get in step 3 has similar properties of original matrix, i.e, all the 1s in the matrix are present contiguously in each row. The determinant of matrix of size 1 will be either 1 or 0. So by mathematical induction we can prove that if matrix of size N - 1 can have values {-1, 0, 1} only, then matrix of size N can also have values {-1, 0, 1} only, because it just gets multiplied by (-1)^(row number + column number) or 0.\n\nNow the important part is if we directly shift all the elements of one heap to another we will have a complexity of $O(N * N * logN)$, because there will be $O(N * N)$ shifts in the worst case, and each shift takes O(logN) time. So rather than shifting all elements from heap[1] to heap[X + 1] directly, lets maintain pointers to heaps for each column and then we can just shift all the elements in the smaller heap (whichever is the one, i.e either the heap pointed by 1st column or $X$ + 1th column) to the larger heap, and point $X$ + 1th column to larger heap. This ensures that there will be only $N * log N$ shifts in the worst case, because whenever an element is shifted from one heap to the other, the size of the heap it is present will become more than doubled, so any particular element will be shifted at most $log N$ times.\n\nWriting a heap class will be very helpful in implementation.\n\n# AUTHOR'S AND TESTER'S SOLUTIONS:\n\nTo be posted soon.\n\nThis question is marked \"community wiki\".\n\n92852428\naccept rate: 11%\n\n19.8k350498541\n\n 1 the proof for det(A)= Pp-Np answered 17 Apr '15, 19:12 11\u25cf1 accept rate: 0% 2 This comes directly from the definition of the determinant, which is a sum of products of elements of permutations multiplied by their \"sign\", where the \"sign\" is 1 if it has an even number of inversions, -1 if it has an odd number of inversions. To be more precise, let's consider each of the N! permutations and the elements A(i,p(i)). If not all of its elements are 1 (i.e. black), then the product of their elements is zero, so they don't count when computing the determinant. If all their elements are 1 then the product of the elements is 1, and they contribute to the determinant by -1 or +1. (17 Apr '15, 19:47) I was unaware of this definition of Determinant , which seeing on net seems to be quite a popular definition . :) (17 Apr '15, 19:52) @mugurelionut yes, that's true! It was difficult to see this and relate it to the problem. Thanks for the explanation. Besides, you are very competitive. (18 Apr '15, 15:11)\n 0 Nice problem :) answered 17 Apr '15, 01:29 148\u25cf5 accept rate: 0%\n 0 but i am not getting that this is in HARD categeries of problem,,because of finding determinent of matrix or many of person dont know this predefined defnation,,,plz tell me how can i get logic in competition for this question if i have never heard this propertie?? answered 18 Apr '15, 22:52 1\u25cf1 accept rate: 0%\n toggle preview community wiki:\nPreview\n\nBy Email:\n\nMarkdown Basics\n\n\u2022 *italic* or _italic_\n\u2022 **bold** or __bold__\n\u2022 image?![alt text](\/path\/img.jpg \"title\")\n\u2022 numbered list: 1. Foo 2. Bar\n\u2022 to add a line break simply add two spaces to where you would like the new line to be.\n\u2022 basic HTML tags are also supported\n\u2022 mathemetical formulas in Latex between \\$ symbol\n\nQuestion tags:\n\n\u00d715,852\n\u00d71,359\n\u00d7186\n\u00d742\n\u00d741\n\u00d715\n\u00d76\n\nquestion asked: 15 Apr '15, 03:11\n\nquestion was seen: 2,861 times\n\nlast updated: 01 Jun '16, 20:21","date":"2019-03-23 12:42:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8039058446884155, \"perplexity\": 670.0184567526444}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912202804.80\/warc\/CC-MAIN-20190323121241-20190323143241-00157.warc.gz\"}"} | null | null |
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{"url":"https:\/\/math.stackexchange.com\/questions\/901117\/operatornamegcdab-ab-1-if-a-and-b-are-relatively-prime","text":"# $\\operatorname{gcd}(ab,a+b)=1$ if $a$ and $b$ are relatively prime\n\nI'm trying to show that if $\\operatorname{gcd}(a,b) = 1$, then $\\operatorname{gcd}(ab,a+b)=1$.\n\nI've tried to use the gcd properties: $$\\operatorname{gcd}(a,b)=1 \\implies \\operatorname{gcd}(a,a+b)=1\\\\\\operatorname{gcd}(a,b)\\implies\\operatorname{gcd}(ab,b^2)=b$$ but I got stuck. Any hint will help.\n\n\u2022 $\\gcd(ab,a+b) = \\gcd(ab - (a+b)b,a+b)$ \u2013\u00a0Daniel Fischer Aug 17 '14 at 15:24\n\nHint:\n\nSupose $k:=\\gcd(ab,a+b) > 1$. Let $p$ be a prime that divides $k$.\n\nThen $p$ divides $ab$ wich means that it divides $a$ or $b$ (but not both because $\\gcd(a,b) = 1$.\n\nCan $p$ divide $a+b$?\n\n\u2022 No. because you will always be left out with a part (a or b) which p can't divide. so p cannot be greater than 1? \u2013\u00a0user170198 Aug 17 '14 at 15:48\n\nIf integer $d$ divides $ab,a+b;$\n\n$d$ must divide $(a+b)b-ab=b^2$ and $(a+b)a-ab=a^2$\n\n$\\implies d$ must divide $(a^2,b^2)=(a,b)^2$\n\nBut, $(a,b)=1$\n\nA good method to tackle such problems would be to start by assuming that there is a prime $p$ dividing both $ab$ and $a+b$.\n\nIf $p$|$ab$ then $p$|$a$ or $p$|$b$. WLOG assume the former to hold. Then, $p$ will not divide $b$ otherwise $gcd(a,b)=1$ does not hold.\n\nBut $p$ divides $a$ and $p$ divides $a+b$. It follows that $p$ divides $b$ also. This is a contradiction to our hypothesis, so our assumption is false.\n\n$\\begin{eqnarray}{\\bf Hint}\\ \\ &&{\\rm prime}\\ p\\mid (xy,\\,x+y)\\\\ \\iff &&\\qquad\\ \\ \\, p\\mid\\ xy,\\, x+y\\\\ \\iff && {\\rm mod}\\ p\\!:\\ \\color{#c00}{xy\\equiv 0}\\equiv x+y\\\\ \\iff && {\\rm mod}\\ p\\!:\\ [\\color{#c00}{x\\equiv 0\\ \\ {\\rm or}\\ \\ y\\equiv 0}]\\ \\ {\\rm and}\\ \\ y\\equiv -x,\\ \\ {\\rm \\color{#c00}{by}\\ unique\\ factorization}\\\\ \\iff && {\\rm mod}\\ p\\!:\\ \\ x\\equiv 0\\equiv y \\end{eqnarray}$\n\ni.e. the union of $\\rm\\color{#c00}{both\\ axes}$ intersects line $\\,y\\equiv -x\\,$ at the origin (viewed geometrically).\n\n\u2022 See here for a generalization. \u2013\u00a0Bill Dubuque Sep 19 '19 at 18:02\n\nProof by contrapositive:\n\nLet (ab, a+b)= d > 1.\n\nThen $d \\vert ab$ and $d \\vert a+b$.\n\nWLOG, let $d \\vert a$.\n\nSince $d \\vert a$ and $d \\vert a+b$, $d \\vert b$ [easy to show]\n\nTherefore, (a,b) is not 1.\n\nHint $$a^2=a(a+b)-ab$$ $$b^2=b(a+b)-ab$$\n\nUse this to show that for all $a,b$ we have $\\mbox{gcd}(ab, a+b)$ divides $\\mbox{gcd}(a^2,b^2)$.\n\nWe get following two conclusions : $gcd(a,\\;a+b)=1$, $\\;\\;gcd(b,\\;a+b)=1$, thus $gcd(ab, a+b)=1$ otherwise one(or both) of the two conclusions will be wrong.","date":"2020-01-25 14:24:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8746660947799683, \"perplexity\": 306.2774064656837}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251672537.90\/warc\/CC-MAIN-20200125131641-20200125160641-00069.warc.gz\"}"} | null | null |
Q: Vector title attribute can't display? var vectormap = new OpenLayers.Layer.Vector('Overlay',
{
styleMap: new OpenLayers.StyleMap(
{
"default": new OpenLayers.Style({
externalGraphic: 'girl.png',
graphicWidth: 50,
graphicHeight: 50,
graphicYOffset: -24,
title: 'marker'}),
"select": new OpenLayers.Style({
externalGraphic: 'boy.png',
graphicWidth: 16,
graphicHeight: 26,
graphicYOffset: -24,
title: 'marker_select'})
})
});
featurearray = new OpenLayers.Feature.Vector(new OpenLayers.Geometry.Point(113.950,33.019));
vectormap.addFeatures([ featurearray ]);
This is my vector layer, but a bit wrong, my png image title attribute can not be displayed when the mouse to move up, I checked the official website the "OSM with Marker and Popup.html" code, exactly the same, you know why.
A: It looks like you are using canvas but title option does not supported by the canvas renderer. Try to configure you vector layer following way:
var vectormap = new OpenLayers.Layer.Vector('Overlay',
{
styleMap: new OpenLayers.StyleMap(
{
"default": new OpenLayers.Style({
externalGraphic: 'girl.png',
graphicWidth: 50,
graphicHeight: 50,
graphicYOffset: -24,
title: 'marker'}),
"select": new OpenLayers.Style({
externalGraphic: 'boy.png',
graphicWidth: 16,
graphicHeight: 26,
graphicYOffset: -24,
title: 'marker_select'})
})
renderers: ["SVG", "VML", "Canvas"]
});
| {
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} | 7,937 |
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Home/Celebrity Net Worth/Actors/Corey Haim Net Worth
Corey Haim Net Worth
Corey Haim Net Worth 2020: Wiki Biography, Married, Family, Measurements, Height, Salary, Relationships
292 2 minutes read
Wiki Biography
Corey Ian Haim was born on 23 December 1971, in Toronto, Canada to an Israeli-born mother and a Canadian father, and died of pneumonia on 10 March 2010, in Burbank, California. Corey was an actor, perhaps best remembered as a 1980s teen idol from Hollywood movies, his most famous role probably being in "The Lost Boys" (1987), an American horror comedy film, in which he played alongside Corey Feldman. Their mutual appearance turned out to be quite successful, so they later became 1980s icons and appeared together in seven movies as well as the American reality show "The Two Coreys"
Have you ever wondered how rich Corey Haim was before he died? According to sources, it has been estimated that at the time of his death, Corey Haims net worth was $5,000. He earned his net worth mostly through his prosperous acting career during the 1980s, although his engagement in a few successful movies during the 1990s, also added up to Corey's net worth.
Corey Haim Net Worth $5,000
Corey Haim was raised in Toronto; his parents divorced when he was eleven. As a child, Haim was quite shy, and in order to overcome this problem, he started attending drama classes. While accompanying his sister to an audition, Corey accidentally got into the film industry, when he was given a part in "The Edison Twins", a Canadian children's television program aired from 1982 to 1986. Since he wasn't too much interested in acting, he continued practicing other activities such as ice hockey, which eventually resulted in him being scouted for the AA Thunderbirds hockey team. Haim's education at the North York's Zion Heights Junior High lasted until 8th grade, by which time he had already started his career as a child actor.
His debut film role happened in the 1984 thriller "Firstborn" in which he starred with Sarah Jessica Parker and Robert Downey Jr. Corey's minor roles in films "Secret Admirer" and "Murphy's Romance" during the year 1985 eventually lead to the leading role in "Silver Bullet", a film adaptation of Stephen King's novella, which brought increased income to his net worth. The young actor then gained recognition for his role in the television movie "A Time To Live"(1985), in which he portrayed Liza Minnelli's dying son. This earned him his first Young Artist Award. However, Haim's true breakthrough role came a year later when he starred as a titular character in "Lucas", beside Charlie Sheen and Winona Ryder, further adding to his net worth, and also earning him the nomination for an Exceptional Performance by a Young Actor in a Feature Film – Comedy or Drama at the Young Artist Awards.
In 1987, Corey starred in an American horror comedy film "The Lost Boys", which was well received by most critics and regarded as an '80s classic. His role in this film, secured Haims another Young Artist Award nomination, as Best Young Male Superstar in a Motion Picture. Some of his other notable appearances include his roles in the horror film "The Watchers" (1988), teen adventure film "License to Drive", a romantic comedy "Dream a Little Dream" (1989), slapstick comedy film "Snowboard Academy" and many others. All added to his net worth.
Haim's addiction to alcohol and drugs resulted in his poor social life, and probably explains his very modest net worth at the time of his death. He never married or had any children, and was said to be quite lonely. Corey first became addicted to alcohol and drugs at the age of 15, eventually resulting in early death at the age of 38.
Structural Info
Full Name Corey Haim
Net Worth $5 thousand
Died March 10, 2010, Burbank, California, United States
Height 1.73 m
Profession Musician, Film producer, Actor, Painter
Nationality Canadian
Parents Judy Haim, Bernie Haim
Siblings Daniel Lee Haim, Carol Haim
IMDB www.imdb.com/name/nm0000433/
Movies The Lost Boys, License to Drive, Lucas, Dream a Little Dream, Silver Bullet, Blown Away, Murphy's Romance, Prayer of the Rollerboys, Firstborn, Lost Boys: The Tribe, Crank: High Voltage, Fast Getaway, Anything for Love, Secret Admirer, Dream a Little Dream 2, Watchers, Oh, What a Night, The Double 0...
TV Shows The Two Coreys, Roomies, The Edison Twins, 100 Greatest Kid Stars
1 In early years, his mouth was mostly always open
1 [on battle with drug abuse] It's one day at a time. It's one second at a time-it's really one millisecond at a time sometimes. But you gotta grind your teeth and bear it. There's no way to get around it. It's just time, effort and support-and you gotta reach out and ask for help
2 [on his cameo in Crank: High Voltage (2009)] My first day of filming, I went in my camper for a minute, and I just started crying, because I was, like, home. And I'm just so grateful. It's a cameo. But, like, Ricky Schroder did a cameo in Crimson Tide (1995), and then boom! He got NYPD Blue.
3 [on effects of molestation] It's something that will be addressed in my inner soul for the rest of my life, and it's something that truly affects me, and I opened up a can of worms, so to speak. Every day I opened up, like, a can of sardines. It's something I've addressed. Psychiatrists can be helpful. They have the medications and blah-blah-blah. But I don't want any of that, man. I've dealt with this, and I'm dealing with this-second by second, minute by minute, day by day. Everything's cool. It's just like, It happened, it's over, and move on. Let's move on to the next subject.
4 This is where you make new friends and you start living and learning. So I'm a late bloomer, I'm 36, but hey, I'm learning. Every day's a learning experience, right?
5 I want to be the guy they talk about when they talk about comebacks. I want people to learn from me, see I'm human, and understand that I make mistakes just like they do, but it doesn't have to consume you. You've got to walk through the raindrops, and that's totally what I am trying to do.
6 I was working on The Lost Boys (1987) when I smoked my first joint. But a year before that, I was starting to drink beer on the set of the film Lucas (1986). I lived in Los Angeles in the '80s, which was not the best place to be. I did cocaine for about a year and a half, then it led to crack. I started on the downers which were a hell of a lot better than the uppers because I was a nervous wreck. But one led to two, two led to four, four led to eight, until at the end it was about 85 a day - the doctors could not believe I was taking that much. And that was just the valium - I'm not talking about the other pills I went through.
1 Was in a relationship with Victoria Beckham in 1995, which ended on mutual terms.
2 He was in a relationship with actress Alyssa Milano from 1987 to 1990.
3 Haim was engaged to model Cindy Guyer in 2000. He proposed to Guyer two days after they met at a Chicago autograph show.
4 Was offered the role of Charlie Fox in The Mosquito Coast (1986) and Chris Chambers in Stand by Me (1986). His father, who was acting as his manager at the time, turned down both in favor of Corey having a starring role in Lucas (1986). Both of those roles eventually went to the late River Phoenix.
5 Back when he was a kid in Thornhill Ontario, Corey Haim used to break dance with the other kids at recess, in school.
6 Best friend Corey Feldman chose not to attend Haim's funeral at the request of Haim's family, because they wanted to keep the funeral as low-key as possible. Feldman instead issued an open letter on the Internet which he wrote to his deceased friend.
7 Was buried at Thornhill memorial chapel in Toronto on March 16th, 2010. Because Haim was broke at the time of his death, a celebrity memorabilia site that he sometimes sold items to, Startifacts, had donated $20,000 to cover the cost of the transportation and headstone. The check was subsequently canceled when it was found the funeral home was covering the cost already.
8 Formerly engaged to actress Tiffany Shepis. [Oct. 30, 2008].
9 He and Corey Feldman were known as the Two Coreys at the peak of their careers in the 1980s.
10 He starred in eight movies with Corey Feldman: The Lost Boys (1987), License to Drive (1988), Dream a Little Dream (1989), Blown Away (1993), Last Resort (1994), Dream a Little Dream 2 (1995), Busted (1997) and Lost Boys: The Tribe (2008).
11 Auditioned for the role of Dick Grayson/Robin in Batman Forever (1995).
12 Ranked #26 on VH1's 100 Greatest Kid Stars.
13 Bill Egert's character Sean Haim in the movie Detour (2002) was named after him, as Egert's favorite actor.
14 Was engaged to Nicole Eggert, whom he credits with helping save his life on a certain occasion: while they were filming one of several movies together, she noticed that Haim was suffering from a narcotic "rush." Eggert promptly drove him to the local hospital where Corey was detoxed.
15 Was asked by best friend and frequent co-star Corey Feldman - while filming the first season of The Surreal Life (2003) - to be his best man at his wedding to Susie Feldman; but unfortunately, due to prior commitments he was not able to attend.
16 Brooke McCarter managed him up to the mid 1990s until Corey's drug problem caused a fallout.
17 Was engaged to Holly Fields (not the boxer) in 1996, but broke up.
18 In 2004, he became the subject of a single by the Irish band The Thrills titled "Whatever Happened To Corey Haim?".
19 Filed for Chapter 11 bankruptcy protection in July 1997 in Los Angeles, California, listing debts that included nearly $104,000 to the I.R.S., $100,000 in California taxes and a variety of medical expenses. The document stated that he had no bank account and very few assets, including a 1987 BMW, $100 in cash, $750 worth of clothing, and $7,500 in residuals and royalty rights. He also listed a $31,000 pension plan.
20 Parents Bernie and Judy divorced when Corey was 11 years old. They were married for 18 years.
21 His dad, Bernie Haim, worked as a sales rep. His mom, Judy Haim, worked as a computer operator.
22 Got into acting because his mom put him into acting lessons to help him deal with his shyness and it later became a career because his older sister Carol Haim was auditioning for roles.
23 His father is Canadian and his mother was born in Israel; both are from Jewish families. Corey had a Bar Mitzvah ceremony.
24 Favorite actress: Cybill Shepherd. Favorite Actor: James Dean.
The Dead Sea 2014 Oso (credit only)
Decisions 2011 Det. Lou Andreas
New Terminal Hotel 2010 Jasper Crash
The Hostage Game 2010 Tom MacLean
Shark City 2009 Chip Davis
Crank: High Voltage 2009 Randy
Trade In 2009 Corey Haim
Lost Boys: The Tribe 2008 Video Sam Emerson
Universal Groove 2007 Jim
Dickie Roberts: Former Child Star 2003 Corey Haim
The Backlot Murders 2002 Video Tony
Without Malice 2000 TV Movie Marty
Big Wolf on Campus 2000 TV Series Corey Haim
Wishmaster 2: Evil Never Dies 1999 Video Museum Burglar (uncredited)
Merlin 1998 TV Movie Wilf
PSI Factor: Chronicles of the Paranormal 1998 TV Series Research Project Intern
Demolition University 1997 Video Lenny Slater
Busted 1997 Video Clifford
Shooter on the Side 1996
Snowboard Academy 1996 Video Chris Barry
Never Too Late 1996 Max
Demolition High 1996 Video Lenny Slater
Fever Lake 1996 Video Albert
Life 101 1995 Video Ramsy
Dream a Little Dream 2 1995 Video Dinger Holfield
Fast Getaway II 1994 Video Nelson Potter
Last Resort 1994/II Video Dave
Double Switch 1993 Video Game Eddie
Just One of the Girls 1993 Video Chris Calder
Blown Away 1993 TV Movie Rich
Oh, What a Night 1992 Eric
The Double 0 Kid 1992 Video Lance Elliot
Dream Machine 1991 Barry Davis
Fast Getaway 1991 Video Nelson
Prayer of the Rollerboys 1990 Griffin
Dream a Little Dream 1989 Dinger
Watchers 1988 Travis
License to Drive 1988 Les Anderson
The Lost Boys 1987 Sam
Roomies 1987 TV Series Matthew Wiggins
Lucas 1986 Lucas
Murphy's Romance 1985 Jake Moriarity
A Time to Live 1985 TV Movie Peter Weisman
Silver Bullet 1985 Marty Coslaw
Secret Admirer 1985 Jeff (as Cory Haim)
The Edison Twins 1984-1985 TV Series Larry
Firstborn 1984 Brian Livingston
The Two Coreys 2007 TV Series executive producer
Demolition University 1997 Video executive producer
Demolition High 1996 Video executive producer
Life 101 1995 Video associate producer
Fast Getaway 1991 Video associate producer
Dickie Roberts: Former Child Star 2003 "Child Stars on Your Television"
Guerrilla 2017 Short dedicatee completed
The Girl 2011 in memory of
Emerging Past 2011 Video special thanks
Lost Boys: The Thirst 2010 Video dedicated to the memory of
Space Daze 2005 Video special thanks
Corey Feldman Interview 2005 Video documentary short special thanks
Corey Haim Interview 2005 Video documentary short special thanks
Shark City: Behind the Scenes 2010 Video documentary short Himself
Entertainment Tonight Canada 2008 TV Series Himself
The Two Coreys 2007-2008 TV Series Himself
eTalk Daily 2008 TV Series Himself
Entertainment Tonight 2008 TV Series Himself
Bloodsucking Cinema 2007 TV Movie documentary Himself
Larry King Live 2007 TV Series Himself
Biography 2007 TV Series documentary Himself
Bullrun: Cops, Cars & Superstars III 2006 TV Series Himself
Robot Chicken 2006 TV Series Himself / Teacher / Man
The Greatest 2005 TV Series documentary Himself
Corey Haim Interview 2005 Video documentary short
The Lost Boys: A Retrospective 2004 Video documentary short Himself
Child Stars 2002 TV Movie Himself
E! True Hollywood Story 2001 TV Series documentary Himself
Late Show with David Letterman 1999 TV Series Himself - Cameo
The 37th Annual Thalians Ball 1992 TV Movie Himself
Me, Myself and I 1989 Video documentary short Himself
Steven Spielberg: An American Cinematheque Tribute 1989 TV Movie Himself
The Arsenio Hall Show 1989 TV Series Himself
The Starlight Annual Foundation Benefit 1988 TV Special Himself
Archive Footage
An Open Secret 2014/I Documentary Himself (uncredited)
Lost Boys: The Thirst 2010 Video Sam Emerson (uncredited)
100 Greatest Teen Stars 2006 TV Mini-Series Himself
50 Cutest Child Stars: All Grown Up 2005 TV Movie documentary Himself
I Love the '80s 2002 TV Series documentary Himself
Buy Me That! A Kids' Survival Guide to TV Advertising. 1989 TV Movie Himself
Won Awards
1989 Young Artist Award Young Artist Awards Best Young Actor in a Motion Picture Comedy or Fantasy License to Drive (1988)
1987 Young Artist Award Young Artist Awards Exceptional Young Actor Starring in a Television Special or Movie of the Week A Time to Live (1985)
Nominated Awards
1992 Saturn Award Academy of Science Fiction, Fantasy & Horror Films, USA Best Performance by a Younger Actor Prayer of the Rollerboys (1990)
1990 Saturn Award Academy of Science Fiction, Fantasy & Horror Films, USA Best Performance by a Younger Actor Watchers (1988)
1988 Saturn Award Academy of Science Fiction, Fantasy & Horror Films, USA Best Performance by a Younger Actor The Lost Boys (1987)
1988 Young Artist Award Young Artist Awards Best Young Male Superstar in Motion Pictures The Lost Boys (1987)
1987 Young Artist Award Young Artist Awards Exceptional Performance by a Young Actor Starring in a Feature Film - Comedy or Drama Lucas (1986)
1986 Young Artist Award Young Artist Awards Exceptional Performance by a Young Actor - Motion Picture Firstborn (1984)
2nd Place Awards
1989 Bravo Otto Germany Bravo Otto Best Actor (Schauspieler)
Known for movies
License to Drive (1988)
as Les Anderson
as Lucas
Crank: High Voltage (2009)
as Randy
IMDB Wikipedia
Abe Silverstein Adam Park Advertising agency Blown Away Burbank Burbank Leader California Canada Charlie Sheen Corey Feldman Corey Haim Costco Drake (entertainer) Dream a Little Dream Elder abuse Encino Liza Minnelli Los Angeles Lost Boys: The Tribe Nutella Old age Robert Downey Jr Sarah Jessica Parker The Goonies The Lost Boys The Two Coreys Toronto Winona Ryder
Facebook Twitter Google+ LinkedIn StumbleUpon Pinterest Reddit
James Denton Net Worth
Shenae Grimes Net Worth
Irwin Winkler Net Worth
Shane Lynch Net Worth
Husbands/Wifes of Married Wiki
Skyler Samuels Net Worth
Keenen Ivory Wayans Net Worth
Bob Woodward Net Worth
Diane Keaton Net Worth
Taeyeon Net Worth
Net Worth Post · Privacy Policy · Terms of Use · DMCA Policy · Links Removal Request | {
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Solution: Consider the following solutions:0.010 m Na 3PO4 in water0.020 m CaBr 2 in water0.020 m KCl in water0.020 m HF in water (HF is a weak acid.)Which solution would have the highest vapor pressure at 28˚C?
Which solution would have the highest vapor pressure at 28˚C? | {
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It is quite obvious that overuse of the arms of youngsters is a major problem at ALL levels of youth baseball. Coaches need to understand the value of proper mechanics, teach viable pre and post game routines, the danger of throwing "breaking balls", and instill the importance of reporting pain in the arm to both them and parents. Parents should oversee the pitch counts, coaching philosophies, and be sure to protect the players from arm abuse. Many adults are so intent on winning the game and projecting their youngsters to the pros, that they do not see the downside of throwing and pitching too much or with a flawed set of mechanics.
Learn the proper throwing/pitching mechanics.
NO CURVE BALLS until the arm is fully developed and the player is throwing from 60'6".
Watch for overuse. Use strict pitch counts, taking a full day off between pitching dates, etc. Common sense rules!
Develop arm strength with stretching routines, long toss, lunges, roll-ups, and other light wrist and forearm lifting drills. Throw 3-5 times per week for 10-15 minutes in the off-season. Increase as it gets closer to the start of the next season.
NEVER pitch a full game on consecutive days.
No NOT throw or pitch with pain in the arm, elbow and/or shoulder. See a doctor if pain persists.
Forget the radar gun..most are inaccurate and velocity is over-rated.
Emphasize the mental aspect of pitching and focus on control and changing speeds/location/etc.
Avoid throwing a whiffle ball or other light balls as these can strain the arm.
Let the arm have proper rest and keep a journal as to how many innings/pitches/etc you throw. Ice after each pitching session(even practice).
See a doctor if you have any numbness or shooting arm pains..NOW!
Focus on COMMAND Drills(being able to throw or pitch to and exact location).
Keep in mind that arm abuse can affect your later life and even eliminate the ability to playing catch with your own children. KWTP!
This entry was posted in Handouts & Drills, Pitching on August 5, 2014 by admin.
A pitcher must be physically and mentally prepared to pitch in a game. This will require a number of exercises, conditioning repetitions, and mental considerations to be performed prior to the actual game! Below are listed suggested plans/recommendations for pre-game and post-game preparations.
Stretch your entire body and arm BEFORE you throw a ball. NEVER throw to warm up!!
Arm exercises and tubular resistance reps with an elastic cord.
Do mechanics work(9-12-6, etc) for first 5 minutes without a ball.
Throw warm-up pitches at 50% from 40 feet using entire arm motion.
Throw only 4-seam fastballs while warming up.
Throw 10-15 pitches from behind the mound(10 feet) to a standing catcher.
After the warm up pitches, throw 35-40 pitches from the mound with catcher in the squat. 50% in stretch and 50% in full wind up.
Throw inside, outside, high and low in the zone, using all your pitches.
End of the warm up: Last 10-15 pitches at 100% with pitches you will use in the 1st inning!!
Mental Focus: Clear your mind and focus on the game. Do not concern yourself with factors YOU can not control: weather, field, umpire, fans, etc.
Review signs with your catcher and basic pitch strategy. Review with coach.
HOME GAME: Run 3-4 foul poles. Walk one more!
Begin warm up throwing 30 minutes before the game, so you finish 5-10 minutes prior.
Go to dugout and put on jacket and sit and relax. Review game plan and have a few sips of water.
AWAY GAME: Start warm up throwing 20 minutes before the game starts so you will be ready at the end of your team's at bats. Go to the dugout with 2 outs and sit and have a few sips of water.
ALWAYS put on your jacket after your warm up…no matter the weather!!
ALWAYS ice your arm after you pitch(unless playing another position).
20 minutes on shoulder and 10 minutes on elbow!! Apply 30 minutes after pitching NO MATTER how many pitches you throw!!
Put on your jacket after icing or if at home, put on a long sleeve jacket or shirt.
ALWAYS run 6-8 poles after the game. This will help circulate your blood and rid your body of lactic acid that builds up during the game.
Extensive arm stretching and elastic resistance tube exercises.
Throw 30-35 pitches on a mound to a catcher in a squat.
Warm up playing catch. Exaggerate mechanics and lob the ball.
Throw at 100% and mix pitches.
FOCUS on your general target, between the catcher's shoulders and knees. Last 8 pitches to a specific glove target.
Tube resistance drills and exercises.
Run 5-6 40 yard dashes and light agility drills.
Eat a good dinner…FIVE COLORS!!
Get a very good night sleep: 8 hours and NO sleepovers!!
Being a pitcher is hard work…it is up to YOU to PREPARE: be in good shape, have a plan, and follow a good, sound pre-game, post-game, and between games routine!! GAGPTH!
Arm, shoulder and elbow, injuries are on the rise in youth baseball rather than decreasing!! There are more Tommy John surgeries and labrum repairs in youth ball than in professional baseball!! Listed below are the common causes found in the American Sports Medical Institute(ASMI) data from treating and rehabbing thousands of youngsters.
1. Lack of a long term throwing and conditioning program. There should be at least one month of progressive throwing before actually pitching in a game. Preparing the arm to pitch is very important. A solid pre-season throwing program should include stretching, form running, and throwing short to long (increasing the distance periodically). Youth players need to throw (catch, bull pen, etc) and long-toss in order to build arm strength and endurance.
2. Over-use. This is caused by throwing too many pitches in a single outing/game. This may cause throwing with arm fatigue and will increase the risk of serious injury. Pitch counts are vital! It is not about the innings but the actual pitches thrown.
3. Overload. This is pitching without proper recovery time. Generally this is a result of pitching for more than one team, too many starts in a week, too much pitching in the off-season, and lack of arm care/maintenance between pitching starts. Pitchers should have a scheduled program throughout the season with days of rest and recovery.
4. Improper pitching mechanics. If a player does not have proper mechanics, the chance of injury is exponential. The body and arm MUST be in the optimum position during the delivery. The head alignment, landing, posture, trunk flex, and balance all must be considered.
5. Over-exertion of the pitching arm. This is a result of a player attempting to throw 100% all the time. Pitchers should vary their throwing and generally be at the 90% effort level.
Be careful of the arm and develop a common sense regimen for proper pitching!!
This entry was posted in Handouts & Drills, Pitching, Throwing on June 30, 2014 by admin.
Pitching a baseball is not only a skill but also an art. It is the most important defensive skill, and pitching will win most games. Hitting is the essence of the game but pitching wins games.
In order to be a successful pitcher you MUST learn and practice the proper fundamentals of pitching. Many young players have a knack for getting the ball over the plate and just need to develop proper technique and arm strength. The key to good, solid pitching is CONTROL! Control is developed over time by learning the correct mechanics and hours of throwing/pitching. As with hitting, good pitching is the result of perfect practice. Remember: It takes patience, hard work, focus/concentration to learn how to pitch a baseball.
NOTE: Proper arm care is a MUST! Do not throw or pitch with a sore arm.
1. ATTITUDE: "King of the Hill"; I can throw strikes!
PRACTICE YOUR "FUNDIES"!! Fundamentals!! Focus on the catcher's mitt/target. Do not over-practice…let your arm rest! The very essence of success in getting batters out is CONSISTENCY with LOCATION and changing speeds. RUN: Legs and the core body are the secret to success. VISUALIZE!!
This entry was posted in Handouts & Drills, Pitching on June 30, 2014 by admin.
1. The plate is only 17" wide: Actually it is 13" as the middle 4" is "clobberville", hence the plate is a very small target for the pitcher to hit. In fact, some umps make it even smaller!! The size of the plate makes it very easy for a good hitter to protect. Hitters: Establish your strike zone and GYPTH!
2. NO Pitcher has perfect control and/or command: In each plate appearance, a hitter will get at least ONE very good pitch to hit…find YOURS! Take a pitch or two; getting deep into the count puts pressure on the pitcher. The more pitches you see the easier it is to time the pitch.
3. Pitcher's effectiveness suffers as his pitch count rises: Arms tire and the legs lose push, therefore, control wanes and he makes more mistakes. Swinging early helps the pitcher! NEVER allow a pitcher a three(3) pitch inning!
4. Most pitchers love to work the outside corners: This creates more room for misses/error and these pitches are harder to hit. Increase plate coverage and hit the outside pitch to the opposite field. Do not try to pull the ball, but let it travel more and hit it with a short, quick swing the other way.
5. Everyone loves a Home Run hitter: Pitchers love to see the batter who swings with an upper cut and one who has a long swing. They will throw off speed and high(above the hands) to get the easy pop up. Pitchers hate the "chippers" who just "chip away".
A pitchers WORST nightmare: He has to come to you with a pitch. He must throw the ball through a very small window. He will make a mistake during your at bat. You expect the outside pitch. The short, quick swing will lead to his short quick stay on the bump!!!
Hitters: Work the count and GYPTH!!!
This entry was posted in Handouts & Drills, Pitching on May 9, 2014 by admin. | {
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Dr. Richelle Tanner
Climate change ecophysiology.
This page is framed for public communication purposes. Please see CURRENT PROJECTS for specific research topics.
Breaking it down word-by-word, "ecophysiology" is really two topics in one. Physiology is the study of how the body operates, and responds to the environment. It is the physical expression of the relationship between selection and the environment! Ecophysiology refers to a specific subset of physiology where we look at how animal (or plant) physiology is changed by ecological and environmental relationships. Within these ecological and environmental relationships with animal physiology, I'm specifically interested in how climate change plays a role. If climate change is the "cause", the "effect" I'm interested in is physiology. I define climate change as any environmental phenomenon associated with the human-induced increase in carbon dioxide in the atmosphere. This includes environmental changes like rising average temperatures, ocean acidification, and increased frequency of extreme events (cold snaps, heat waves, flooding, hurricanes, etc.).
How do I study it?
Physiology has many methods in the toolbox, both in the lab and in the field. I track natural populations in the wild, I raise animals in the lab, and I use these animals to look at many levels of biological organization. This means I look at how often individuals reproduce, how they respond to climate change stressors in their swimming speed/muscle function/heart rate/etc., and what is happening at the biochemical level to proteins, genes, and potential modifiers of these. Through all of these methods, I can build a picture of how an animal may respond to certain environmental conditions during different parts of its life and through multiple generations of a species. I also use these findings to build predictive mathematical models of how ecological interactions and ecosystems in general will shift with changes in climate.
What ecological and evolutionary questions can I address?
As climate change is ongoing, there are so many opportunities to look forward to future shifts and also explore current physiological responses. My main focus currently is looking at how all of these physiological traits vary within a single species, so that we can better understand the available variation that selection can act upon when the environment changes. I can look at how these single species responses translate to ecological interactions - within a food web or ecosystem, species rely on each other to function properly. What happens when one or more has a change in their physiology and consequently their functioning within the ecosystem? I can also look at whether certain species are more likely to adapt to climate change because of evolutionary history. For example, species living in highly fluctuating environments might fare well in future fluctuating environments that we're expecting. Is physiological adaptation more likely for certain groups of species and why? Does this relate to patterns we've seen in the fossil record for the most successful taxa through major environmental changes?
An easy method to look at how sea hares respond to climate change stressors in foot muscle function. We raise animals in the lab at multiple temperatures representing different climate scenarios, and then test their response to heat waves by heating them until muscle function is lost. This is called the critical thermal maximum. When they lose muscle function, they can't eat or move around - they recover from this experiment fully, but this response tells us about how sea hares can stop functioning in their food web or ecosystem when heat waves strike.
Nudibranchs, a special topic
© 2019 Richelle Tanner | {
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Q: Math.pow and Method Calling I am writing a method to identify a polynomial functions from a given sequence of numbers.
public void updateSequence(Term t) {
for (int i = 0; i <= sequence.length; i++) {
sequence[i] = sequence[i] - t(c) * Math.pow(i, t(e));
}
}
I am having some problems with this method. I am unsure how to call a method called getCoefficient from another class called Term which is basically a method containing the following statement
return coefficient;
and then use that in the method above. I need to find the term which is basically coefficient * i ^exponent and then save it back into the sequence to be run through the iteration again.
These are the content of the term class.
public class Term {
term = coefficient * x ^ exponent
private double coefficient;
private int exponent;
public Term(double c, int e) {
coefficient = c;
exponent = e;
}
// returns the coefficient
public double getCoefficient() {
return coefficient;
}
// returns the exponent
public int getExponent() {
return exponent;
}
// returns the term as a String for display
// see the sample file for the layout required
public String display() {
// TODO
String tocompile = "tocompile";
return tocompile;
}
}
| {
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College Basketball – Jordan Walker's 19 points leads men's basketball past Cincinnati Christian in 106-63 exhibition win
BSN Staff
MSU win over Cincinnati Christian
MOREHEAD, Ky. – Morehead State men's basketball defeated Cincinnati Christian 101-63 in the Eagles' lone exhibition test of the season on Sunday at Johnson Arena.
Morehead State was led by freshman point guard Jordan Walker, who logged 19 points, eight rebounds, and four assists in 20 minutes off the bench. The 6-foot-1 Indianapolis native headed an Eagles bench attack that outscored Cincinnati Christian 67-6.
"I was very pleased with the new guys," said head coach Sean Woods. "I thought they came in and played solid, hit open shots, and took care of the basketball. These young guys are really refreshing to watch."
Morehead State outgunned its opponent on 9-for-18 (.500) shooting from three-point range, including 3-for-4 from Walker. The Eagles rounded out the afternoon with 58 points in the paint to Cincinnati Christian's 10, and converted 24 points off 17 forced turnovers.
Sophomore guard Malik Maitland poured in 11 of his 13 points in the first half, and dished out four assists in 20 minutes of action. Ty'Quan Bitting (12 points, seven rebounds) and Djimon Henson (10 points, four rebounds, four assists) capped off the double-digit scoring for the Eagles.
Morehead State broke off a 21-0 run that highlighted a 52-29 lead at halftime, and ended the contest with 10 consecutive points over the last 3:29. All 14 players on the roster made their way onto the scoring column.
"For the most part, we were solid as a rock," Woods said. "The older guys have to be major factors, and these new guys have got to be prepared to step in to play."
The Eagles open regular season play on Friday, Nov. 11 when Kentucky Christian visits Johnson Arena at 7:45 p.m.
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DARRELL MILLER, January 26, 1946 – July 27, 2021. BRENDON D. MILLER – July... | {
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{"url":"http:\/\/math.sjtu.edu.cn\/Conference\/2015WCA\/talk.php?ZhuYan","text":"2015 Workshop on Combinatorics and Applications at SJTU\n\nApril 21 -- 27, Shanghai Jiao Tong University\n\nDate: Tuesday, April 21, 2015\n\nTime: 11:45 - 12:15\n\nVenue: Large Meeting Room, Math Building\n\nTitle: Tight Relative $2e$-Designs in Association Schemes\n\nAbstract:\n\nRelative $t$-designs is defined on Q-polynomial association scheme and we call it tight if it satisfies the Fisher type lower bound. We will mainly review some results about tight relative $2$-designs $(Y,w)$ on two shells $X_{r_1} \\cup X_{r_2}$ in binary Hamming association scheme $H(n,2)$ and Johnson association scheme $J(v,k)$. The good feature of $H(n,2)$ is that the distance set $\\{ \\alpha_1,\\alpha_2,\\gamma \\}$ and $\\frac{w_2}{w_1}$ are uniquely expressed in terms of $n,r_1,r_2,N_{r_1}$. This implies coherent configuration is attached to $Y$. (However, for $J(v,k)$, this property is difficult to prove in general .) There exist many tight relative $2$-designs in $H(n,2)$ with both constant weight and $w_2 \\neq w_1$. So far, we are unable to find such designs with $w_2 \\neq w_1$ in $J(v,k)$. Now we are working on the existence of tight relative $4$-designs on two shells in $H(n,2)$. In this case, we can not determine all the feasible parameters. But it is proved that $Y \\cap X_{r_i}, i=1,2$ should be combinatorial $3$-design. Finally this problem is related to the existence of some combinatorial designs. This is joint work with Eiichi Bannai and Etsuko Bannai.\n\nSlides: View slides\n\nWebmaster","date":"2019-12-11 06:03:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8223011493682861, \"perplexity\": 605.2498008729111}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540529955.67\/warc\/CC-MAIN-20191211045724-20191211073724-00148.warc.gz\"}"} | null | null |
Hang Tuah station, also known as BBCC - Hang Tuah station due to the new transit hub, is an interchange station in the Pudu district of Kuala Lumpur, Malaysia, between the Ampang and Sri Petaling Lines (formerly known as STAR) and the KL Monorail. Seamless physical and fare integration was achieved on 1 March 2012 when the "paid-up" or restricted areas of both the LRT and monorail stations, which previously operated as two separate stations, were linked up, allowing passengers to transfer without needing to buy new tickets for the first time since the monorail became operational in 2003.
Located right beside of Bukit Bintang City Centre (BBCC), the station is in the vicinity of the Methodist Boys School, Victoria Institution, Bukit Aman (Kuala Lumpur Contingent Police Headquarters) and Stadium Negara.
History
Hang Tuah LRT station
The LRT station was formed in 1996 with the completion of the Hang Tuah LRT station, a rapid transit station along a reused stretch of a long abandoned railway line, used by STAR (now Ampang Line) trains traveling along the Ampang-bound line or the Putra Heights-bound line. The station was one of the first 13 STAR rapid transit stations to open in the city, part of "Phase I" of "System I". The surface level station consists of two sheltered platforms connected to a sheltered ticketing concourse above, linked only via staircases, and exiting to Jalan Hang Tuah.
Hang Tuah KL Monorail station
The area saw the addition of the Hang Tuah KL Monorail station, a separate station that provides access to Kuala Lumpur Monorail (KL Monorail) services, with the opening of the line on 31 August 2003. The station, like virtually all other KL Monorail stations, is elevated and built over Jalan Hang Tuah, but was not initially directly integrated to the nearby existing LRT station until 2012. The station had two exits that lead to either side of the road, and is adjacent to and located northeast from the LRT station. The station is only accessible via stairways and escalators. The station's length is also longer in comparison to certain KL Monorail stations.
The Hang Tuah monorail station is one of four Kuala Lumpur Monorail stations that serves the Kuala Lumpur Golden Triangle locality, the other three being the Raja Chulan station, the Bukit Bintang station and the Imbi station. The station is also one of two KL Monorail stations that is designated as an interchange to and from the Ampang Line and Sri Petaling Line, the other being the Titiwangsa station.
Integrated Hang Tuah Station
The proximity of the monorail station with the LRT station meant both stations are designated in transit maps as interchange stations between the two lines, and is subsequently use heavily primarily to reach Bukit Bintang and surrounding areas via the monorail line from the Ampang Line and the Sri Petaling Line. Since 2013, the two stations have been physically linked so it is no longer required to exit one station or to use the public sidewalk outside the station to reach the other station or to change lines (monorail to LRT or vice versa).
Station structure
The platforms of the LRT station were significantly revamped and remodelled when the station was physically linked to the nearby monorail station in 2012. These platforms have also been lengthened to accommodate longer trains. The elevated KL Monorail's station is located northeast to the LRT station. The concourse of the stations are located at the street level and level 1 respectively.
Exits and Entrances
This station has three signed entrances. Two originally part of the Hang Tuah Monorail station and a common signed access to the main ticketing hall for all three rail services. A new entrance connecting the station complex to the Bukit Bintang City Centre transit hub has been built for the development.
Connection to MRT
Though not an interchange, Hang Tuah station is located close to the Merdeka MRT station and is accessible by a 600-metre walk. Merdeka MRT station itself is integrated with Plaza Rakyat LRT station.
Around the station
Bukit Bintang City Centre (BBCC)
Victoria Institution
Al-Bukhari Foundation Mosque
Merdeka 118 precinct
118 Mall
Kenanga Wholesale City
Notes
References
Ampang Line
Kuala Lumpur Monorail stations
Railway stations opened in 1996 | {
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Can not sign onto my Microsoft Surface.
I have signed onto my surface for so long using my pin I forgot my password. My surface done a update and now the sign on screen is asking for a password and not a pin number. It shows my gmail account on the sign in screen and when I put that passwords in it will not let me in. I can sign into my gmail on another computer with no problem. Help. I now have a expensive door stop.
Re: Can not sign onto my Microsoft Surface. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,924 |
Palmarès
Club
Atlético Madrid: 2016-2017, 2017-2018, 2018-2019
Atlético Madrid: 2016
Nazionale
2018
2010
Individuali
Guanto d'oro: 1
Mondiale Under-17 di Trinidad e Tobago 2010
Note
Altri progetti
Collegamenti esterni
Calciatrici della Nazionale spagnola | {
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Niña bonita es una telenovela venezolana producida y emitida por Venevisión en 1988, primer rol protagónico de Ruddy Rodríguez al lado de Luis José Santander.
Argumento
Un triángulo amoroso, un conflicto en el cual están involucradas tres personas, dos jóvenes hermanas y un joven médico casado con una de ellas, pero a la vez enamorado de la otra sin conocer el parentesco que las une. Ángela, que tras estar tres años fuera de su país, llega a la Isla de Margarita y prepara su regreso sorpresivo a la capital y a su familia. Casualmente, en la misma isla, se encuentra con el doctor Francisco León en un congreso médico.
El encuentro se produce mientras Ángela, buceando en las fabulosas aguas del Caribe, sufre un accidente y el doctor León logra auxiliarla al subir a la superficie, y allí comienza todo. Encuentros llenos de humor y romance, impregnados de pasión. Quedan en encontrarse al día siguiente y esto no llega a producirse. Ángela acude a ver a su hermana Emilia y a sus padres, quienes le preparan una fiesta de bienvenida, donde su hermana le presenta a su esposo: nada menos que el doctor Francisco León. Humor, drama y pasión se unen en una simbiosis perfecta, para hacer de esta historia, algo "muy fuerte".
Elenco
Ruddy Rodríguez - ''Ángela Santana De León Soler / Paulina Santana de León Soler
Luis José Santander - Francisco Antonio León
Abril Méndez - Emilia
Henry Galué - Danilo
Fernando Flores - Aquiles
Raúl Xiques - Eligio Santana
Eduardo Gadea Pérez
Ramón Hinojosa - Filiberto
Helianta Cruz - Ana Elisa Valparaiso
Francis Helen -Carlota Fuenmayor,la abogada de Eligio Santana
Laura Zerra - Alcira
Mirtha Borges - Dolorita
Betty Ruth - Altagracia
Chela D'Gar - María Renata Izaguirre
Jimmy Verdum - Igreja
Mauricio González - Gaspar
José Rubens - Sr. Uzcategui
Vicente Tepedino - Héctor, el hijo de Aquiles
Carlos Subero - Pío
Vilma Otazo - Violeta
Martha Pabón - Virginia
María Elena Heredia - Julie
Héctor Clotet
Marcos Campos - padre Sebastián
Juan Carlos Gardié - Manolo
Marisela Leandro - Debora
Marilyn Sánchez
Ricardo Montaner - Arnaldo Blanco
Ernesto Balzi - Ernesto Martínez
José Ruiz - policía
Rafael Romero - Chucho
Yaneth Pérez - Cecilia,la sobrina de Dolores
María Elena Coello - Cathy
Carlos Davila - Carlos
Sixto Blanco - Aristides
José María Bauce - Alejandro Fuentes, el abogado
Yomally Almenar - Claudia, la secretaria de Aquiles Blanco
Adela Romero - Angélica, la novia de Héctor Blanco
Luis Aular - Rodrigo
Zulma López - Elena
Mayra Chardiet - Miguelina
Producción
Original de: César Miguel Rondon
Tema musical: "Tan Enamorados"
Intérprete: Ricardo Montaner
Musicalización: Frank Aguilar
Coordinación: Isidro Riera
Escenografía: Rolando Salazar
Edición: Carlos izquierdo
Producción: Raúl Díaz
Producción ejecutiva: Carlos Suárez
Dirección: Reinaldo Lancaster
Telenovelas de Venezuela
Telenovelas de 1988 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,487 |
Q: How to implement RecyclerView with section header depending on category? I want to implement sections in my list. I have a list of tasks. List has a custom adapter which extends recyclerview swipe adapter as I have implemented swipe gesture to the recyclerview.
So now tasks list is shown together with completed and pending tasks. Each list item has a check box which shows task is completed or pending.
If check box is checked then task is completed and vise versa. Now I want to make two sections in this with header. One For completed tasks and another for pending tasks.
So completed tasks should be shown inside completed section and vise versa. Also if the task is unchecked i.e pending and if user checks the check box then the item should get removed from the pending section and should get added to the completed section and vise versa.
I checked with one library for sections.
https://github.com/afollestad/sectioned-recyclerview
But when I tried to implement the library I got the error that adapter can not extend two classes as I have extended recyclerview swipe library before.
Adapter :
public class IAdapter extends RecyclerSwipeAdapter<IAdapter.ItemViewHolder> {
public ArrayList<Task> items;
Context conext;
public int _mId;
List<Task> itemsPendingRemoval = new ArrayList<>();
public IAdapter(Context context, ArrayList<Task> item) {
this.conext=context;
this.items=item;
}
@Override
public int getItemCount() {
return items.size();
}
public static class ItemViewHolder extends RecyclerView.ViewHolder {
Task task;
CheckBox cb;
SwipeLayout swipeLayout;
TaskTableHelper taskTableHelper;
ItemViewHolder(final View itemView) {
super(itemView);
taskTableHelper= new TaskTableHelper(itemView.getContext());
swipeLayout = (SwipeLayout) itemView.findViewById(R.id.swipe);
cb = (CheckBox) itemView.findViewById(R.id.checkbox);
}
}
@Override
public void onBindViewHolder(final ItemViewHolder itemViewHolder,final int i) {
itemViewHolder.cb.setText(items.get(i).getTitle());
itemViewHolder.task = items.get(i);
int taskId = itemViewHolder.task.getId();
itemViewHolder.task = itemViewHolder.taskTableHelper.getTask(taskId);
int status = itemViewHolder.task.getStatus();
if(status == 0)
{
itemViewHolder.cb.setTextColor(Color.WHITE);
}
else {
itemViewHolder.cb.setChecked(true);
itemViewHolder.cb.setTextColor(Color.parseColor("#B0BEC5"));
}
itemViewHolder.cb.setOnCheckedChangeListener(new CompoundButton.OnCheckedChangeListener() {
@Override
public void onCheckedChanged(CompoundButton buttonView, boolean isChecked) {
if (isChecked) {
itemViewHolder.cb.setTextColor(Color.parseColor("#B0BEC5"));
itemViewHolder.task.setStatus(1);
itemViewHolder.taskTableHelper.updateStatus(itemViewHolder.task);
}
else
{
itemViewHolder.cb.setTextColor(Color.WHITE);
itemViewHolder.task.setStatus(0);
itemViewHolder.taskTableHelper.updateStatus(itemViewHolder.task);
}
}
});
final Task item = items.get(i);
itemViewHolder.swipeLayout.addDrag(SwipeLayout.DragEdge.Right,itemViewHolder.swipeLayout.findViewById(R.id.bottom_wrapper_2));
itemViewHolder.swipeLayout.setShowMode(SwipeLayout.ShowMode.LayDown);
itemViewHolder.swipeLayout.setOnDoubleClickListener(new SwipeLayout.DoubleClickListener() {
@Override
public void onDoubleClick(SwipeLayout layout, boolean surface) {
Toast.makeText(conext, "DoubleClick", Toast.LENGTH_SHORT).show();
}
});
itemViewHolder.swipeLayout.findViewById(R.id.trash2).setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
mItemManger.removeShownLayouts(itemViewHolder.swipeLayout);
items.remove(i);
notifyItemRemoved(i);
notifyItemRangeChanged(i, items.size());
mItemManger.closeAllItems();
itemViewHolder.taskTableHelper.deleteTask(item);
_mId = item.getAlertId();
cancelNotification();
Toast.makeText(view.getContext(), "Deleted " + itemViewHolder.cb.getText().toString() + "!", Toast.LENGTH_SHORT).show();
}
});
itemViewHolder.swipeLayout.findViewById(R.id.done).setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
itemViewHolder.task.setStatus(1);
itemViewHolder.taskTableHelper.updateStatus(itemViewHolder.task);
itemViewHolder.cb.setChecked(true);
Toast.makeText(conext, "Task Completed.", Toast.LENGTH_SHORT).show();
}
});
itemViewHolder.swipeLayout.getSurfaceView().setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
boolean mEditMode;
int id = item.getId();
int priority = item.getTaskPriority();
String title = item.getTitle();
String alertDate = item.getAlertDate();
String alertTime = item.getAlertTime();
String dueDate = item.getDueDate();
String dueTime = item.getDueTime();
_mId = item.getAlertId();
int listId = item.getList();
mEditMode = true;
Intent i = new Intent(conext, AddTaskActivity.class);
i.putExtra("taskId", id);
i.putExtra("taskTitle", title);
i.putExtra("taskPriority", priority);
i.putExtra("taskAlertTime", alertTime);
i.putExtra("taskAlertDate", alertDate);
i.putExtra("taskDueDate", dueDate);
i.putExtra("taskDueTime", dueTime);
i.putExtra("taskListId", listId);
i.putExtra("EditMode", mEditMode);
i.putExtra("AlertId",_mId);
conext.startActivity(i);
}
});
mItemManger.bindView(itemViewHolder.itemView, i);
}
@Override
public ItemViewHolder onCreateViewHolder(ViewGroup viewGroup,int position) {
View itemView = LayoutInflater.
from(viewGroup.getContext()).
inflate(R.layout.card_layout, viewGroup, false);
return new ItemViewHolder(itemView);
}
@Override
public void onAttachedToRecyclerView(RecyclerView recyclerView) {
super.onAttachedToRecyclerView(recyclerView);
}
public void remove(int position) {
Task item = items.get(position);
if (itemsPendingRemoval.contains(item)) {
itemsPendingRemoval.remove(item);
}
if (items.contains(item)) {
items.remove(position);
notifyItemRemoved(position);
}
}
public void cancelNotification()
{
AlarmManager alarmManager = (AlarmManager)conext.getSystemService(Context.ALARM_SERVICE);
Intent intent = new Intent(conext, NotificationReceiver.class);
PendingIntent pendingIntent = PendingIntent.getBroadcast(conext,_mId, intent, 0);
alarmManager.cancel(pendingIntent);
}
@Override
public int getSwipeLayoutResourceId(int position) {
return R.id.swipe;
}
}
EDIT:
onBindViewHolder method after extending sectionedRecyclerview adapter:
@Override
public void onBindViewHolder(ItemViewHolder itemViewHolder, int section, int i, int absolutePosition) {
itemViewHolder.cb.setText(items.get(i).getTitle());
itemViewHolder.task = items.get(i);
int taskId = itemViewHolder.task.getId();
itemViewHolder.task = itemViewHolder.taskTableHelper.getTask(taskId);
int status = itemViewHolder.task.getStatus();
if(status == 0)
{
itemViewHolder.cb.setTextColor(Color.WHITE);
}
else {
itemViewHolder.cb.setChecked(true);
itemViewHolder.cb.setTextColor(Color.parseColor("#B0BEC5"));
}
itemViewHolder.cb.setOnCheckedChangeListener(new CompoundButton.OnCheckedChangeListener() {
@Override
public void onCheckedChanged(CompoundButton buttonView, boolean isChecked) {
if (isChecked) {
itemViewHolder.cb.setTextColor(Color.parseColor("#B0BEC5"));
itemViewHolder.task.setStatus(1);
itemViewHolder.taskTableHelper.updateStatus(itemViewHolder.task);
}
else
{
itemViewHolder.cb.setTextColor(Color.WHITE);
itemViewHolder.task.setStatus(0);
itemViewHolder.taskTableHelper.updateStatus(itemViewHolder.task);
}
}
});
final Task item = items.get(i);
itemViewHolder.swipeLayout.addDrag(SwipeLayout.DragEdge.Right,itemViewHolder.swipeLayout.findViewById(R.id.bottom_wrapper_2));
itemViewHolder.swipeLayout.setShowMode(SwipeLayout.ShowMode.LayDown);
itemViewHolder.swipeLayout.setOnDoubleClickListener(new SwipeLayout.DoubleClickListener() {
@Override
public void onDoubleClick(SwipeLayout layout, boolean surface) {
Toast.makeText(conext, "DoubleClick", Toast.LENGTH_SHORT).show();
}
});
itemViewHolder.swipeLayout.findViewById(R.id.trash2).setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
mItemManger.removeShownLayouts(itemViewHolder.swipeLayout);
items.remove(i);
notifyItemRemoved(i);
notifyItemRangeChanged(i, items.size());
mItemManger.closeAllItems();
itemViewHolder.taskTableHelper.deleteTask(item);
_mId = item.getAlertId();
cancelNotification();
Toast.makeText(view.getContext(), "Deleted " + itemViewHolder.cb.getText().toString() + "!", Toast.LENGTH_SHORT).show();
}
});
itemViewHolder.swipeLayout.findViewById(R.id.done).setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
itemViewHolder.task.setStatus(1);
itemViewHolder.taskTableHelper.updateStatus(itemViewHolder.task);
itemViewHolder.cb.setChecked(true);
Toast.makeText(conext, "Task Completed.", Toast.LENGTH_SHORT).show();
}
});
itemViewHolder.swipeLayout.getSurfaceView().setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View v) {
boolean mEditMode;
int id = item.getId();
int priority = item.getTaskPriority();
String title = item.getTitle();
String alertDate = item.getAlertDate();
String alertTime = item.getAlertTime();
String dueDate = item.getDueDate();
String dueTime = item.getDueTime();
_mId = item.getAlertId();
int listId = item.getList();
mEditMode = true;
Intent i = new Intent(conext, AddTaskActivity.class);
i.putExtra("taskId", id);
i.putExtra("taskTitle", title);
i.putExtra("taskPriority", priority);
i.putExtra("taskAlertTime", alertTime);
i.putExtra("taskAlertDate", alertDate);
i.putExtra("taskDueDate", dueDate);
i.putExtra("taskDueTime", dueTime);
i.putExtra("taskListId", listId);
i.putExtra("EditMode", mEditMode);
i.putExtra("AlertId",_mId);
conext.startActivity(i);
}
});
mItemManger.bindView(itemViewHolder.itemView, i);
}
How can I do this? Can anyone help with this please?
Thank you..
A: The most simple way to split your recycler view into sections is by using a layout with the header and the item already in place and then changing the visibility if the header is the same.
Layout:
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="wrap_content">
<TextView
android:id="@+id/tvHeader"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_gravity="start"
android:padding="16dp"
android:textAppearance="?android:attr/textAppearanceMedium"
android:textColor="@color/primary"
android:textStyle="bold"
tools:text="A" />
<TextView
android:id="@+id/tvName"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:layout_below="@+id/tvHeader"
android:background="?android:attr/selectableItemBackground"
android:padding="16dp"
android:textAppearance="?android:attr/textAppearanceMedium"
tools:text="Adam" />
</RelativeLayout>
Adapter (2018 Kotlin Edition):
class ContactAdapter @Inject constructor() : RecyclerView.Adapter<ContactAdapter.ViewHolder>() {
var onItemClick: ((Contact) -> Unit)? = null
var contacts = emptyList<Contact>()
override fun getItemCount(): Int {
return contacts.size
}
override fun onCreateViewHolder(parent: ViewGroup, viewType: Int): ViewHolder {
return ViewHolder(LayoutInflater.from(parent.context).inflate(R.layout.item_contact, parent, false))
}
override fun onBindViewHolder(holder: ViewHolder, position: Int) {
val name = contacts[position].name
holder.header.text = name.substring(0, 1)
holder.name.text = name
// if not first item, check if item above has the same header
if (position > 0 && contacts[position - 1].name.substring(0, 1) == name.substring(0, 1)) {
holder.headerTextView.visibility = View.GONE
} else {
holder.headerTextView.visibility = View.VISIBLE
}
}
inner class ViewHolder(itemView: View) : RecyclerView.ViewHolder(itemView) {
val headerTextView: TextView = itemView.tvHeader
val nameTextView: TextView = itemView.tvName
init {
itemView.setOnClickListener {
onItemClick?.invoke(contacts[adapterPosition])
}
}
}
}
Might be helpful as well: RecyclerView itemClickListener in Kotlin
Old Java Adapter Version:
public class RecyclerAdapter extends RecyclerView.Adapter<RecyclerAdapter.DataViewHolder> {
private List<Contact> mData;
@Inject
public RecyclerAdapter() {
mData = new ArrayList<>();
}
public void setData(List<Contact> data) {
mData = data;
}
public Contact getItem(int position){
return mData.get(position);
}
@Override
public DataViewHolder onCreateViewHolder(ViewGroup parent, int viewType) {
View itemView = LayoutInflater.from(parent.getContext()).inflate(R.layout.item_contact, parent, false);
return new DataViewHolder(itemView);
}
@Override
public void onBindViewHolder(final DataViewHolder holder, int position) {
Contact contact = mData.get(position);
holder.headerTextView.setText(contact.getName().substring(0, 1));
holder.nameTextView.setText(contact.getName());
// if not first item check if item above has the same header
if (position > 0 && mData.get(position - 1).getName().substring(0, 1).equals(contact.getName().substring(0, 1))) {
holder.headerTextView.setVisibility(View.GONE);
} else {
holder.headerTextView.setVisibility(View.VISIBLE);
}
}
@Override
public int getItemCount() {
return mData.size();
}
public class DataViewHolder extends RecyclerView.ViewHolder {
@BindView(R.id.text_header)
TextView headerTextView;
@BindView(R.id.text_name)
TextView nameTextView;
public DataViewHolder(View itemView) {
super(itemView);
ButterKnife.bind(this, itemView);
}
}
}
A: You can implement it with the library SectionedRecyclerViewAdapter as I explained in this post.
In order to implement the SwipeLayout, don't extend RecyclerSwipeAdapter, extend SectionedRecyclerViewAdapter and implement the SwipeLayout in ItemViewHolder / onBindItemViewHolder as you have done.
A: You can do it youself by hard codding.There ar smart ways to do this. follow these links. and choose one for you.
https://github.com/afollestad/sectioned-recyclerview
https://github.com/truizlop/SectionedRecyclerView
http://android-pratap.blogspot.in/2015/12/sectioned-recyclerview-in-android_1.html
You can search more by "sectioned recyclerViews android libraries"
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{"url":"http:\/\/mathhelpforum.com\/advanced-statistics\/55058-statistics-joint-probability-distribution-problem.html","text":"# Math Help - Statistics: Joint Probability Distribution Problem\n\n1. ## Statistics: Joint Probability Distribution Problem\n\nThe following table gives the joint probability function of return on stocks (X) and returns on bonds (Y).\n\nWhat is the expected value of return on stocks? This is (.3)(-10)+(.7)(10)=4\nWhat is the expected value of return on bonds? This is (.4)(0)+(.6)(5)=3\nWhat is the variance of returns on stocks?\nWhat is the variance of returns on bonds?\nWhat is the covariance of returns on stocks and bonds?\n\nI'm not exactly sure how to proceed with calculating variance. I think I'm just misapplying the formula for this seemingly easy problem but my answers just aren't matching up. Can someone help?\n\n2. Originally Posted by ashinynicklel\nThe following table gives the joint probability function of return on stocks (X) and returns on bonds (Y).\n\nWhat is the expected value of return on stocks? This is (.3)(-10)+(.7)(10)=4\nWhat is the expected value of return on bonds? This is (.4)(0)+(.6)(5)=3\nWhat is the variance of returns on stocks?\nWhat is the variance of returns on bonds?\nWhat is the covariance of returns on stocks and bonds?\n\nI'm not exactly sure how to proceed with calculating variance. I think I'm just misapplying the formula for this seemingly easy problem but my answers just aren't matching up. Can someone help?\n$Var(X) = E(X^2) - [E(X)]^2$\n\nLet X be the random variable value of return on stocks.\n\n$E(X) = 4$, as you already know. $E(X^2) = (0.3)(-10)^2 + (0.7)(10)^2 = 30 + 70 = 100$.\n\nTherefore $Var(X) = 100 - 4^2 = 84$.\n\n3. Okay thank you I found the formula but wasn't sure how to apply it.\n\nSo this is the formula for covariance:\n\nThis isn't clear to me but again, how do I apply the formula for covariance? Namely what is E(X*Y)?\n\nSorry, I'm not really familiar with using expected values to calculate variance\/covariance.\n\n4. edit: got it. thanks again.","date":"2016-02-14 04:46:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 4, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7547647953033447, \"perplexity\": 935.9117671655616}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-07\/segments\/1454701168998.49\/warc\/CC-MAIN-20160205193928-00107-ip-10-236-182-209.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/blog.modelworks.ch\/page\/2\/","text":"# Using Powershell to clean up the animations your LaTeX file\n\nIn a previous post, I talked how to animate your Beamer presentations and how to print them as slides and article for the audience. One problem with a presentation with animations is that every element of an animation means one more slide. If you want to give the presentation to the audience so they can write on their slides using their ipad or touch screen notebook, they will complain that only one of the animated slides is complete.\n\nThe easiest way to remedy this, replace all overlays, \\pause, etc. from your LaTeX file and print it like this. One could do this easily in Emacs and write a macro for this. This time I chose another way to do this using Powershell, the awesome windows scripting language.\n\nFirst, I define a function that will show me a file picker. Then I will use this file picker to get the LaTeX-file that should be cleaned up. From the file chosen, I save the path in a variable $FolderPath and get the name of the file. This information is used to save the file as the original name with \u201c_NA\u201d (for no animation). Then I read the content and replace all occurences of \\pause, <+-.> <digit-> and \\only by a space using regular expressions. After this I save the file in the original place using the _NA-addition. The only thing I had problems with was the encoding (Powershell saved it in the wrong format), so I had to add \u201c\u2013Encoding Default\u201d as option. That\u2019s all. Of course, this could be done with any scripting language. # Tricks with Beamer: Animations and Print-Outs If you use Beamer for presentations, you might want to use \u201canimations\u201d: either show list elements successively or build up a diagram. Beamer has some nice possibilities for doing this: \u2022 In lists (enumerate, itemize) you can use the overlays. Just add <1-> after the first \\item and this item will show up first. For the second item you just add <2->, etc. You can also let them disappear again by giving a range (e.g. <2-3>, which means that this item will appear at the second and third slide, but will disappear again after that. If you want all the list elements to appear one after another and you don\u2019t want do write down the elements one by one, you can use [<+->] just after your \\begin{itemize}. \u2022 For other parts you can use \\pause. \u2022 For diagrams made with tikz, you can enclose the parts of the diagram using \\only<1->{ draw commands } with the same syntax as the overlays mentioned above. If I give a lecture, I use a trick by Tom Rutherford, to print out my presentation in slides- and article form. My lecture is always in the file lecture.tex. I than have two additional tex files: beamer.tex and print.tex for producing pdf-files in slide- and article format. This way, students can either use the slides and write on them using their ipad, or use the article and print it out for in class. The files look like this for the slides: and for the article: And here are the first pages: # Backing up your subversion repository on a remote Windows server using batch files My subversion repository is on a server by Webfaction (probably the best and cheapest way to host your repository. I have 100 Gigabyte space). I used to have a backup script that would save the backup on the same server, which then was transferred to my Windows 8 computer. Because I wanted to replace the complete backup with a new one, I ran into problems, as the backup procress was taking to much of the server CPU. Support pointed me to svnradmin, which is a tool comming with the command tools of tortoisesvn (otherwise you will find these tools on Collabnet). I searched the net and found some nice scripts to run this process automatically. I have two batch files. The first one runs the complete backup which is run once, the second one is a daily backup, that checks if there are new revsions and then starts to backup these. The first batch file (fullbackup.bat) looks like this: I agree: this looks rather cryptic. The first line takes the first 3 items from the output of the date\/t DOS-command is something like: 16.07.2015 (depending on your settings, this might be differt). The for command now reads this and separates this information in three parts (i,j,k) where the delimiter is given as \u201c.\u201d. This information is assigned to the variable \u201cdatesvn\u201d and will be stored as 20150716. The second line does the same kind of trick with the time. The third line gets the actual revision on my computer. I than set the filename where everything should be saved to the directory the batch file resides (in this case, it would be something like \u201cd:\\svnbackup\\SVN_20150716_1305_rev1_3995.dmp\u201d. The next line does the actual dump using svnrdump. Now I have a full backup I can build on. The last line saves the number of the last revision to a file. This whole process can take a couple of hours, if you have a big repository. You can also adjust this script, so you split the backup for revisions ranges like 1:1000, 1001:2000, etc. The second batch file will check if there are new revisions and if necessary start an incremental backup: It looks for the last revison in the text file, checks if there are new revisions and then runs the same kind of procedure as before. This batch file can be run on a regular basis using the task scheduler from Windows. # Automatic operator formatting mode in Emacs Emacs has a nice new mode called electric-operator developed by David Spepherd. It helps when you write code by formatting all operators in a predefined way. For example, in R it adds spaces around the operator signs, when you write 1+1, this is automatically converted into 1 + 1. You can define your own way of formatting for other modes. For gams-mode you add the following to your .emacs file First you set gams-mode as a mode on which electric operator can work.Then you add some rules. After that you set a hook so electric-operator is automatically started when you use Gams (or ESS). A special formatting is necessary for the comments in GAMS as they start with the same symbol as the multiplication. With the help of David, I could add a function gams-mode-* that changes the \u201c*\u201d used as a comment in \u201c* \u201c and otherwise in \u201c * \u201c. It checks if the \u201c*\u201d symbol is at the beginning of the line and in that case will not add a space in front of it. It probably needs some more tweaking, and I will keep you posted on improvements. # Note taking with LaTeX for learning Summarizing a text book is easy in LaTeX, but sometimes I want to learn the summarized text. Reading the summary over and over again is not very efficient, so I came up with the typical solution, where you can hide the main part of the summary and can check your knowledge by asking yourself questions based on keywords in the margin. Here is an example, where I started summarizing the R package data.table: Now I use hide the main text and can ask myself the questions: \u2022 Express the data.table in SQL-form? \u2022 How to create a data.table? Add the following to your latex file (note that I load also the packages lstlisting and some other packages not mentioned here): Some remarks: \u2022 I changed the vertical distance of the paragraphs, so it automatically skips a line (if you summarize with lots of text this is probably not a good thing, but if you summarize R commands like in the example, you don\u2019t want to have to add linebreaks after every paragraph. \u2022 I removed the indent for new paragraphs for the same reason. \u2022 I added the package color so I can show the margin notes in the color red (using the renewcommand for the marginfont. In the text you can now add a margin note by adding: # Column guide in Emacs Many professional editors have a column guide. For example, below you see in the Powershell editor a veritcal line at column 80. This marker comes in handy, if you want to print your files. In Emacs I usually check for the column 80 in the status line. However, I usually forget to check, and when I print out stuff or publish, I got line breaks where I don\u2019t want them. Passing the 80th column for comments is not a problem, as a simple Ctrl-q forces Emacs to break the lines nicely and add comment symbols at the beginning. But after hitting Ctrl-q, Emacs produces this: There is, however, a package, that mimics column guide seen in other editors. The package is called \u201cfill-column-indicator\u201d. Installing this package will do the trick. After installing, you only have to add the following lines to your .emacs file: I added two hooks, so the column rule appears automatically in Gams and ess. More information can be found here: http:\/\/www.emacswiki.org\/emacs\/FillColumnIndicator # Using file templates in Emacs A good practice in modeling is to place information related to the project your are working on, the subject, the data and your personal information like E-mail address at the top of your file. In Emacs you can define skeleton functions that will prompt for the information and put it in the file. Below is an example for a file with as comment symbol \u201c**: (the$Id: $is for my version control system, that automatically will add the information on the last commit in this line). Below you see how to make a function using skeleton. As I don\u2019t want to repeat myself, the function asks as an input the comment symbol(s) that should be used. I now can use the function for different programming languages (R uses \u201c#\u201d , C++ uses two forward slashes, GAMS \u201c*\u201d, etc.). You can add the time and date by using the command \u201ccurrent-time-string\u201d. All other inputs that have skeleton-read in front of it, are also prompted for. If you need the template for your whole project team, you could just make the author and contact field also input-dependent. More information can be found at: SkeletonMode. (note that this is part of Emacs since version 22). PS. The function puts an extra comment symbol at the end (not shown). This is a bug in my function. Any ideas how to get rid of this are welcome. # Using Org-mode to keep track of your project files For every project I work on I have a org-file. Every file has the following structure: \u2022 Tasks \u2022 Repeat \u2022 Wait \u2022 Notes \u2022 Calls \u2022 Reading I use remember-mode to capture tasks, notes and calls. Under the heading \u201cRepeat\u201d are tasks that are repeated on a regularly base. Under \u201cWait\u201d are reminders for things I wait for (like a reply on an e-mail, an order, a call, etc.). These waits are also captured using org-remember (I have written about that in an earlier post). Under the heading \u201cReading\u201d I have links to papers I am currently reading (using Bibtex; see an earlier post on that). Sometimes it is nice to have information on the files in one of the project directories, if possible with links. This can be done with org-fstree. Just add (require \u2018org-fstree) in your .emacs-file after installing. You now can add the following line somewhere in your org-project file: #+BEGIN_FSTREE: d:\/inbox\/Gams This will give me all the files in the directory d:\/inbox\/Gams with a direct link (this is my folder for answering questions in the Gams forums). You can update this by putting the cursor in this line and hitting C-c C-c. The documentation gives you information on the options: \u201c#+BEGIN_FSTREE: <dir> :<optionname1> <optionvalue1> :<optionname2> <optionvalue2> \u2026 Options are: \u2022 :non-recursive t , to suppress recursion into directories \u2022 :exclude-regexp-name <list of regexp strings> , exclude file\/directory names matching either of the given regexp expressions \u2022 :exclude-regexp-name (\u201c.*\\\\.pdf$\u201d \u201c.*\\\\.zip$\u201d), excludes files\/directories ending with either \u201c.pdf\u201d or \u201c.zip\u201d \u2022 :exclude-regexp-name (\u201c^\\\\.git$\u201c) , excludes files\/directories named \u201c.git\u201d\n\u2022 \u2013 :exclude-regexp-fullpath <list of regexp strings>, same as :exclude-regexp-name but matches absolute path to file\/directory\n\u2022 :show-only-matches t , only files that are being linked to show up\n\u2022 :only-directories t , only directories are listed\n\u2022 :only-regular-files t , only regular files are listed\n\u2022 :dynamic-update t , [EXPERIMENTAL] dynamically update a subtree on visibility cycling.\n\u2022 :no-annotations t, suppresses the search and display of file annotations\n\n# Following up on Outlook E-Mails in Org-Mode\n\nThe combination of Org-Mode and Remember-Mode helps me to organize my projects. One special task category is \u201cWAIT\u201d, if I have to wait for some input from somebody else or waiting for a delivery to arrive. An example is ordering a book at Amazon. As soon as I make the order, I generate a \u201cWAIT\u201d which would look like this\n\n** WAIT [#C] [2014-09-16 Tu] Book on Modeling by Morgan\n\nIn this case, I ordered the book on September 9th and it has not a high priority. Once a week I check this category in a customized agenda view which sorts the WAITs according to their priority. If necessary, I take action (for example resending an E-Mail).\n\nOne nuisance is that I use Outlook for my E-Mails and Org-Mode for my tasks.If I send an E-Mail and are waiting for an answer, or if I receive an E-Mail with information on my order, I can\u2019t generate a direct link between the E-Mail and the WAIT-task and I loose time looking for the original E-Mail.\n\nI googled around and found the following solution by John Hilliard here (there is org-outlook, but I did not find a good explanation, how that works). He suggests to write a macro in Outlook that looks like this\n\nThe macro gets the unique id for the email message and writes an Org Mode link to that message to the clip board.\n\nNote that in order to write a macro, you have to activate the \u201cDeveloper\u201d tab in Outook,\n\nwhich can be done by right-clicking on the tabs and choosing \u201cCustomizing the Ribbon\u201d. After defining the macro, you can add it to the quick access toolbar (same procedure as adding the developer tab), and choose a nice icon\n\nThere might be an error, when running the macro, but you John nicely describes how to get rid of that one:\n\n\u201cWhen I first ran the macro, I actually got an error saying \u201cCompile error: User-defined type not defined.\u201d That was a little mysterious. In order to fix that error:\n\nClick \u201cTools\u201d then \u201cReferences\u201d in the menu.\n\nCheck \u201cMicrosoft Forms 2.0 Object Library\u201d in the list of available references.\n\n\u2022 This option wasn\u2019t available to me at first so I clicked \u201cBrowse\u201d\n\u2022 Then I opened \u201cc:\\windows\\system32\\FM20.DLL\u201d\n\nThen I clicked \u201cOK\u201d\n\nI finally added the code for opening Outlook in my .emacs file.\n\n(defcustom org-outlook-location (w32-short-file-name \u201cc:\/Program Files\/Microsoft Office 15\/root\/office15\/OUTLOOK.exe\u201d)\n\u201c* Microsoft Outlook 2013 location.\u201d\n:type \u2018string\n:group \u2018org-outlook)\n\nNow, I can select a message, click on the macro symbol which copies the link, jump to Emacs (I have a script for that) and insert the link. The WAIIT-Task would look like this:\n\nIf I click on the Message-link, Outlook opens the message.\n\n# Using Preview for LaTeX documents in Emacs\n\nEditing LaTeX documents with lots of equations can be sometimes hard if you want to refer to the equations in the text. You can use, if you have a big monitor, put the compiled file to the right of your editor, but on a notebook this is not a good option. If you use AucTex in Emacs, you can use the preview mode.\n\nHere is the text without preview in Emacs And here it is with preview.\n\nYou can set the display options in the Option Group \u201cAucTex\u201d under \u201cPreview Appearance\u201d. One more thing: I always compile to pdf (not dvi). In order to generate the preview, you have to set this option back (C-c C-t C-p).","date":"2019-06-16 11:23:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6208057403564453, \"perplexity\": 2503.1678227569046}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998100.52\/warc\/CC-MAIN-20190616102719-20190616124719-00556.warc.gz\"}"} | null | null |
Vice President – Healthcare Division Message
Groups Vision
Thumbay Research Institute for Precision Medicine (TRIPM)
Leading the Innovation
The newly created Thumbay Research Institute for Precision Medicine will establish an interdisciplinary basic and translational cancer and diabetes research program to meet growing challenges facing UAE health care providers dealing with the increase in cancer burden and diabetes disease. It will focus its activities on comprehensive, translational research and personalised medicine, aims to become a centre of excellence and a leader in developing personalised medicine in the region. The centre will concentrate its activities on the development of state-of-the-art research technology platforms (liquid biopsy, NGS, functional genomics, imaging, gene editing) fundamental to position the centre in the international scene research field and as a regional leader in personalized cancer immunotherapy. Our ultimate goal is to position GMU among the leading research universities and a national model for medical education in the region by integrating innovation through research.
The relevance of the Centre for the University and the Hospital
The TRIPM will serve as a bridge linking scientists, faculty members and clinicians who will work together in a synergistic manner in the frame of a well-focused program in order to reach scientific excellence in the research domain. We are building the most favorable environment for investigators and teams at the level of science, structure and equipment in order to make the TRIPM the most suitable environment for cancer research and diagnosis and a challenging initiative for seamless collaboration between research, education, healthcare and industry.
Thumbay Institute of Population Health
The Institute of Population Health focuses on postgraduate studies and research in the fields of public health, Epidemiology, Evidence Based Medicine, Big Data Analysis, Evidence Based Medicine & Policy and Global Health.
Thumbay Institute Of Health Workforce Development
The Institute of Health Workforce Development builds on and expand the existing "Center for Advanced Simulation in Healthcare" (CASH) and the "Center for Continuing Education and Community Outreach" (CCE&CO) and the new "Center for Health Professions Education and Research" (CHPER). Its main goals is to respond to national, regional and international shortage and need for competent health workforce, "Doctors, Pharmacist, Dentist, Nurses, Paramedics and All Allied Health". In addition to Postgraduate programs, the institute will offer in collaboration with the collage of Medicine, 'Health Proffessions Education Programs', Master Diploma and Certificate and conduct research in the field of medical education, simulation and training.
BA Center for Online Health Professions Education and Training
Established with the aim of extending world-class health education to health professionals, individuals and communities worldwide, the center designs and delivers online professional education through a range of certificate programs.
Thumbay Medicity, Al Jurf, Ajman
info@thumbaymedicity.com
© Thumbay Medicity | Thumbay Group | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,922 |
Q: Is there a better way to populate userform comboboxes with the data from the active row when it initializes? Excel 2013 The code below works PERFECTLY in Excel 2010, but for some reason hardly ever works right in Excel 2013.
The point is that when a cell is double clicked (code not here, works fine in both 2010 and 2013 tho), a userform will pop up with the data of the "active row." When I do this in excel 2013, the userform will pop up, but it will either have data from a different row or all comboboxes will be blank. There are no debugging issues or notifications.
Here is what works in 2010 and not in 2013:
Private Sub UserForm_Initialize()
Call UnProtect
Dim r As Integer
r = ActiveCell.Row
StatusBox.Clear
With StatusBox
.AddItem "New"
.AddItem "In Process"
.AddItem "Waiting on Material/Parts"
.AddItem "Re-Assigned"
.AddItem "Complete"
End With
'When the userform pops up, it automatically has the
'data from the row that has been double clicked
LocationValue.Value = Open_Orders.Cells(r, 2).Value
AssetValue.Value = Open_Orders.Cells(r, 3).Value
DescriptionValue.Value = Open_Orders.Cells(r, 10).Value
CommentBox.Value = Open_Orders.Cells(r, 11).Value
StatusBox.Value = Open_Orders.Cells(r, 8).Value
'LocationValue, AssetValue, DescriptionValue, CommentBox, and StatusBox are all
'either a combobox or textbox
Call Protect
End Sub
I have only been using vba for a short time, so I'm sure this is the completely wrong way to do it. Is there a better way?
EDIT: Here is the code that is placed in the sheet called Open_Orders:
Private Sub Worksheet_BeforeDoubleClick(ByVal Target As Range, Cancel As Boolean)
Cancel = True
Sheets("Open_Orders").Protect Password:="password", userinterfaceonly:=True
Dim TheRow As Integer
TheRow = LastRow(ActiveSheet.Range("A1:K1"))
'^Function found in "LastRowFunction" module
'Determines the last row that contains data in it
Set rng = Range("B2:K" & TheRow)
'If you click a cell with data in it then show the userform
If Not Intersect(Target, rng) Is Nothing Then Load UserComment
UserComment.Show
Sheets("Open_Orders").Activate
End Sub
UserComment is the userform that pops up, and is supposed to have the data from the active row.
A: When you write:
Call UnProtect
Dim r As Integer
r = ActiveCell.Row
StatusBox.Clear
Your UnProtect Macro will probably return a row that is not the one you want when you write:
If Not Intersect(Target, rng) Is Nothing Then Load UserComment
UserComment.Show
Sheets("Open_Orders").Activate
I also find these very lines somehow not very clear, as the UserComment.Show will work anyway since you didn't specify and End Ifthere. I'd write:
If Not Intersect(Target, rng) Is Nothing Then
UserComment.Show
Sheets("Open_Orders").Activate
End If
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 965 |
\section{Introduction}
The axial charges $g_A$ of baryon states are essential quantities for the understanding
of both the electroweak and strong interactions within the Standard Model of
elementary-particle physics. They do not only govern weak processes, such as the
$\beta$ decay, but also intertwine the weak and strong interactions. This is most
clearly reflected by the Goldberger-Treiman relation, which in case of the $N$ reads
$g_A=f_{\pi}g_{\pi NN}/M_N$~\cite{Goldberger:1958tr}. Given the $\pi$ decay constant
$f_{\pi}$ and the nucleon mass $M_N$, the $\pi NN$ coupling constant $g_{\pi NN}$
just turns out to be proportional to $g_A$. Thus the relevance of $\pi$ degrees of
freedom in (low- and intermediate-energy) hadronic physics is intimately tied to the axial
charges: Whenever $g_A$ becomes sizable, $\pi$ degrees of freedom should matter sensibly.
Therefore $g_A$ can also be viewed as an indicator of the phenomenon
of spontaneous breaking of chiral symmetry (SB$\chi$S) of non-perturbative
quantum chromodynamics (QCD),
which is manifested by the non-vanishing value of the light-flavor chiral
condensate $\left<0|q\bar q|0\right>^{1/3}\approx -235$ MeV. The axial charges
thus constitute important parameters for low-energy effective theories. Any
reasonable model of non-perturbative QCD should yield the $g_A$ of correct sizes.
In fact, the axial charges may be considered as benchmark observables
for the nucleon, and more comprehensively the baryon, structures.
Best known is, of course, the axial charge of the $N$, as its experimental value
can be deduced from the ratio of the axial to the vector coupling
constants $g_A/g_V=1.2695\pm0.0029$~\cite{Amsler:2008zzb}; usually this is done under
the assumption of conserved vector currents (CVC), which implies $g_V=1$. The deviation of
$g_A$ from 1, the axial charge of a point-like particle, can be attributed,
according to the Adler-Weisberger sum rule~\cite{Adler:1965ka,Weisberger:1965hp}, to the
differences between the $\pi^+ N$ and $\pi^- N$ cross sections in pion-nucleon scattering.
Unfortunately, axial charges of other baryon (ground) states are not known directly
from experiment.
The axial charges are also 'measured' in lattice QCD. An increasing number of results
has recently become available, even from full-QCD lattice calculations. A recent summary
of the lattice-QCD results for the $g_A$ of the nucleon is presented in
ref.~\cite{Renner_2009}. The axial charges of hyperons have been studied by Lin et
al.~\cite{Lin:2007ap} as well as Erkol et al.~\cite{Erkol:2009ev} and Engel et
al.~\cite{Engel:2009nh} in (2+1)- and
2-flavor lattice QCD, respectively. There have also been a number of other attempts
to explain the axial charges of the $N$ and the other baryons. We mention only the
more modern ones through chiral perturbation theory
($\chi$PT) (see the recent review by Bernard~\cite{Bernard:2007zu} or, for example,
ref.~\cite{Jiang:2009sf}), chiral unitary
approaches~\cite{Nacher:1999vg}, and relativistic constituent quark models
(RCQM)~\cite{Glozman:2001zc,Boffi:2001zb,Merten:2002nz}.
Recently, also the axial charges of the $N$ resonances have come into the focus of
interest, because of the question of restoration of chiral symmetry higher in the
baryon (as well as meson) spectra. Specifically, it has been argued that the magnitudes
of $g_A$ should become small for almost degenerate parity-partner $N$ resonances,
indicating the onset of chiral-symmetry restoration with higher excitation
energies~\cite{Glozman:2007ek,Glozman:2008vg}. As the $g_A$ values of $N$
resonances are not known from phenomenology and can hardly be measured experimentally,
this remains a highly theoretical question. However, the problem can be explored
with the use of lattice QCD. Corresponding first results have already become
available, but only for two of the $N$ resonances, namely, $N$(1535) and
$N$(1650)~\cite{Takahashi:2008fy}. Both of them have spin $J=\frac{1}{2}$ and
parity $P=-1$. Since there is not yet any lattice-QCD result for
positive-parity states, the above issue relating to parity-doubling remains
unresolved from this side.
The problem of $g_A$ of the $N$ resonances has most recently also been studied
within the RCQM~\cite{Choi2010}. The axial charges of all the $N$ resonances
up to $\sim$1.9 GeV and $J^P=\frac{1}{2}^\pm, \frac{3}{2}^\pm, \frac{5}{2}^\pm$
have been calculated with $N$ resonance wave functions stemming from realistic RCQM
with Goldstone-boson-exchange (GBE) as well as one-gluon-exchange (OGE) hyperfine
interactions. One has found the remarkable result that, especially in case of the
GBE RCQM, the magnitudes of the axial charges need not be small, even if the
energy levels of the opposite-parity partners become (almost) degenerate at
increased excitation energies, e.g. the $J^P=\frac{5}{2}^\pm$ resonances
$N$(1680) and $N$(1675). Thus the issue of possible chiral-restoration phenomena
reflected by the axial charges remains tantalizing until further insights become
available (e.g. from lattice QCD or alternative attempts).
Another question related to the axial charges of the $N$ resonances concerns the
role of $\{QQQQ\bar Q\}$ components. It has been argued that sizable admixtures of
$\{QQQQ\bar Q\}$ are needed in order to reproduce an almost vanishing $g_A$ of the
$N$(1535) resonance~\cite{An:2008tz,Yuan:2009st}. However, these results are usually
obtained in a simplistic non-relativistic approach. Meanwhile it is known that a
RCQM with realistic $\{QQQ\}$ wave functions can easily explain a practically
vanishing $g_A$ of $N$(1535)~\cite{Choi2010}, in perfect congruency with the
predictions obtained from lattice QCD~\cite{Takahashi:2008fy}, and there is no
need for considerable $\{QQQQ\bar Q\}$ admixtures in this case. Moreover, the
correct sizes of the axial charges of the $N$ ground state and the $N$(1535)
as well as $N$(1650) resonances can simultaneously and consistently be reproduced
within a RCQM with only $\{QQQ\}$ configurations~\cite{Choi2010}.
In the context of hyperons the axial charges are also important to learn about the role
of $SU(6)$ flavor-symmetry breaking. In particular, in the case of conserved $SU(3)_F$
the axial charges of the $N$, $\Sigma$, and $\Xi$ ground states are connected by the
following simple relations~\cite{Gaillard:1984ny,Dannbom:1996sh}
\begin{equation}
g_A^N=F+D\,, \hspace{3mm} g_A^{\Sigma}=\sqrt{2}F\,, \hspace{3mm} g_A^{\Xi}=F-D\,,
\label{axcharges}
\end{equation}
which follow through $SU(3)$ Clebsch-Gordan coefficients in the decomposition of the
axial form factor into the functions $F$ and $D$ relating to the octet components in
$SU(3)$~\cite{de Swart:1963gc}. Note that we adopt the convention of $g_A/g_V$ being
positive for the $N$ (like in ref.~\cite{Gaillard:1984ny}) contrary to the
PDG~\cite{Amsler:2008zzb}; this then determines also the signs of all other baryon
axial charges according to Eq.~(\ref{axcharges}).
In the present paper we present results from a comprehensive study of the axial charges
of octet and decuplet ground states $N$, $\Sigma$, $\Xi$, $\Delta$, $\Sigma^*$,
and $\Xi^*$ as well as their resonances along RCQMs. In particular, we employ the RCQMs
whose quark-quark hyperfine interactions derive from OGE~\cite{Theussl:2000sj} and
GBE dynamics~\cite{Glozman:1997fs}; in the latter case we consider both the
version with only the spin-spin interaction from pseudoscalar exchange
(psGBE)~\cite{Glozman:1997ag} as well as the extended version that includes all
force components (i.e. central, tensor, spin-spin, and spin-orbit) from pseudoscalar,
scalar, and vector exchanges (EGBE)~\cite{Glantschnig:2004mu}.
The calculations are performed in the framework of Poincar\`e-invariant quantum mechanics.
In order to keep the numerical computations manageable, we have to restrict the axial
current operator to the so-called spectator model (SM). It means that the weak-interaction
gauge boson couples only to one of the constituent quarks in the baryon. This approximation
has turned out to be very reasonable already in a previous study of the axial and induced
pseudoscalar form factors of the nucleon~\cite{Glozman:2001zc}, where the SM was employed
specifically in the point form (PF) of relativistic quantum mechanics~\cite{Melde:2004qu}.
It has also been used in studies of the electromagnetic structure of the $N$,
reproducing both the proton and neutron form factors in close agreement with the
experimental data~\cite{Boffi:2001zb,Wagenbrunn:2000es,Berger:2004yi,Melde:2007zz}.
In the following chapter we explain the formalism for the calculation of the matrix
elements of the axial current operator and give the definition of the axial charges
for the different baryon ground and resonant states. Subsequently we
present the results and compare them to experimental data as well as to results from
other approaches, notably from lattice QCD and $\chi PT$. In the final chapter
we draw our conclusions.
\section{Formalism}
In hadronic physics the (diagonal) baryon axial charges $g_A^B$ govern such processes
like $n\rightarrow pe^-\bar \nu_e$, $\Sigma^-\rightarrow \Sigma^0 e^-\bar \nu_e$,
$\Xi^-\rightarrow \Xi^0 e^-\bar \nu_e$ etc. They can generally be calculated through
semileptonic decays $B_1 \rightarrow B_2 \ell\bar \nu$ with strangeness change
$\Delta S=0$. The axial charge is conveniently defined through the value of the axial
form factor $G_A(Q^2)$ at $Q^2=0$, where $Q^2=-q^2$ is the four-momentum transfer.
The axial form factor $G_A(Q^2)$ can be deduced from the relativistically invariant
matrix element of the axial current operator $\hat A^\mu_+ (Q^2)$ sandwiched
between the eigenstates of baryons $B_1$ and $B_2$. Here, the subscript $+$ refers
to the isospin-raising ladder operator
$\tau_+ = \frac{1}{2}\left(\tau_1 + i\tau_2\right)$, with $\tau_i$ being
the usual Pauli matrices. In the specific case of the neutron $\beta$ decay the matrix
element of $\hat A^\mu_+(Q^2=0)$ reads
\begin{equation}
\left<p\left|\hat A^\mu_+\right|n\right>=
g_A^N \bar U_p(P,J'_3)\gamma^\mu \gamma_5 \frac{\tau_+}{2}
U_n(P,J_3) \,,
\label{n-p}
\end{equation}
where $U_n$ and $U_p$ are the neutron and proton spinors, depending on the
four-momentum $P$ and helicities $J_3$ and $J'_3$, respectively; $\gamma^\mu$
and $\gamma_5$ are the usual Dirac matrices. Alternatively, the matrix element in
Eq.~(\ref{n-p}) can also be expressed as
\begin{equation}
\left<p\left|\hat A^\mu_3\right|p\right>=
g_A^N \bar U_p(P,J'_3)\gamma^\mu \gamma_5 \frac{\tau_3}{2}
U_p(P,J_3)
\label{p-p}
\end{equation}
or
\begin{equation}
\left<n\left|\hat A^\mu_3\right|n\right>=
g_A^N \bar U_n(P,J'_3)\gamma^\mu \gamma_5 \frac{\tau_3}{2}
U_n(P,J_3) \,.
\label{n-n}
\end{equation}
In the spirit of the latter relations we may express the axial charge $g_A^B$ of any
baryon $B=N, \Delta, \Sigma, \Xi, ...$ and its resonances more generally. Let us denote
the baryon states by $\left|B;P,J,J_3\right>$, i.e. as simultaneous
eigenstates of the four-momentum operator $\hat P^\mu$, the intrinsic-spin operator
$\hat J$ and its $z$-projection $\hat J_3$. Since $\hat P^\mu$ and the invariant mass
operator $\hat M$ commute, these eigenstates can be obtained by solving the eigenvalue
equation of $\hat M$
\begin{equation}
\hat M \left|B;P,J,J_3\right>=M \left|B;P,J,J_3\right> \, ,
\end{equation}
Then the axial charge $g_A^B$ of any baryon state $B$ with
$J=\frac{1}{2}, \frac{3}{2}, \frac{5}{2}$ is given by the matrix elements of the axial
current operator $\hat A_3^{\mu}$ for zero momentum transfer $Q^2$ as
\begin{eqnarray}
&&\left<B;P,\frac{1}{2},J_3'\left|{\hat A}^{\mu}_3\right|B;P,\frac{1}{2},J_3\right>=\nonumber\\
&&\hspace{1.1cm}C_B\bar U_B(P,J_3')g_A^B \gamma^{\mu}\gamma_5 \frac{\tau_3}{2} U_B(P,J_3) \, ,\nonumber\\
&&\left<B;P,\frac{3}{2},J_3'\left|{\hat A}^{\mu}_3\right|B;P,\frac{3}{2},J_3\right>=\nonumber\\
&&\hspace{1.1cm}C_B\bar U_B^\nu(P,J_3')g_A^B \gamma^{\mu}\gamma_5 \frac{\tau_3}{2}
U_{B;\nu}(P,J_3) \, ,\nonumber\\
&&\left<B;P,\frac{5}{2},J_3'\left|{\hat A}^{\mu}_3\right|B;P,\frac{5}{2},J_3\right>=\nonumber\\
&&\hspace{1.1cm}C_B\bar U_B^{\nu\lambda}(P,J_3')g_A^B \gamma^{\mu}\gamma_5 \frac{\tau_3}{2}
U_{B;\nu\lambda}(P,J_3) \, ,
\end{eqnarray}
where the coefficients $C_B$ are specified by
\begin{equation}
C_N=2C_{\Delta}=\frac{1}{\sqrt{2}}C_{\Sigma}=C_{\Xi}=1 \,.
\end{equation}
Here, $U_B(P,J_3)$ are the usual Dirac spinors for $J=\frac{1}{2}$ baryons and
$U_{B;\nu}(P,J_3)$ as well as $U_{B;\nu\lambda}(P,J_3)$ are the Rarita-Schwinger
spinors~\cite{Rarita:1941mf} for $J=\frac{3}{2}$ and $J=\frac{5}{2}$ baryons,
respectively, with the normalizations as given in Appendix A.
Omitting from now on the denotation after $B$ we can write the matrix elements of
$\hat A^\mu_3$ for any ground and resonance states as
\begin{eqnarray}
&&\left<P,J,J_3'\right|{\hat A^\mu_3 (Q^2=0)}\left|P,J,J_3\right>=
\nonumber \\
&&
2M\sum_{\sigma_i\sigma'_i}{\int{
d^3{\vec k}_1 d^3{\vec k}_2 d^3{\vec k}_3}}
\frac{\delta^3(\vec k_1+\vec k_2+\vec k_3)}{2\omega_1 2\omega_2 2\omega_3} \nonumber \\
&&
\times\Psi^\star_{PJJ_3'}\left({\vec k}_1,{\vec k}_2,{\vec k}_3;
\sigma'_1,\sigma'_2,\sigma'_3\right) \nonumber \\
&&
\times\left<k_1,k_2,k_3;\sigma'_1,\sigma'_2,\sigma'_3\right|\hat{A}^{\mu}_3
\left|k_1,k_2,k_3;\sigma_1,\sigma_2,\sigma_3\right>
\nonumber\\
&&
\times\Psi_{PJJ_3}\left({\vec k}_1,{\vec k}_2,{\vec k}_3;
\sigma_1,\sigma_2,\sigma_3\right) \, .
\label{transampl}
\end{eqnarray}
The $\Psi$'s are the momentum-space representations of the baryon eigenstates for
$\vec P=0$, i.e. the rest-frame wave functions of the baryon ground and resonance states with corresponding mass $M$ and total angular momentum $J$ and $z$-projections $J_3$
as well as $J_3'$. Here they are expressed as functions of the individual quark
three-momenta $\vec k_i$, which sum up to $\vec P=\vec k_1+\vec k_2+\vec k_3=0$;
$\omega_i=\sqrt{m^2_i+\vec k^2_i}$ is the energy of quark $i$ with mass $m_i$, and the
individual-quark spin orientations are denoted by $\sigma_i$.
The integral on the r.h.s. of Eq.~(\ref{transampl}) is evaluated along the SM what
amounts to the matrix element of the axial current operator
$\hat{A}^{\mu}_a$ between (free) three-particle states
$\left|k_1,k_2,k_3;\sigma_1,\sigma_2,\sigma_3\right>$ to be assumed in the form
\begin{multline}
\left<k_1,k_2,k_3;\sigma'_1,\sigma'_2,\sigma'_3\right|
{\hat A}^{\mu}_3
\left|k_1,k_2,k_3;\sigma_1,\sigma_2,\sigma_3\right>
= \\
3\left<k_1,\sigma'_1\right|\hat{A}^{\mu}_{3,{\rm SM}}
\left|k_1,\sigma_1\right>2\omega_2 2\omega_3
\delta_{\sigma_{2}\sigma'_{2}}\delta_{\sigma_{3}\sigma'_{3}}
\label{eq:axcurr1}\, .
\end{multline}
For point-like quarks this matrix element involves the axial current operator
of the active quark 1 (with quarks 2 and 3 being the spectators) in the form
\begin{equation}
\left<k_1,\sigma'_1\right|\hat{A}^{\mu}_{3,{\rm SM}}
\left|k_1,\sigma_1\right>=
{\bar u}\left(k_1,\sigma'_1\right)g_A^q \gamma^\mu
\gamma_5 \frac{{\tau}_3}{2} u\left(k_1,\sigma_1\right) \, ,
\label{eq:axcurr2}
\end{equation}
where $u\left(k_1,\sigma_1\right)$ is the spinor of a quark with flavor $u$ or $d$
and $g_A^q=1$ its axial charge. A pseudovector current analogous to the one in
Eq.~(\ref{eq:axcurr2}) was recently
also used in the calculation of $g_{\pi NN}$ and the strong $\pi NN$ vertex form factor
in ref.~\cite{Melde:2008dg}.
For the calculation of the axial charges $g_A$ we can use either one of the components
$\mu=i=1, 2, 3$ of the axial current operator $\hat{A}^{\mu}_{3,{\rm SM}}$ in
Eq.~(\ref{eq:axcurr2}). The expression on the r.h.s. then specifies to
\begin{eqnarray}
&&{\bar u}\left(k_1,\sigma'_1\right) \gamma^i\gamma_5 \frac{{\tau}_3}{2} u\left( k_1,\sigma_1\right)=\nonumber\\
&&
2\omega_1\chi^*_{\frac{1}{2},\sigma'_1}
\Biggl\{\left[1-\frac{2}{3}\left(1-\kappa\right) \right]\sigma^i \nonumber\\
&&
+\sqrt{\frac{5}{3}}\frac{\kappa^2}{1+\kappa}
\left[\,\left[\vec{v}_{1}\otimes\vec{v}_{1}\right]_2\otimes\vec{\sigma}\right]_1^i\Biggl\}
\frac{{\tau}_3}{2} \chi_{\frac{1}{2},\sigma_1}\,,
\label{eq:ga}
\end{eqnarray}
where $\kappa=1/\sqrt{1+v_1^2}$ and $\vec v_1=\vec k_1/m_1$.
Herein $\sigma^i$ is the $i$-th component of the usual Pauli matrix
$\vec \sigma$ and $v_1$ the magnitude of the three-velocity $\vec v_1$. The symbol
$\left[.\otimes .\right]_k^i$ denotes the $i$-th component of a tensor product
$\left[.\otimes .\right]_k$ of rank $k$. We note that a similar formula was already
published before by Dannbom et al.~\cite{Dannbom:1996sh}, however, restricted to the
case of total orbital angular momentum $L=0$. Our expression holds for any $L$, thus
allowing to calculate $g_A$ for the most general wave function of a baryon ground or
resonances state specified by intrinsic spin and parity $J^P$.
\section{Results}
\renewcommand{\arraystretch}{1.4}
\begin{table*}
\centering
\caption{Axial charges $g_A^B$ of octet and decuplet ground states as predicted by the
EGBE~\cite{Glantschnig:2004mu},
psGBE~\cite{Glozman:1997ag}, and OGE~\cite{Theussl:2000sj} RCQMs in comparison to
experiment~\cite{Amsler:2008zzb} and lattice QCD results from Lin and Orginos
(LO)~\cite{Lin:2007ap} and
Erkol, Oka, and Takahashi (EOT)~\cite{Erkol:2009ev} as well as $\chi$PT results from
Jiang and Tiburzi (JT)~\cite{Jiang:2008we,Jiang:2009sf}; also given is the
nonrelativistic limit (NR) from the EGBE RCQM.}
\begin{tabular}{ccccccccc}
\hline
\hline
& Exp & EGBE & psGBE & OGE &LO&EOT&JT& NR\tabularnewline
\hline
N & 1.2695$\pm$0.0029 & 1.15 & 1.15 & 1.11 & 1.18$\pm$0.10&1.314$\pm$0.024&1.18& 1.65\tabularnewline
$\Sigma$ & - & 0.65 & 0.65 & 0.65 & 0.636$\pm$0.068$^\dagger$
&0.686$\pm$0.021$^\dagger$&0.73& 0.93\tabularnewline
$\Xi$ & - & -0.21 & -0.22 & -0.22 & -0.277$\pm$0.034&-0.299$\pm$0.014$^\ddagger$&-0.23$^\ddagger$& -0.32\tabularnewline
\hline
$\Delta$ & - & -4.48 & -4.47 & -4.30 &-&-&$\sim\,$-4.5& -6.00 \tabularnewline
$\Sigma^{*}$ & - & -1.06 & -1.06 & -1.00 & -&-&-& -1.41\tabularnewline
$\Xi^{*}$ & - & -0.75 & -0.75 & -0.70 & -&-&-& -1.00\tabularnewline
\hline
\hline
\end{tabular}
\\
\begin{flushleft}
$^\dagger$ Due to another definition of $g_A^{\Sigma}$ this numerical value is
different by a $\sqrt{2}$ from the one quoted in the original paper.\\
$^\ddagger$ Due to another definition of $g_A^{\Xi}$ this value has a sign opposite
to the one in the original paper.
\end{flushleft}
\label{EGBE}
\end{table*}
In Table~\ref{EGBE} we present the RCQM results from our calculations for the axial
charges $g_A^B$ of the octet and decuplet ground states
$B=N$, $\Sigma$, $\Xi$, $\Delta$, $\Sigma^*$, and $\Xi^*$. Except for the $N$
there are no direct
experimental data for $g_A$ (from $\Delta S=0$ decays). The predictions for $g_A^N$
by all three RCQMs come close to the experimental value, with all of them falling slightly
below it. This is also the trend of present-day lattice-QCD calculations~\cite{Renner_2009};
only the EOT result seems to represent a notable exception, even if we take the
theoretical uncertainties into account (in Table~\ref{EGBE} we have chosen to quote the
EOT result corresponding to their calculation with the smallest quark mass of 35 MeV).
In addition, also the
JT prediction obtained from $\chi$PT remains below the experimental value. There might be
a variety of reasons why the different approaches underestimate the $g_A^N$. However,
one should also bear in mind that the phenomenological value of $g_A^N\sim\,$1.27 is
supposed under the conjecture of conserved vector currents. What concerns the RCQMs,
and in particular the {\mbox psGBE} RCQM, interestingly, it has recently been
found~\cite{Melde:2004qu}
that also the $\pi NN$ coupling constant turns out to be too small, namely
$\frac{f^2_{\pi NN}}{4\pi}=$ 0.0691, as compared to the phenomenological value of
about 0.075~\cite{Bugg:2004cm}. It remains to be clarified if in case of the
RCQMs these undershootings of both the $g_A^N$ and $f^2_{\pi NN}$, which are related
by the Goldberger-Treiman relation, have to be interpreted as lacking $\pi-$dressing
effects.
In the last column of Table~\ref{EGBE} we quote also the nonrelativistic limit of the
prediction by the EGBE RCQM (i.e. for the limit $\kappa \rightarrow$ 1 in Eq. (\ref{eq:ga})).
It deviates grossly from the relativistic result, indicating that a consideration of
axial charges within a nonrelativistic approach is unreliable. This conclusion is
further substantiated by considering the axial charges of $N$ resonances, for which
indeed no experimental data are available but lattice-QCD results have recently been
produced. While the relativistic predictions of especially the EGBE RCQM agree very
well with the lattice-QCD data in case of both the $N$(1535) and $N$(1650) resonances,
the nonrelativistic limits deviate here too~\cite{Choi2010}.
For the $g_A^B$ of the octet and decuplet ground states the RCQMs yield very similar
results. While the predictions of the psGBE and the EGBE are essentially the same,
differences occur only for the OGE RCQM, but they remain within at most $\sim\,$6\%.
In case of the octet states $\Sigma$ and $\Xi$ we can also compare to lattice-QCD
as well as $\chi$PT results. The comparison of the RCQM predictions to the former
is quite satisfying, as the figures agree rather well. Except for $g_A^{\Sigma}$
practically the same is true with regard to the $\chi$PT results of JT. Again,
the results from the nonrelativistic limit of the EGBE RCQM fall short; as in
the case of the $N$ the corresponding values are always bigger (in absolute value)
than all of the other results.
For the decuplet ground states $\Delta$, $\Sigma^*$, and $\Xi^*$
there are neither experimental data nor lattice-QCD
results. Only for the $\Delta$ we may compare with a $\chi$PT prediction, showing
again a striking similarity. For the other cases of $\Sigma^*$ and $\Xi^*$ we have
here produced first predictions and one has still to await results from other approaches.
\begin{table}[h]
\centering
\caption{Mass eigenvalues and axial charges $g_A^N$ of the $N$ ground state and the
low-lying $N$ resonances as predicted by the EGBE, the psGBE, and the OGE RCQMs.}
\begin{tabular}{lcp{3mm}crp{3mm}crp{3mm}cr}
\hline\hline \\[-3mm]
\multicolumn{2}{c}{} && \multicolumn{2}{c}{EGBE} && \multicolumn{2}{c}{psGBE} && \multicolumn{2}{c}{OGE}\tabularnewline
\hline
State & $J^{p}$ && Mass & $g_{A}$ && Mass & $g_{A}$ && Mass & $g_{A}$\tabularnewline
\hline\\[-2mm]
$N$(939) & $\frac{1}{2}^{+}$ && 939 & 1.15 && 939& 1.15 && 939 & 1.11\tabularnewline
$N$(1440) & $\frac{1}{2}^{+}$ && 1464 & 1.16 && 1459 & 1.13 && 1578 & 1.10\tabularnewline
$N$(1520) & $\frac{3}{2}^{-}$ && 1524 & -0.64 && 1519 & -0.21 && 1520 & -0.15\tabularnewline
$N$(1535) & $\frac{1}{2}^{-}$ && 1498 & 0.02 && 1519 & 0.09 && 1520 & 0.13\tabularnewline
$N$(1650) & $\frac{1}{2}^{-}$ && 1581 & 0.51 && 1647 & 0.46 && 1690 & 0.44\tabularnewline
$N$(1700) & $\frac{3}{2}^{-}$ && 1608 & -0.10 && 1647 & -0.50 && 1690 & -0.47\tabularnewline
$N$(1710) & $\frac{1}{2}^{+}$ && 1757 & 0.35 && 1776 & 0.37 && 1860 & 0.32\tabularnewline
$N$(1720) & $\frac{3}{2}^{+}$ && 1746 & 0.35 && 1728 & 0.34 && 1858 & 0.25\tabularnewline
$N$(1675) & $\frac{5}{2}^{-}$ && 1676 & 0.84 && 1647 & 0.83 && 1690 & 0.80\tabularnewline
$N$(1680) & $\frac{5}{2}^{+}$ && 1689 & 0.89 && 1728& 0.83 && 1858 & 0.70\tabularnewline
\hline
\hline
\end{tabular}
\label{N}
\end{table}
Next we come to discuss the axial charges of nucleon and other baryon resonances.
As mentioned in the Introduction, especially the nucleon resonances have recently
attracted interest, mainly because the issue of chiral-symmetry restoration
higher in the baryon spectra has been raised~\cite{Glozman:2007ek,Glozman:2008vg}
and because first lattice-QCD calculations have been performed~\cite{Takahashi:2008fy}.
Certainly the results of the latter have still to be taken with care, as they
correspond to relatively high quark masses. For the case of the nucleon we have
presented resonance axial charges from RCQMs already
in a previous paper~\cite{Choi2010}; for completeness we repeat them here in
Table~\ref{N}. While for details we refer to ref.~\cite{Choi2010}, we mention
as the main characterization of these results that \\[2mm]
$i$) the RCQM predictions perfectly agree with the lattice-QCD results
for the $N$(1535) and $N$(1650) resonances, i.e. in the two cases for which
lattice-QCD calculations have so far become available, \\[2mm]
$ii$) the small, practically vanishing, $g_A$ of $N$(1535) can be reproduced with
$\{QQQ\}$ configurations alone, \\[2mm]
$iii$) the predictions of different RCQMs are generally very similar except for the
$J^P=\frac{3}{2}^-$ resonances $N$(1520) and $N$(1700), \\[2mm]
$iv$) a relativistic description is necessary and a simple $SU(6)\times O(3)$
nonrelativistic quark model is not reliable, and \\[2mm]
$v$) there is no tendency of the axial charges of high-lying parity partners to
assume particularly small values.
\begin{table}[h]
\centering
\caption{Same as Table~\ref{N} but for the octet $\Sigma$ and decuplet $\Sigma^*$ states.}
\begin{tabular}{lcp{3mm}crp{3mm}crp{3mm}cr}
\hline\hline \\[-3mm]
\multicolumn{2}{c}{} && \multicolumn{2}{c}{EGBE} && \multicolumn{2}{c}{psGBE} && \multicolumn{2}{c}{OGE}\tabularnewline
\hline
State & $J^{p}$ && Mass & $g_{A}$ && Mass & $g_{A}$ && Mass & $g_{A}$\tabularnewline
\hline\\[-2mm]
$\Sigma$(1193) & $\frac{1}{2}^{+}$ && 1194 & 0.65 && 1182 & 0.65 && 1121 & 0.65\tabularnewline
$\Sigma$(1560) & $\frac{1}{2}^{-}$ && 1672 & -0.15 && 1678& -0.07&& 1655 & 0.01\tabularnewline
$\Sigma$(1620) & $\frac{1}{2}^{-}$ && 1740 & 0.62 && 1736& 0.58&& 1770 & 0.54\tabularnewline
$\Sigma$(1660) & $\frac{1}{2}^{+}$ && 1664 & 0.69 && 1619 & 0.64 && 1755 & 0.64 \tabularnewline
$\Sigma$(1670) & $\frac{3}{2}^{-}$ && 1681 & -0.92 && 1678& -0.48 && 1655 & -0.24 \tabularnewline
$\Sigma$(1775) & $\frac{5}{2}^{-}$ && 1765 & 1.06 && 1736& 1.03&& 1770 & 0.97 \tabularnewline
$\Sigma$(1880) & $\frac{1}{2}^{+}$ && 1903 & 0.38 && 1912 & 0.42 && 1980 & 0.17\tabularnewline
$\Sigma$(1940) & $\frac{3}{2}^{-}$ && 1725 & -0.45 && 1736& -0.83&& 1770 & -0.78 \tabularnewline
\hline
$\Sigma^*$(1385) & $\frac{3}{2}^{+}$ && 1365 & -1.06 && 1389& -1.06&& 1311 & -1.00\tabularnewline
$\Sigma^*$(1690) & $\frac{3}{2}^{+}$ && 1812 & -1.05 && 1865& -1.03&& 1932 & -0.99 \tabularnewline
$\Sigma^*$(1750) & $\frac{1}{2}^{-}$ && 1761 & -0.08 && 1759 & -0.13 && 1718 & -0.18\tabularnewline
\hline
\hline
\end{tabular}
\label{Sigma}
\end{table}
The RCQM predictions for the axial charges of the octet $\Sigma$ and decuplet
$\Sigma^*$ resonances are quoted in Table~\ref{Sigma}. The gross pattern of the
results is like the one of the $N$ resonances. First of all the different
{\mbox RCQMs} yield similar values for the axial charges except for the cases of
$\Sigma$(1670) and $\Sigma$(1940), which are again $J^P=\frac{3}{2}^-$ resonances
and are to be assigned to the same octets as $N$(1520) and $N$(1700), respectively,
according to a recent identification of baryon resonances~\cite{Melde:2008yr}
(see also~\cite{PDG2010}). For both $\Sigma$(1670) and $\Sigma$(1940) the differences
among the predictions prevail also in the comparison between the EGBE and the
psGBE RCQMs hinting to considerable influences from tensor and/or spin-orbit forces,
just as in the corresponding two $N$ resonances. All of the RCQMs
produce very small values for the axial charges of $\Sigma$(1560). Notably,
this state falls into the same octet as $N$(1535)~\cite{Melde:2008yr,PDG2010},
whose $g_A$ is also extremely small (see Table~\ref{N}). A similar small axial
charge is found for the decuplet $\Sigma^*$(1750). In general, however, we do not
observe the axial charges to become small as we go up to higher resonances.
The results for the axial charges of the octet $\Xi$ and decuplet $\Xi^*$ resonances
are collected in Table~\ref{Xi}. Here, all the RCQMs produce similar predictions,
where the axial charges of the octet resonances remain rather small with values
ranging from -0.2 to -0.4.
\begin{table}[h]
\centering
\caption{Same as Table~\ref{N} but for the octet $\Xi$ and decuplet $\Xi^*$ states..}
\begin{tabular}{lcp{3mm}crp{3mm}crp{3mm}cr}
\hline\hline \\[-3mm]
\multicolumn{2}{c}{} && \multicolumn{2}{c}{EGBE} && \multicolumn{2}{c}{psGBE} && \multicolumn{2}{c}{OGE}\tabularnewline
\hline
State & $J^{p}$ && Mass & $g_{A}$ && Mass & $g_{A}$ && Mass & $g_{A}$\tabularnewline
\hline\\[-2mm]
$\Xi$(1318) & $\frac{1}{2}^{+}$ && 1355 & -0.21 && 1348 & -0.22 && 1193 & -0.22\tabularnewline
$\Xi$(1690) & $\frac{1}{2}^{+}$ && 1813 & -0.23 && 1806 & -0.22 && 1826 & -0.22\tabularnewline
$\Xi$(1820) & $\frac{3}{2}^{-}$ && 1807 & -0.38 && 1792 & -0.40 && 1751 & -0.23 \tabularnewline
\hline
$\Xi^*$(1530) & $\frac{3}{2}^{+}$ && 1512 & -0.75 && 1528 & -0.75 && 1392 & -0.70 \tabularnewline
\hline
\hline
\end{tabular}
\label{Xi}
\end{table}
Finally, in Table~\ref{Delta} the axial charges of the $\Delta$ resonances are given.
Again, all of the RCQMs yield similar predictions. Only it is remarkable that the
axial charges especially of the $J^P=\frac{3}{2}^+$ states are rather big in
absolute value. If we consider the $\Delta$(1232) ground state, its $g_A$ is at
least three times larger than the one of the $N$ ground state. Remarkably a ratio
of about the same size has recently been found between the $\pi N\Delta$ and the
$\pi NN$ strong coupling constants~\cite{Melde:2008dg}. The smallest $g_A$ is
found for $\Delta$(1620). It should be noted that it falls into the same decuplet
as the $\Sigma^*$(1750), whose $g_A$ was also seen as the smallest among the
$\Sigma^*$ resonances (cf. Table~\ref{Sigma}).
\begin{table}[h]
\centering
\caption{Same as Table~\ref{N} but for the $\Delta$ states.}
\begin{tabular}{lcp{3mm}crp{3mm}crp{3mm}cr}
\hline\hline \\[-3mm]
\multicolumn{2}{c}{} && \multicolumn{2}{c}{EGBE} && \multicolumn{2}{c}{psGBE} && \multicolumn{2}{c}{OGE}\tabularnewline
\hline
State & $J^{p}$ && Mass & $g_{A}$ && Mass & $g_{A}$ && Mass & $g_{A}$\tabularnewline
\hline\\[-2mm]
$\Delta$(1232) & $\frac{3}{2}^{+}$ && 1231 & -4.48 && 1240 & -4.47 && 1231 & -4.30\tabularnewline
$\Delta$(1600) & $\frac{3}{2}^{+}$ && 1686 & -4.41 && 1718 & -4.33 && 1855 & -4.20 \tabularnewline
$\Delta$(1620) & $\frac{1}{2}^{-}$ && 1640 & -0.76 && 1642 & -0.75 && 1621 & -0.74\tabularnewline
$\Delta$(1700) & $\frac{3}{2}^{-}$ && 1639 & -1.68 && 1642 & -1.66 && 1621 & -1.58\tabularnewline
\hline
\hline
\end{tabular}
\label{Delta}
\end{table}
\section{Conclusions}
We have presented results from a comprehensive and consistent study of axial charges
$g_A^B$ of octet and decuplet baryon ground and resonant states with RCQMs. The
dynamical models differ mainly with regard to their hyperfine $Q$-$Q$ interactions,
which stem from OGE and psGBE as well as EGBE. Whenever a comparison is possible
with either experimental data or established theoretical results (especially from
lattice QCD and $\chi$PT), the RCQM predictions turn out to be quite reasonable.
The values deduced from a nonrelativistic approximation in general differ grossly,
indicating that a relativistic approach to the axial charges is mandatory. The RCQMs
considered here rely on $\{QQQ\}$ configurations only. Nevertheless, the $g_A^B$
results never fall short but rather produce a consistent picture. Already for the
ground states one finds a scatter of $g_A^B$ values from small to large. Through
the Goldberger-Treiman relation one may thus expect smaller or larger $\pi$-dressing
effects depending on the baryon state. Particularly big are the axial charges of the
$\delta$ ground and first excited states, much in congruency with the relatively
large $\pi N\Delta$ coupling constant. The axial charges of some baryon resonances
are rather sensitive to tensor and/or spin-orbit forces in the hyperfine interaction.
These resonances fall into the same flavor multiplets. In general the pattern observed
from the predictions for $g_A^B$ is congruent with the classification of baryons
into flavor multiplets as found recently. From the RCQMs predictions presented
here, no particular trend is observed for the axial charges of $N$ and other
baryon resonances to become small, when the excitation energy is increased.
Certainly, the consideration of baryon axial charges remains an exciting field,
and one is eager to see additional experimental data as well as more theoretical
results from different approaches to QCD.
\begin{acknowledgments}
The authors are grateful to L.Ya. Glozman for valuable incentives regarding specific
aspects of this work and to the Graz lattice-QCD group for several clarifying discussions
about respective calculations.
This work was supported by the Austrian Science Fund, FWF, through the Doctoral
Program on {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N08).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,720 |
Jean-François Roux (ur. 9 września 1920 w Cherbourgu, zm. 25 stycznia 2008) – francuski prawnik, urzędnik konsularny i dyplomata.
Uczęszczał do Szkoły Bossuet (École Bossuet), Liceum Ludwika Wielkiego (Lycée Louis-le-Grand) i Wydziałów Prawa i Filologii Uniwersytetu w Paryżu (Facultés de droit et des lettres à Paris).
W 1944 wstąpił do francuskiej służby zagranicznej, a następnie studiował w Narodowej Szkole Administracji (École nationale d'administration) (1946-1947). W 1948 powrócił do MSZ, pełniąc rolę obserwatora Komisji Nadzwyczajnej Narodów Zjednoczonych ds. Bałkanów w Salonikach, konsula w Gdańsku (1950-1953), pracownika biura kadr MSZ (1954-1955), I sekretarza Stałej Misji przy ONZ w Nowym Jorku (1955-1961), konsula gen. w Kenitrze (1961-1963), doradcy ministra prac publicznych i transportu (1964-1965) i sekretarza stanu ds transportu (1966-1967), I radcy w Bernie (1968-1971), konsula gen. w Los Angeles (1972-1975), I radcy w Bonn (1975-1978), konsula gen. w Bombaju (1978-1980) i w Londynie (1980-1984), ministra pełnomocnego (1983), prezydenta delegacji francuskiej w Komisji Międzynarodowej Pirenejów (1985-), prezydenta 34 sesji Komisji w Madrycie (1986). Nadano mu rangę ministra pełnomocnego I kl. (1987). W 1988 odszedł na emeryturę.
Otrzymał tytuł oficera Legii Honorowej.
Zobacz też
Konsulat Francji w Gdańsku
Linki zewnętrzne
Who's who in France - hasło Jean-François Roux
Stosunki polsko-francuskie
Urzędnicy konsularni w Gdańsku
Urodzeni w 1920
Zmarli w 2008
Ludzie urodzeni w Cherbourg-en-Cotentin | {
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{"url":"https:\/\/marcofrasca.wordpress.com\/tag\/tetraquark\/","text":"## f0(500) and f0(980) are not\u00a0tetraquarks\n\n27\/06\/2014\n\nLast week I have been in Giovinazzo, a really beautiful town near Bari in Italy. I participated at the QCD@Work conference. This conference series is now at the 7th edition and, for me, it was my second attendance. The most striking news I heard was put forward in the first day and represents a striking result indeed. The talk was given by Maurizio Martinelli on behalf of LHCb Collaboration. You can find the result on page 19 and on an arxiv paper . The question of the nature of f0(500) is a vexata quaestio since the first possible observation of this resonance. It entered in the Particle Data Group catalog as f0(600) but was eliminated in the following years. Today its existence is no more questioned and this particle is widely accepted. Also its properties as the mass and the width are known with reasonable precision starting from a fundamental work by Irinel Caprini, Gilberto Colangelo and Heinrich Leutwyler (see here). The longstanding question around this particle and its parent f0(980) was about their nature. It is generally difficult to fix the structure of a resonance in QCD and there is no exception here.\n\nThe problem arose from famous papers by Jaffe on 1977 (this one and this one) that using a quark-bag model introduced a low-energy nonet of states made of four quarks each. These papers set the stage for what has been the current understanding of the f0(500) and f0(980) resonances. The nonet is completely filled with all the QCD resonances below 1 GeV and so, it seems to fit the bill excellently.\n\nSomeone challenged this kind of paradigm and claimed that f0(500) could not be a tetraquark state (e.g. see here and here but also papers by Wolfgang Ochs and Peter Minkowski disagree with the tetraquark model for these resonances). The answer come out straightforwardly from LHCb collaboration: Both f0(500) and f0(980) are not tetraquark and the original view by Jaffe is no more supported. Indeed, people that know the Nambu-Jona-Lasinio model should know quite well where the f0(500) (or $\\sigma$ ) comes from and I would also suggest that this model can also accommodate higher states like f0(980).\n\nI should say that this is a further striking result coming from LHCb Collaboration. Hopefully, this should give important hints to a better understanding of low-energy QCD.\n\n$\\overline{B}^0\\rightarrow J\/\u03c8\u03c0^+\u03c0^-$ decays arXiv arXiv: 1404.5673v2\nIrinel Caprini, Gilberto Colangelo, & Heinrich Leutwyler (2005). Mass and width of the lowest resonance in QCD Phys.Rev.Lett.96:132001,2006 arXiv: hep-ph\/0512364v2\nJaffe, R. (1977). Multiquark hadrons. I. Phenomenology of Q^{2}Q[over \u00af]^{2} mesons Physical Review D, 15 (1), 267-280 DOI: 10.1103\/PhysRevD.15.267\nJaffe, R. (1977). Multiquark hadrons. II. Methods Physical Review D, 15 (1), 281-289 DOI: 10.1103\/PhysRevD.15.281\nG. Mennessier, S. Narison, & X. -G. Wang (2010). The sigma and f_0(980) from K_e4+pi-pi, gamma-gamma scatterings, J\/psi,\nphi to gamma sigma_B and D_s to l nu sigma_B Nucl.Phys.Proc.Suppl.207-208:177-180,2010 arXiv: 1009.3590v1\n\nMarco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2\n\n## The question of\u00a0X(3872)\n\n13\/10\/2009\n\nX(3872) is a resonance observed a few years ago at Belle and Tevatron and what hit immediately physicists imagination was that it has roughly two times the mass of $D^0$ meson. This would imply that it could be a neat example of hadron molecule being a combination of two couples of quarks. As you may know, there is a lot of activity in QCD to understand if tetraquarks exist or not and notable physicists are involved in this quest. Several proposals emerged showing how tetraquarks can be the answer to the spectrum of light unflavored mesons. X(3872) could be a particular tetraquark state with diquarks combining with a very low binding energy (about 0.25 MeV) forming a molecule. Whatever its nature, this resonance appears quite exotic indeed. But a recent paper (see here) sheds some light about what this particle cannot be. The authors derive some bounds on the production cross section of it showing that is not plausible to consider this particle as a diquark state. They carry on simulations of production of the resonance proving that is unlikely the formation in S-wave of a molecular $D^0\\bar D^{*0}$ state. The paper appeared in this days in Physical Review Letters (see here).\n\nThe interest arisen on this particle at the time makes important this article giving a significant clarification about the direction to take to have an understanding of its very nature.\n\n## A striking confirmation\n\n17\/04\/2009\n\nOn arxiv today it is appeared a paper by Stephan Narison,\u00a0 Gerard Mennessier and Robert Kaminski (see here). Stephan Narison is the organizer of QCD Conferences series and I attended one of this, QCD 08, last year. Narison is located in Montpellier (France) and, together with other researchers, is carrying out research aimed to an understanding of low-energy phenomenology of QCD. So, there is a strong overlapping between their work and mine. Their tools are QCD spectral sum rules and low energy theorems and the results they obtain are quite striking. Narison has written a relevant handbook of QCD (see here) that is a worthwhile tool for people aimed to work with this theory.\n\nThe paper gives further support to the idea that the resonance f0(600)\/$\\sigma$ is indeed a glueball. Currently, researchers have explored another possibility, that this particle is a four quark state. Narison, Mennessier and Kaminski consider that, if this would be true, being this a state with u and d quarks, coupling with K mesons should be suppressed. This would imply that, in a computation for the rates of $\\sigma$ decays, the contribution coming in the case of K mesons in the final state should be really small. But, for a glueball state, these couplings for $\\pi\\pi$ and $KK$ decays should be almost the same.\n\nIndeed, they get the following\n\n$|g^{os}_{\\sigma\\pi +\\pi -}|\\simeq 6 GeV, r_{\\sigma\\pi K}\\equiv \\frac{g^{os}_{\\sigma K+K-}}{g^{os}_{\\sigma\\pi +\\pi -}} \\simeq 0.8$\n\nthat is quite striking indeed. They do the same for f0(980) and, even if they get a similar result, they draw no conclusion about the nature of this resonance.\n\nThis, together with the small decay rate in $\\gamma\\gamma$, gives a really strong support to the conclusion that $\\sigma$ is indeed a glueball. At this stage, we would like to see an improved support from lattice computations. Surely, it is time to revise some theoretical computations of the gluon propagator.\n\nUpdate: I have received the following correction to above deleted sentence by Stephan Narison. This is the right take:\n\nOne should take into account that the sigma to KK is suppressed due to phase space BUT the coupling to KK is very strong. The non-observation of sigma to KK has been the (main) motivation that it can be pi-pi or 4-quark states and nobody has payed attention to this (unobserved) decay.\n\n## Wonderful QCD!\n\n28\/11\/2008\n\nOn Science this week appeared a milestone paper showing two great achievements by lattice QCD:\n\n\u2022 QCD gives a correct description of low energy phenomenology of strong interactions.\n\u2022 Most of the ordinary mass (99%) is due to the motion of quarks inside hadrons.\n\nThe precision reached has no precedent. The authors are able to get a control of involved\u00a0 errors in such a way to reach an agreement of about 1% into the computation of nucleon masses. Frank Wilczek gives here a very readable account of these accomplishments and is worthwhile reading. These results open a new era into this kind of method to extract results to be compared with experiments for QCD and give an important confirmation to our understanding of strong interactions. But I would like to point out Wilczek\u2019s concern: Until we will not have a theoretical way to obtain results from QCD in the low energy limit, we will miss a great piece of understanding of physics. This is a point that I discussed largely with my posts in this blog but it is worthwhile repeating here coming from such an authoritative voice.\n\nAn interesting point about these lattice computations can be made by observing that again no $\\sigma$ resonance is seen. I would like to remember that in these computations entered just u, d and s quarks as the authors\u2019 aims were computations of bound states of such quarks. Some authoritative theoretical physicists are claiming that this resonance should be a tetraquark, that is a combination of u and d quarks and their antiparticles. What we can say about from our point of view? As I have written here some time ago, lattice computations of the gluon propagator in a pure Yang-Mills theory prove that this can be fitted with a Yukawa form\n\n$G(p)=\\frac{A}{p^2+m^2}$\n\nbeing $m\\approx 500 MeV$. This is given in Euclidean form. This kind of propagators says to us that the potential should be Yukawa-like, that is\n\n$V(r)=-A\\frac{e^{-mr}}{r}$\n\nif this is true no tetraquark state can exist for lighter quarks. The reason is that a Yukawa-like potential heavily damps any van der Waals kind of residual potential. But, due to asymptotic freedom, this is no more true for heavier quarks c and b\u00a0 as in this case the potential is Coulomb-like and, indeed, such kind of states could have been seen at Tevatron.\n\nWe expect that the glueball spectrum should display itself in the observed hadronic spectrum. This means that a major effort in lattice QCD computations should be aimed in this direction now that such a deep understanding of known hadronic states has been reached.","date":"2018-02-21 10:49:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 14, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7711656093597412, \"perplexity\": 1184.2936925078095}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891813608.70\/warc\/CC-MAIN-20180221103712-20180221123712-00411.warc.gz\"}"} | null | null |
Q: How to delete a contributor on GitHub repository page? I have multiple GitHub accounts, say A and B, and I'm in a project at the remote repository of A.
Yesterday, I was surprised to find that there was somehow the account name of B on the Contributors list at the repository page of A.
I checked the commit log and I found that there was somehow the name of B signed for some of the commits. So I used git filter-repo command to forcibly rewrite the Author name and Committer name and email address from those of B to those of A.
Then, I forcibly pushed it with git push -f origin and I confirmed that the commit log of a remote repository of A was completely rewritten as I expected.
However, there still remains B's name in the Contributors list at the repository of A.
Anyway, I want to delete B's name from Contributors list at A's repository page, because I don't want to let anybody know that the owner of A account is identical to that of B account.
I would appreciate if you give me some advice.
A: I tested this, GitHub appears to treat the list of "Contributors" as append-only: whenever it sees a new one, it adds it to the list, but it never removes from that list. See this repo which lists someone who never committed in the Contributors list, despite only having one commit (that they didn't author). I did this by pushing a commit as myself, then pushing one as torvalds, then force-pushing back to my commit. Despite this, GitHub still shows torvalds as a contributor.
Go to the full "Contributors" graph in the insights tab, and check if B listed there. The data for this graph is always calculated using the actual commit graph. If B is listed there, then GitHub still thinks B did make some commits. In that case, you should double-check that you fixed all commits belonging to B. Click on "N commits" to see what commits GitHub thinks B did:
If no commits are attributed to B, contact GitHub support (I think Remove data from a repository I own or control would be the right topic to select) and ask them to re-generate the Contributors list.
| {
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/*
* To change this template, choose Tools | Templates
* and open the template in the editor.
*/
package com.lacv.marketplatform.daos.impl;
import com.lacv.marketplatform.daos.CommerceJpa;
import com.lacv.marketplatform.entities.Commerce;
import com.dot.gcpbasedot.dao.JPAAbstractDao;
import javax.persistence.EntityManager;
import javax.persistence.PersistenceContext;
import javax.sql.DataSource;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.stereotype.Repository;
/**
*
* @author lacastrillov
*/
@Repository
public class CommerceJpaController extends JPAAbstractDao<Commerce> implements CommerceJpa {
@Autowired
public void init(DataSource dataSource){
super.setDataSource(dataSource);
}
@Override
@PersistenceContext(unitName ="MarketPlatformPU")
public void setEntityManager(EntityManager entityManager) {
this.entityManager= entityManager;
}
}
| {
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Home > News > Neymar could still face Real, no operation planned- Coach
Neymar could still face Real, no operation planned- Coach
Neymar still has a "small chance" of playing against Real Madrid in the Champions League and there are no plans for an operation on his cracked bone, Paris Saint-Germain coach Unai Emery said on Tuesday.
"For Neymar there has been no decision to have an operation," said Emery, denying an earlier report in the Brazilian press that the world's most expensive player would go under the knife to fix the cracked bone in his foot.
"Neymar is the first to want to play every match, he's very focussed on Real. I think there's a small chance that he'll be ready for the match," added Emery.
PSG trail 3-1 from the last 16 first leg in Madrid two weeks ago and the loss of Neymar would be a huge blow to their chances of overturning that deficit at the Parc des Princes on March 6 and reaching the Champions League quarter-finals.
Neymar suffered a cracked metatarsal in his foot and a twisted ankle in an innocuous looking incident against Marseille on Sunday night.
And while Spaniard Emery said there is a possibility he'll recover by next week, he was more pessimistic than he had been on Sunday night.
"Today, after analysing Neymar, it looks less likely than it did on Sunday that he'll be ready for this match," said Emery.
However, he was clear in his dismissal of a report published by Brazilian newspaper Globo Esporte on Tuesday morning that claimed a decision had already been reached for Neymar to undergo an operation, and that he would be out until May.
– 'Tranquility and patience' –
"Last night (Monday) the club issued an official statement: it said how the player was medically," said Emery.
"Things (like operations) aren't decided like that," he added, clicking his fingers.
"You need discussions with the doctor, the player, his entourage.
"To make such an important decision you need tranquility and a bit of patience.
"Right now, I don't think you need anything else. There's a lot of information from outside, but the truth is what I told you, which is what the doctor told me."
While PSG's primary concern is having the world's most expensive player available for their Champions League match next week, in Brazil it is Neymar's participation at the World Cup that is most pressing.
The 26-year-old captained Brazil to their first ever Olympic gold medal 18 months ago on home soil and is seen as crucial to their hopes in Russia in June and July.
Were he to undergo an operation now and return to fitness in May, that would suit Brazil, who could then rely on a fully fit and fresh Neymar for the World Cup. Four years ago, Neymar missed the semifinal against Germany with injury and Brazil were thrashed 7-1.
Globo had said Neymar would have a "pin" attached to his metatarsal bone and would be fit to return in May.
Recovery from metatarsal injuries usually take weeks, if not months.
Just last year, Neymar's international team-mate Gabriel Jesus was out of action for two months after breaking a metatarsal playing for Manchester City.
Former England stars David Beckham and Wayne Rooney spent six weeks on the sidelines with metatarsal injuries in the past.
Earlier on Tuesday, a number of medical experts who were spoken to all set a downbeat tone.
"It would be very risky as it's a fragile bone in the foot which is difficult to protect," said Jean-Marcel Ferrat, a former doctor with the France national football team.
"(Playing against Real) could compromise the rest of his season."
Parisian osteopath Alain Gosp Server said such injuries normally take "three weeks to heal".
"It will all depend on his treatment but what's sure is that he wouldn't be at 100 percent (in eight days)," added Gosp Server.
The post Neymar could still face Real, no operation planned- Coach appeared first on Vanguard News.
Read About What Happened To Yusuf Buhari After His Motorbike Accident
Delta 2019: Delta PDP group endorses Gov. Okowa for re-election | {
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#include <aws/workspaces-web/model/IdentityProviderSummary.h>
#include <aws/core/utils/json/JsonSerializer.h>
#include <utility>
using namespace Aws::Utils::Json;
using namespace Aws::Utils;
namespace Aws
{
namespace WorkSpacesWeb
{
namespace Model
{
IdentityProviderSummary::IdentityProviderSummary() :
m_identityProviderArnHasBeenSet(false),
m_identityProviderNameHasBeenSet(false),
m_identityProviderType(IdentityProviderType::NOT_SET),
m_identityProviderTypeHasBeenSet(false)
{
}
IdentityProviderSummary::IdentityProviderSummary(JsonView jsonValue) :
m_identityProviderArnHasBeenSet(false),
m_identityProviderNameHasBeenSet(false),
m_identityProviderType(IdentityProviderType::NOT_SET),
m_identityProviderTypeHasBeenSet(false)
{
*this = jsonValue;
}
IdentityProviderSummary& IdentityProviderSummary::operator =(JsonView jsonValue)
{
if(jsonValue.ValueExists("identityProviderArn"))
{
m_identityProviderArn = jsonValue.GetString("identityProviderArn");
m_identityProviderArnHasBeenSet = true;
}
if(jsonValue.ValueExists("identityProviderName"))
{
m_identityProviderName = jsonValue.GetString("identityProviderName");
m_identityProviderNameHasBeenSet = true;
}
if(jsonValue.ValueExists("identityProviderType"))
{
m_identityProviderType = IdentityProviderTypeMapper::GetIdentityProviderTypeForName(jsonValue.GetString("identityProviderType"));
m_identityProviderTypeHasBeenSet = true;
}
return *this;
}
JsonValue IdentityProviderSummary::Jsonize() const
{
JsonValue payload;
if(m_identityProviderArnHasBeenSet)
{
payload.WithString("identityProviderArn", m_identityProviderArn);
}
if(m_identityProviderNameHasBeenSet)
{
payload.WithString("identityProviderName", m_identityProviderName);
}
if(m_identityProviderTypeHasBeenSet)
{
payload.WithString("identityProviderType", IdentityProviderTypeMapper::GetNameForIdentityProviderType(m_identityProviderType));
}
return payload;
}
} // namespace Model
} // namespace WorkSpacesWeb
} // namespace Aws
| {
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\section{\label{sec:level1}First-level heading:\protect\\ The line
break was forced \lowercase{via} \textbackslash\textbackslash}
This sample document demonstrates proper use of REV\TeX~4.2 (and
\LaTeXe) in mansucripts prepared for submission to APS
journals. Further information can be found in the REV\TeX~4.2
documentation included in the distribution or available at
\url{http://journals.aps.org/revtex/}.
When commands are referred to in this example file, they are always
shown with their required arguments, using normal \TeX{} format. In
this format, \verb+#1+, \verb+#2+, etc. stand for required
author-supplied arguments to commands. For example, in
\verb+\section{#1}+ the \verb+#1+ stands for the title text of the
author's section heading, and in \verb+\title{#1}+ the \verb+#1+
stands for the title text of the paper.
Line breaks in section headings at all levels can be introduced using
\textbackslash\textbackslash. A blank input line tells \TeX\ that the
paragraph has ended. Note that top-level section headings are
automatically uppercased. If a specific letter or word should appear in
lowercase instead, you must escape it using \verb+\lowercase{#1}+ as
in the word ``via'' above.
\subsection{\label{sec:level2}Second-level heading: Formatting}
This file may be formatted in either the \texttt{preprint} or
\texttt{reprint} style. \texttt{reprint} format mimics final journal output.
Either format may be used for submission purposes. \texttt{letter} sized paper should
be used when submitting to APS journals.
\subsubsection{Wide text (A level-3 head)}
The \texttt{widetext} environment will make the text the width of the
full page, as on page~\pageref{eq:wideeq}. (Note the use the
\verb+\pageref{#1}+ command to refer to the page number.)
\paragraph{Note (Fourth-level head is run in)}
The width-changing commands only take effect in two-column formatting.
There is no effect if text is in a single column.
\subsection{\label{sec:citeref}Citations and References}
A citation in text uses the command \verb+\cite{#1}+ or
\verb+\onlinecite{#1}+ and refers to an entry in the bibliography.
An entry in the bibliography is a reference to another document.
\subsubsection{Citations}
Because REV\TeX\ uses the \verb+natbib+ package of Patrick Daly,
the entire repertoire of commands in that package are available for your document;
see the \verb+natbib+ documentation for further details. Please note that
REV\TeX\ requires version 8.31a or later of \verb+natbib+.
\paragraph{Syntax}
The argument of \verb+\cite+ may be a single \emph{key},
or may consist of a comma-separated list of keys.
The citation \emph{key} may contain
letters, numbers, the dash (-) character, or the period (.) character.
New with natbib 8.3 is an extension to the syntax that allows for
a star (*) form and two optional arguments on the citation key itself.
The syntax of the \verb+\cite+ command is thus (informally stated)
\begin{quotation}\flushleft\leftskip1em
\verb+\cite+ \verb+{+ \emph{key} \verb+}+, or\\
\verb+\cite+ \verb+{+ \emph{optarg+key} \verb+}+, or\\
\verb+\cite+ \verb+{+ \emph{optarg+key} \verb+,+ \emph{optarg+key}\ldots \verb+}+,
\end{quotation}\noindent
where \emph{optarg+key} signifies
\begin{quotation}\flushleft\leftskip1em
\emph{key}, or\\
\texttt{*}\emph{key}, or\\
\texttt{[}\emph{pre}\texttt{]}\emph{key}, or\\
\texttt{[}\emph{pre}\texttt{]}\texttt{[}\emph{post}\texttt{]}\emph{key}, or even\\
\texttt{*}\texttt{[}\emph{pre}\texttt{]}\texttt{[}\emph{post}\texttt{]}\emph{key}.
\end{quotation}\noindent
where \emph{pre} and \emph{post} is whatever text you wish to place
at the beginning and end, respectively, of the bibliographic reference
(see Ref.~[\onlinecite{witten2001}] and the two under Ref.~[\onlinecite{feyn54}]).
(Keep in mind that no automatic space or punctuation is applied.)
It is highly recommended that you put the entire \emph{pre} or \emph{post} portion
within its own set of braces, for example:
\verb+\cite+ \verb+{+ \texttt{[} \verb+{+\emph{text}\verb+}+\texttt{]}\emph{key}\verb+}+.
The extra set of braces will keep \LaTeX\ out of trouble if your \emph{text} contains the comma (,) character.
The star (*) modifier to the \emph{key} signifies that the reference is to be
merged with the previous reference into a single bibliographic entry,
a common idiom in APS and AIP articles (see below, Ref.~[\onlinecite{epr}]).
When references are merged in this way, they are separated by a semicolon instead of
the period (full stop) that would otherwise appear.
\paragraph{Eliding repeated information}
When a reference is merged, some of its fields may be elided: for example,
when the author matches that of the previous reference, it is omitted.
If both author and journal match, both are omitted.
If the journal matches, but the author does not, the journal is replaced by \emph{ibid.},
as exemplified by Ref.~[\onlinecite{epr}].
These rules embody common editorial practice in APS and AIP journals and will only
be in effect if the markup features of the APS and AIP Bib\TeX\ styles is employed.
\paragraph{The options of the cite command itself}
Please note that optional arguments to the \emph{key} change the reference in the bibliography,
not the citation in the body of the document.
For the latter, use the optional arguments of the \verb+\cite+ command itself:
\verb+\cite+ \texttt{*}\allowbreak
\texttt{[}\emph{pre-cite}\texttt{]}\allowbreak
\texttt{[}\emph{post-cite}\texttt{]}\allowbreak
\verb+{+\emph{key-list}\verb+}+.
\subsubsection{Example citations}
By default, citations are numerical\cite{Beutler1994}.
Author-year citations are used when the journal is RMP.
To give a textual citation, use \verb+\onlinecite{#1}+:
Refs.~\onlinecite{[][{, and references therein}]witten2001,Bire82}.
By default, the \texttt{natbib} package automatically sorts your citations into numerical order and ``compresses'' runs of three or more consecutive numerical citations.
REV\TeX\ provides the ability to automatically change the punctuation when switching between journal styles that provide citations in square brackets and those that use a superscript style instead. This is done through the \texttt{citeautoscript} option. For instance, the journal style \texttt{prb} automatically invokes this option because \textit{Physical
Review B} uses superscript-style citations. The effect is to move the punctuation, which normally comes after a citation in square brackets, to its proper position before the superscript.
To illustrate, we cite several together
\cite{[See the explanation of time travel in ]feyn54,*[The classical relativistic treatment of ][ is a relative classic]epr,witten2001,Berman1983,Davies1998,Bire82},
and once again in different order (Refs.~\cite{epr,feyn54,Bire82,Berman1983,witten2001,Davies1998}).
Note that the citations were both compressed and sorted. Futhermore, running this sample file under the \texttt{prb} option will move the punctuation to the correct place.
When the \verb+prb+ class option is used, the \verb+\cite{#1}+ command
displays the reference's number as a superscript rather than in
square brackets. Note that the location of the \verb+\cite{#1}+
command should be adjusted for the reference style: the superscript
references in \verb+prb+ style must appear after punctuation;
otherwise the reference must appear before any punctuation. This
sample was written for the regular (non-\texttt{prb}) citation style.
The command \verb+\onlinecite{#1}+ in the \texttt{prb} style also
displays the reference on the baseline.
\subsubsection{References}
A reference in the bibliography is specified by a \verb+\bibitem{#1}+ command
with the same argument as the \verb+\cite{#1}+ command.
\verb+\bibitem{#1}+ commands may be crafted by hand or, preferably,
generated by Bib\TeX.
REV\TeX~4.2 includes Bib\TeX\ style files
\verb+apsrev4-2.bst+, \verb+apsrmp4-2.bst+ appropriate for
\textit{Physical Review} and \textit{Reviews of Modern Physics},
respectively.
\subsubsection{Example references}
This sample file employs the \verb+\bibliography+ command,
which formats the \texttt{\jobname .bbl} file
and specifies which bibliographic databases are to be used by Bib\TeX\
(one of these should be by arXiv convention \texttt{\jobname .bib}).
Running Bib\TeX\ (via \texttt{bibtex \jobname})
after the first pass of \LaTeX\ produces the file
\texttt{\jobname .bbl} which contains the automatically formatted
\verb+\bibitem+ commands (including extra markup information via
\verb+\bibinfo+ and \verb+\bibfield+ commands).
If not using Bib\TeX, you will have to create the \verb+thebibiliography+ environment
and its \verb+\bibitem+ commands by hand.
Numerous examples of the use of the APS bibliographic entry types appear in the bibliography of this sample document.
You can refer to the \texttt{\jobname .bib} file,
and compare its information to the formatted bibliography itself.
\subsection{Footnotes}%
Footnotes, produced using the \verb+\footnote{#1}+ command,
usually integrated into the bibliography alongside the other entries.
Numerical citation styles do this%
\footnote{Automatically placing footnotes into the bibliography requires using BibTeX to compile the bibliography.};
author-year citation styles place the footnote at the bottom of the text column.
Note: due to the method used to place footnotes in the bibliography,
\emph{you must re-run Bib\TeX\ every time you change any of your document's footnotes}.
\section{Math and Equations}
Inline math may be typeset using the \verb+$+ delimiters. Bold math
symbols may be achieved using the \verb+bm+ package and the
\verb+\bm{#1}+ command it supplies. For instance, a bold $\alpha$ can
be typeset as \verb+$\bm{\alpha}$+ giving $\bm{\alpha}$. Fraktur and
Blackboard (or open face or double struck) characters should be
typeset using the \verb+\mathfrak{#1}+ and \verb+\mathbb{#1}+ commands
respectively. Both are supplied by the \texttt{amssymb} package. For
example, \verb+$\mathbb{R}$+ gives $\mathbb{R}$ and
\verb+$\mathfrak{G}$+ gives $\mathfrak{G}$
In \LaTeX\ there are many different ways to display equations, and a
few preferred ways are noted below. Displayed math will center by
default. Use the class option \verb+fleqn+ to flush equations left.
Below we have numbered single-line equations; this is the most common
type of equation in \textit{Physical Review}:
\begin{eqnarray}
\chi_+(p)\alt{\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
|{\bf p}|+p_z\\
px+ip_y
\end{array}\right)\;,
\\
\left\{%
\openone234567890abc123\alpha\beta\gamma\delta1234556\alpha\beta
\frac{1\sum^{a}_{b}}{A^2}%
\right\}%
\label{eq:one}.
\end{eqnarray}
Note the open one in Eq.~(\ref{eq:one}).
Not all numbered equations will fit within a narrow column this
way. The equation number will move down automatically if it cannot fit
on the same line with a one-line equation:
\begin{equation}
\left\{
ab12345678abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
\frac{1\sum^{a}_{b}}{A^2}%
\right\}.
\end{equation}
When the \verb+\label{#1}+ command is used [cf. input for
Eq.~(\ref{eq:one})], the equation can be referred to in text without
knowing the equation number that \TeX\ will assign to it. Just
use \verb+\ref{#1}+, where \verb+#1+ is the same name that used in
the \verb+\label{#1}+ command.
Unnumbered single-line equations can be typeset
using the \verb+\[+, \verb+\]+ format:
\[g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \]
\subsection{Multiline equations}
Multiline equations are obtained by using the \verb+eqnarray+
environment. Use the \verb+\nonumber+ command at the end of each line
to avoid assigning a number:
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
\delta_{\sigma_1,-\sigma_2}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_jl_i\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1),
\end{eqnarray}
\begin{eqnarray}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
(N^2-1)\nonumber \\
& &\times \left( \sum_{i<j}\right)
\sum_{\text{perm}}
\frac{1}{S_{12}}
\frac{1}{S_{12}}
\sum_\tau c^f_\tau~.
\end{eqnarray}
\textbf{Note:} Do not use \verb+\label{#1}+ on a line of a multiline
equation if \verb+\nonumber+ is also used on that line. Incorrect
cross-referencing will result. Notice the use \verb+\text{#1}+ for
using a Roman font within a math environment.
To set a multiline equation without \emph{any} equation
numbers, use the \verb+\begin{eqnarray*}+,
\verb+\end{eqnarray*}+ format:
\begin{eqnarray*}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
(N^2-1)\\
& &\times \left( \sum_{i<j}\right)
\left(
\sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}}
\right)
\frac{1}{S_{12}}~.
\end{eqnarray*}
To obtain numbers not normally produced by the automatic numbering,
use the \verb+\tag{#1}+ command, where \verb+#1+ is the desired
equation number. For example, to get an equation number of
(\ref{eq:mynum}),
\begin{equation}
g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \tag{2.6$'$}\label{eq:mynum}
\end{equation}
\paragraph{A few notes on \texttt{tag}s}
\verb+\tag{#1}+ requires the \texttt{amsmath} package.
Place the \verb+\tag{#1}+ command before the \verb+\label{#1}+, if any.
The numbering produced by \verb+\tag{#1}+ \textit{does not affect}
the automatic numbering in REV\TeX;
therefore, the number must be known ahead of time,
and it must be manually adjusted if other equations are added.
\verb+\tag{#1}+ works with both single-line and multiline equations.
\verb+\tag{#1}+ should only be used in exceptional cases---%
do not use it to number many equations in your paper.
Please note that this feature of the \texttt{amsmath} package
is \emph{not} compatible with the \texttt{hyperref} (6.77u) package.
Enclosing display math within
\verb+\begin{subequations}+ and \verb+\end{subequations}+ will produce
a set of equations that are labeled with letters, as shown in
Eqs.~(\ref{subeq:1}) and (\ref{subeq:2}) below.
You may include any number of single-line and multiline equations,
although it is probably not a good idea to follow one display math
directly after another.
\begin{subequations}
\label{eq:whole}
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1).\label{subeq:2}
\end{eqnarray}
\begin{equation}
\left\{
abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
\frac{1\sum^{a}_{b}}{A^2}
\right\},\label{subeq:1}
\end{equation}
\end{subequations}
Giving a \verb+\label{#1}+ command directly after the \verb+\begin{subequations}+,
allows you to reference all the equations in the \texttt{subequations} environment.
For example, the equations in the preceding subequations environment were
Eqs.~(\ref{eq:whole}).
\subsubsection{Wide equations}
The equation that follows is set in a wide format, i.e., it spans the full page.
The wide format is reserved for long equations
that cannot easily be set in a single column:
\begin{widetext}
\begin{equation}
{\cal R}^{(\text{d})}=
g_{\sigma_2}^e
\left(
\frac{[\Gamma^Z(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
+\frac{[\Gamma^Z(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
\right)
+ x_WQ_e
\left(
\frac{[\Gamma^\gamma(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
+\frac{[\Gamma^\gamma(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
\right)\;.
\label{eq:wideeq}
\end{equation}
\end{widetext}
This is typed to show how the output appears in wide format.
(Incidentally, since there is no blank line between the \texttt{equation} environment above
and the start of this paragraph, this paragraph is not indented.)
\section{Cross-referencing}
REV\TeX{} will automatically number such things as
sections, footnotes, equations, figure captions, and table captions.
In order to reference them in text, use the
\verb+\label{#1}+ and \verb+\ref{#1}+ commands.
To reference a particular page, use the \verb+\pageref{#1}+ command.
The \verb+\label{#1}+ should appear
within the section heading,
within the footnote text,
within the equation, or
within the table or figure caption.
The \verb+\ref{#1}+ command
is used in text at the point where the reference is to be displayed.
Some examples: Section~\ref{sec:level1} on page~\pageref{sec:level1},
Table~\ref{tab:table1},%
\begin{table}[b
\caption{\label{tab:table1}%
A table that fits into a single column of a two-column layout.
Note that REV\TeX~4 adjusts the intercolumn spacing so that the table fills the
entire width of the column. Table captions are numbered
automatically.
This table illustrates left-, center-, decimal- and right-aligned columns,
along with the use of the \texttt{ruledtabular} environment which sets the
Scotch (double) rules above and below the alignment, per APS style.
}
\begin{ruledtabular}
\begin{tabular}{lcdr}
\textrm{Left\footnote{Note a.}}&
\textrm{Centered\footnote{Note b.}}&
\multicolumn{1}{c}{\textrm{Decimal}}&
\textrm{Right}\\
\colrule
1 & 2 & 3.001 & 4\\
10 & 20 & 30 & 40\\
100 & 200 & 300.0 & 400\\
\end{tabular}
\end{ruledtabular}
\end{table}
and Fig.~\ref{fig:epsart}.%
\begin{figure}[b]
\includegraphics{fig_1
\caption{\label{fig:epsart} A figure caption. The figure captions are
automatically numbered.}
\end{figure}
\section{Floats: Figures, Tables, Videos, etc.}
Figures and tables are usually allowed to ``float'', which means that their
placement is determined by \LaTeX, while the document is being typeset.
Use the \texttt{figure} environment for a figure, the \texttt{table} environment for a table.
In each case, use the \verb+\caption+ command within to give the text of the
figure or table caption along with the \verb+\label+ command to provide
a key for referring to this figure or table.
The typical content of a figure is an image of some kind;
that of a table is an alignment.%
\begin{figure*}
\includegraphics{fig_2
\caption{\label{fig:wide}Use the figure* environment to get a wide
figure that spans the page in \texttt{twocolumn} formatting.}
\end{figure*}
\begin{table*}
\caption{\label{tab:table3}This is a wide table that spans the full page
width in a two-column layout. It is formatted using the
\texttt{table*} environment. It also demonstates the use of
\textbackslash\texttt{multicolumn} in rows with entries that span
more than one column.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
&\multicolumn{2}{c}{$D_{4h}^1$}&\multicolumn{2}{c}{$D_{4h}^5$}\\
Ion&1st alternative&2nd alternative&lst alternative
&2nd alternative\\ \hline
K&$(2e)+(2f)$&$(4i)$ &$(2c)+(2d)$&$(4f)$ \\
Mn&$(2g)$\footnote{The $z$ parameter of these positions is $z\sim\frac{1}{4}$.}
&$(a)+(b)+(c)+(d)$&$(4e)$&$(2a)+(2b)$\\
Cl&$(a)+(b)+(c)+(d)$&$(2g)$\footnotemark[1]
&$(4e)^{\text{a}}$\\
He&$(8r)^{\text{a}}$&$(4j)^{\text{a}}$&$(4g)^{\text{a}}$\\
Ag& &$(4k)^{\text{a}}$& &$(4h)^{\text{a}}$\\
\end{tabular}
\end{ruledtabular}
\end{table*}
Insert an image using either the \texttt{graphics} or
\texttt{graphix} packages, which define the \verb+\includegraphics{#1}+ command.
(The two packages differ in respect of the optional arguments
used to specify the orientation, scaling, and translation of the image.)
To create an alignment, use the \texttt{tabular} environment.
The best place to locate the \texttt{figure} or \texttt{table} environment
is immediately following its first reference in text; this sample document
illustrates this practice for Fig.~\ref{fig:epsart}, which
shows a figure that is small enough to fit in a single column.
In exceptional cases, you will need to move the float earlier in the document, as was done
with Table~\ref{tab:table3}: \LaTeX's float placement algorithms need to know
about a full-page-width float earlier.
Fig.~\ref{fig:wide}
has content that is too wide for a single column,
so the \texttt{figure*} environment has been used.%
\begin{table}[b]
\caption{\label{tab:table4}%
Numbers in columns Three--Five are aligned with the ``d'' column specifier
(requires the \texttt{dcolumn} package).
Non-numeric entries (those entries without a ``.'') in a ``d'' column are aligned on the decimal point.
Use the ``D'' specifier for more complex layouts. }
\begin{ruledtabular}
\begin{tabular}{ccddd}
One&Two&
\multicolumn{1}{c}{\textrm{Three}}&
\multicolumn{1}{c}{\textrm{Four}}&
\multicolumn{1}{c}{\textrm{Five}}\\
\hline
one&two&\mbox{three}&\mbox{four}&\mbox{five}\\
He&2& 2.77234 & 45672. & 0.69 \\
C\footnote{Some tables require footnotes.}
&C\footnote{Some tables need more than one footnote.}
& 12537.64 & 37.66345 & 86.37 \\
\end{tabular}
\end{ruledtabular}
\end{table}
The content of a table is typically a \texttt{tabular} environment,
giving rows of type in aligned columns.
Column entries separated by \verb+&+'s, and
each row ends with \textbackslash\textbackslash.
The required argument for the \texttt{tabular} environment
specifies how data are aligned in the columns.
For instance, entries may be centered, left-justified, right-justified, aligned on a decimal
point.
Extra column-spacing may be be specified as well,
although REV\TeX~4 sets this spacing so that the columns fill the width of the
table. Horizontal rules are typeset using the \verb+\hline+
command. The doubled (or Scotch) rules that appear at the top and
bottom of a table can be achieved enclosing the \texttt{tabular}
environment within a \texttt{ruledtabular} environment. Rows whose
columns span multiple columns can be typeset using the
\verb+\multicolumn{#1}{#2}{#3}+ command (for example, see the first
row of Table~\ref{tab:table3}).%
Tables~\ref{tab:table1}, \ref{tab:table3}, \ref{tab:table4}, and \ref{tab:table2}%
\begin{table}[b]
\caption{\label{tab:table2}
A table with numerous columns that still fits into a single column.
Here, several entries share the same footnote.
Inspect the \LaTeX\ input for this table to see exactly how it is done.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
&$r_c$ (\AA)&$r_0$ (\AA)&$\kappa r_0$&
&$r_c$ (\AA) &$r_0$ (\AA)&$\kappa r_0$\\
\hline
Cu& 0.800 & 14.10 & 2.550 &Sn\footnotemark[1]
& 0.680 & 1.870 & 3.700 \\
Ag& 0.990 & 15.90 & 2.710 &Pb\footnotemark[2]
& 0.450 & 1.930 & 3.760 \\
Au& 1.150 & 15.90 & 2.710 &Ca\footnotemark[3]
& 0.750 & 2.170 & 3.560 \\
Mg& 0.490 & 17.60 & 3.200 &Sr\footnotemark[4]
& 0.900 & 2.370 & 3.720 \\
Zn& 0.300 & 15.20 & 2.970 &Li\footnotemark[2]
& 0.380 & 1.730 & 2.830 \\
Cd& 0.530 & 17.10 & 3.160 &Na\footnotemark[5]
& 0.760 & 2.110 & 3.120 \\
Hg& 0.550 & 17.80 & 3.220 &K\footnotemark[5]
& 1.120 & 2.620 & 3.480 \\
Al& 0.230 & 15.80 & 3.240 &Rb\footnotemark[3]
& 1.330 & 2.800 & 3.590 \\
Ga& 0.310 & 16.70 & 3.330 &Cs\footnotemark[4]
& 1.420 & 3.030 & 3.740 \\
In& 0.460 & 18.40 & 3.500 &Ba\footnotemark[5]
& 0.960 & 2.460 & 3.780 \\
Tl& 0.480 & 18.90 & 3.550 & & & & \\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Here's the first, from Ref.~\onlinecite{feyn54}.}
\footnotetext[2]{Here's the second.}
\footnotetext[3]{Here's the third.}
\footnotetext[4]{Here's the fourth.}
\footnotetext[5]{And etc.}
\end{table}
show various effects.
A table that fits in a single column employs the \texttt{table}
environment.
Table~\ref{tab:table3} is a wide table, set with the \texttt{table*} environment.
Long tables may need to break across pages.
The most straightforward way to accomplish this is to specify
the \verb+[H]+ float placement on the \texttt{table} or
\texttt{table*} environment.
However, the \LaTeXe\ package \texttt{longtable} allows headers and footers to be specified for each page of the table.
A simple example of the use of \texttt{longtable} can be found
in the file \texttt{summary.tex} that is included with the REV\TeX~4
distribution.
There are two methods for setting footnotes within a table (these
footnotes will be displayed directly below the table rather than at
the bottom of the page or in the bibliography). The easiest
and preferred method is just to use the \verb+\footnote{#1}+
command. This will automatically enumerate the footnotes with
lowercase roman letters. However, it is sometimes necessary to have
multiple entries in the table share the same footnote. In this case,
there is no choice but to manually create the footnotes using
\verb+\footnotemark[#1]+ and \verb+\footnotetext[#1]{#2}+.
\texttt{\#1} is a numeric value. Each time the same value for
\texttt{\#1} is used, the same mark is produced in the table. The
\verb+\footnotetext[#1]{#2}+ commands are placed after the \texttt{tabular}
environment. Examine the \LaTeX\ source and output for
Tables~\ref{tab:table1} and \ref{tab:table2}
for examples.
Video~\ref{vid:PRSTPER.4.010101}
illustrates several features new with REV\TeX4.2,
starting with the \texttt{video} environment, which is in the same category with
\texttt{figure} and \texttt{table}.%
\begin{video}
\href{http://prst-per.aps.org/multimedia/PRSTPER/v4/i1/e010101/e010101_vid1a.mpg}{\includegraphics{vid_1a}}%
\quad
\href{http://prst-per.aps.org/multimedia/PRSTPER/v4/i1/e010101/e010101_vid1b.mpg}{\includegraphics{vid_1b}}
\setfloatlink{http://link.aps.org/multimedia/PRSTPER/v4/i1/e010101}%
\caption{\label{vid:PRSTPER.4.010101}%
Students explain their initial idea about Newton's third law to a teaching assistant.
Clip (a): same force.
Clip (b): move backwards.
}%
\end{video}
The \verb+\setfloatlink+ command causes the title of the video to be a hyperlink to the
indicated URL; it may be used with any environment that takes the \verb+\caption+
command.
The \verb+\href+ command has the same significance as it does in the context of
the \texttt{hyperref} package: the second argument is a piece of text to be
typeset in your document; the first is its hyperlink, a URL.
\textit{Physical Review} style requires that the initial citation of
figures or tables be in numerical order in text, so don't cite
Fig.~\ref{fig:wide} until Fig.~\ref{fig:epsart} has been cited.
\begin{acknowledgments}
We wish to acknowledge the support of the author community in using
REV\TeX{}, offering suggestions and encouragement, testing new versions,
\dots.
\end{acknowledgments}
\section{\label{sec1}Introduction}
Inertial migration of particles in microchannels has caught extensive attention in the last two decades due to its numerous applications in cell sorting, fractionation, filtration, and separation in many clinical practices \cite{gossett2010label, toner2005blood, gossett2012hydrodynamic, karimi2013hydrodynamic}. The presence of lift forces acting on these particles is the chief reason for observing the underlying physical phenomena in microfluidic systems \cite{di2009inertial, martel2014inertial, bazaz2020computational, connolly2020mechanical, stoecklein2018nonlinear}. The importance of these forces has motivated many researchers to analyze or measure them within the microchannel. \citeauthor{di2009particle} have derived the inertial lift on particles and studied the effects of channel Reynolds number and particle size on it; they have shown that by increasing Reynolds, the magnitude of lift coefficient decreases near the wall and increases near the channel center. Also, the particle equilibrium positions shift toward the center as its size increases and its rotational motion is not a key component of the inertial lift \cite{di2009particle}. Using lift force profiles, \citeauthor{prohm2014feedback} have investigated and categorized the particle focusing points and demonstrated that the stable fix points lie on either the diagonal or main axes of the channel cross-section \cite{prohm2014feedback}. A fast numerical algorithm combined with machine learning techniques has been proposed to predict the inertial lift distribution acting on solid particles over a wide range of operating parameters in straight microchannels with three types of geometries by specifying the cross-sectional shape, Reynolds number, and particle size \cite{su2021machine}.
Furthermore, there have been attempts to derive analytical relationships for the observed behaviors. A simple formula using data fitting and least square was obtained to investigate the relationship between the lift and particle size and Reynolds number; according to the proposed criterion, particle focusing does not occur for too small particles or too low Reynolds numbers \cite{wang2017analysis}. \citeauthor{asmolov2018inertial} illustrated that the velocity of finite-size particles near the channel wall is different from that in the undisturbed flow and then reported a generalized expression for the lift force at Re $\leq$ 20 \cite{asmolov2018inertial}. Another study has proposed a generalized formula for the inertial lift acting on a sphere that consists of 4 terms: wall-induced lift, shear-gradient-induced lift, slip-shear lift, and correction of the shear-gradient lift; the authors have further confirmed that wall and shear-gradient are the main features of the lift \cite{liu2016generalized}. Moreover, there are examples of works concentrating on the effect of particle shape. For instance, \citeauthor{zastawny2012derivation} presented the great influence of shape both by changing the experienced values of forces and torques and modifying the Reynolds at which the transition to unsteady flow happens \cite{zastawny2012derivation}. Further extension on previous theories and analytical works resulted in an analytical expression capturing the weak, inertial lift on an arbitrarily-shaped particle moving near a wall \cite{mannan2020weak}.
Most of the studies on lift forces in the microchannels have focused on solid particles or non-deformable objects and have analyzed the effect of parameters such as channel Reynolds, particle size, etc. Therefore, there are very few examples presenting the whole lift force profiles acting on deformable particles such as droplets and bubbles and studying the effect of their corresponding parameters like Capillary number on the force values. For example, \citeauthor{chen2014inertial} have extensively studied the inertial migration of a deformable droplet in a rectangular microchannel, but their presented lift force profile only considers one value for particle Weber number (a measure for particle deformability) \cite{chen2014inertial}. \citeauthor{rivero2018bubble} have divided the underlying physics into different regimes. In the pure inertial regime, they have plotted the inertial lift on a rigid bubble at different Reynolds numbers, and in the pure Capillary regime where the inertial effects are absent, a lift profile is presented for different Capillary numbers \cite{rivero2018bubble}. However, their work lacks a similar profile visualizing the total lift force in the most general nonlinear inertial-capillary regime.
The obtained lift force profiles are mainly the result of either some experimental measurements \cite{di2009particle, stan2011sheathless, zhou2013fundamentals} or applying a feedback control in the numerical code to fix the position of particle \cite{raffiee2019numerical}, capsule \cite{schaaf2017inertial}, or drop \cite{chen2014inertial}. Nevertheless, in this paper, we present a method for lift force calculation at different Capillary numbers that solely depends on the trajectory of the drop. In addition, the importance of exploiting an oscillatory flow in the microchannel for working with sub-micron particles \cite{mutlu2018oscillatory}, having direct control over their focal points and tuning them depending on the flow oscillation frequency \cite{lafzi2020inertial}, and more effective separation and sorting strategies \cite{schonewill2008oscillatory} has already been presented in the literature. Thus, we will expand our lift force analysis to include both steady and oscillatory regimes at various Capillary numbers, where the latter is completely missing in the literature. We will then try to fit analytical expressions to the obtained lift profiles for different cases and present a scheme to predict this expression over a continuous range of input parameters.
\section{\label{sec:level1}Methodology\protect\\}
A single droplet with density and viscosity ratios of one is placed in a laminar flow of an incompressible Newtonian fluid in a microchannel as illustrated in Fig. \ref{geometry}. The drop dynamics is simulated using Front-tracking method \cite{unverdi1992front} as elaborated in detail in our previous work \cite{lafzi2021dynamics}. The pressure gradient in the $x$ direction has a constant magnitude of $P_0$ for the steady flow and a varying strength of ${P_0}cos({\omega}t)$ for the oscillatory flow. The periodic boundary condition is applied in the $x$ direction, and the no-slip condition is applied on the walls in the $y$ and $z$ directions. The axes of symmetry have been avoided. Parameters $W$ and $U_c$ (maximum velocity of the steady case) are used as the characteristic length and velocity, respectively. In other words, $x^*={\frac{x}{W}}$, $u^*={\frac{u}{U_c}}$, $t^*={\frac{t}{{\frac{W}{U_c}}}}$, $P^*={\frac{P}{\mu{\frac{U_c}{W}}}}$, $T^*={\frac{T}{{\frac{W}{U_c}}}}$ (where $T$ is the period), and $\omega^*={\frac{2\pi}{T^*}}$. Three dimensionless parameters describe the dynamics of the drop: (i) Reynolds number, $Re={\frac{{\rho}U_c2W}{\mu}}$, where $\rho$ and $\mu$ are the density and viscosity, respectively, (ii) Capillary number, $Ca={\frac{\mu U_c}{\gamma}}$, in which $\gamma$ is the surface tension, and (iii) the dimensionless oscillation frequency ($\omega^*$). The drop has a constant size of $\frac{a}{W}=0.3$ with a spherical initial shape, and $Re=10$ in our entire study. The numerical grid is generated using $196\times 114 \times 114$ cells in the $x$, $y$, and $z$ directions, respectively, and with 29578 triangular elements for the discretization of the drop.
\begin{figure}[h]
\centering
\includegraphics[height=3in]{problem_setup.png}
\caption{Schematic of the problem setup}
\label{geometry}
\end{figure}
The active dominant forces on the migrating drop in the wall-normal direction are the inertial and deformation-induced lift and lateral drag forces. The direction of the inertial lift at drop locations far from the wall is towards it, and the deformation lift pushes the drop towards the channel center \cite{zhang2016fundamentals}. The direction of the lateral drag is the negative of its migration velocity sign, assuming that the carrier fluid is stationary in the wall-normal direction \cite{zhang2016fundamentals}. Therefore, if we assume the positive direction to be the one from the center to the wall, the force balance on the drop according to Newton's second law is the following:
\begin{equation}\label{force balance}
F_{total}=F_{inertial}-F_{deformation}-F_{drag},
\end{equation}
The drag force is computed based on its definition \cite{ishii1979drag}:
\begin{equation}\label{drag force}
F_{drag}=\frac{1}{2}C_D\rho v_r|v_r|A_d,
\end{equation}
\begin{equation}\label{drag coeff}
C_D=\frac{24}{Re_{rel}}(1+0.1Re_{rel}^{0.75}),
\end{equation}
\begin{equation}\label{Re_rel}
Re_{rel}=Re\frac{v_r}{U_c}\frac{a}{W},
\end{equation}
Where $C_D$ is the drag coefficient, $v_r$ is the relative velocity between the drop and the fluid (which is essentially its migration velocity), and $A_d$ is the frontal projected area of the drop. Equation \ref{drag coeff} is consistent with the findings of \cite{snyder2007statistical, ishii1979drag, zhou2020analyses, kelbaliyev2007development, salibindla2020lift} and those of \cite{ceylan2001new} at a viscosity ratio of one. Although eq. \ref{drag coeff} is derived for steady flows, researchers have shown that the drag coefficient in unsteady flows depends heavily on an unsteady parameter that includes the density ratio \cite{shao2017detailed, aggarwal1995review}. Since the density ratio in the present study is one, the aforementioned unsteady parameter becomes zero, and hence, $C_D$ for unsteady flows (including oscillatory cases) can be approximated as the one for steady flows using this equation. The parameter $A_d$ is calculated based on the projected area of the drop on a plane having a normal vector parallel to its migration velocity. Thus, the value of this parameter varies at different instances. This procedure leads to a more precise computation of the drag force.
Considering a viscosity ratio of one, the deformation-induced lift for a drop that has a distance higher than its diameter from the closest wall leads to the following compact form \cite{chan1979motion, stan2013magnitude}:
\begin{equation}\label{deformation force}
F_{deformation} = 75.4Ca_p\mu V_{avg}a(\frac{a}{W})^2\frac{d}{W},
\end{equation}
Where $Ca_p=Ca{\frac{a}{W}}$ is the drop capillary number, $V_{avg}$ is the average velocity of the carrier fluid across the channel, and $d$ is the distance of the drop from the channel center. The linear dependency of this force with respect to the distance $d$ in the specified region is also confirmed in \cite{rivero2018bubble}.
The lift force analysis in this work is solely based on the drop trajectory. Therefore, to get a lift profile that spans a wide range of $d$, the drop is released from two different initial locations: \begin{itemize}
\item $y^*=0.46$ and $z^*=1$ (the upper release)
\item $y^*=0.98$ and $z^*=1$ (the lower release)
\end{itemize}
The upper release is chosen such that the whole range of studied $d$ falls within the validity domain of the deformation force equation (eq. \ref{deformation force}). This enables us to plug eq. \ref{deformation force} into the force balance equation (eq. \ref{force balance}) to get the inertial force once the total force is calculated as elaborated below. We will compare the inertial force at different $Ca$ values for the steady flows in the results section. The lower release initial location is slightly off from the channel center since it is also an equilibrium point, and if a drop is placed there, it does not move at all \cite{chen2014inertial}. The initial $z$ component for both releases is on the main axis for faster convergence since the drop eventually focuses on the main axes according to our previous work \cite{lafzi2021dynamics}. These different initial locations do not alter the drop equilibrium position \cite{pan2016motion, lan2012numerical, chaudhury2016droplet, razi2017direct}. The results of each parameter computation for both releases will be combined to reflect its overall behavior within the channel cross-section.
The migration velocity and acceleration of the drop is calculated by taking the first and second temporal derivatives from its trajectory numerically. Since time-step varies throughout the simulations to keep the Courant–Friedrichs–Lewy number at 0.9, the following equations are used to obtain the corresponding derivatives \cite{sundqvist1970simple}:
\begin{equation}\label{first derivative}
v_r=\dot{d}_i \approx \frac{-h_i}{(h_{i-1})(h_i + h_{i-1})}d_{i-1} + \frac{h_i - h_{i-1}}{h_i h_{i-1}}d_i + \frac{h_{i-1}}{(h_i)(h_i + h_{i-1})}d_{i+1},
\end{equation}
\begin{equation}\label{second derivative}
\frac{dv_r}{dt}=\ddot{d}_i \approx \frac{2\left[ d_{i+1} + \frac{h_i}{h_{i-1}}d_{i-1} -(1 + {\frac{h_i}{h_{i-1}}} d_i) \right]}{h_i h_{i-1} (1 + {\frac{h_i}{h_{i-1}}})},
\end{equation}
In which $h_i = t_{i+1} - t_i$, $h_{i-1} = t_i - t_{i-1}$, and $d_i$ and $t_i$ denote the distance from center and time at the current step. Both non-uniform finite difference schemes have a second-order accuracy.
Taking the first derivative from the steady flow trajectory at the lowest $Ca$ ($Ca=0.09$) results in a very noisy curve that is impossible to interpret. Therefore, we use an accurate non-linear regression by minimizing the sum of squared errors to fit the trajectories with analytical expressions, from which we can take first and second derivatives analytically. The trajectory from the upper release is very similar to an exponential decay. Therefore, we fit a curve with the following form to it:
\begin{equation}\label{upper release trajectory}
d(t) \approx ce^{bt} + k,
\end{equation}
Where $c$, $b$, and $k$ are all constants that should be determined after completing the optimization. The constant $k$ is essentially the drop equilibrium distance from the center. The trajectory from the lower release looks like the sigmoid logistic function. Consequently, we use the following equation as its analytical general form:
\begin{equation}\label{lower release trajectory}
d(t) \approx \frac{c}{1+e^{-b(t-k)}} + offset,
\end{equation}
Where again, $c$, $b$, $k$, and $offset$ are the regressor constants. The regression fits to both trajectories from the upper and lower release have very high $R^2$ scores of 0.99 as plotted in fig. \ref{Ca=0.09 regression fits}. This figure further confirms that the drop focuses at the same $d^*$ regardless of its initial location.
\begin{figure}[h]
\centering
\includegraphics[width=4.3in]{Ca=0.09_steady.png}
\caption{Regression fits to both steady trajectories at $Ca=0.09$}
\label{Ca=0.09 regression fits}
\end{figure}
Once the migration acceleration is derived following the aforementioned steps, it will be multiplied by $\frac{4}{3}\pi a^3 \rho$, which is the total constant mass of the drop with the initial spherical shape, to get the total force. By subtracting the calculated drag force from the total force, the total lift force can be obtained.
\section{\label{sec:level1}Results and Discussion\protect\\}
In this section, we report the results of a single deformable droplet simulations in the previously introduced microchannel that contains both steady and oscillatory carrier fluid. As we are interested in studying the effects of oscillation frequency and Capillary number on the lift force, we fix the $Re$ at a value of $10$. $Ca$ ranges between $0.09$ and $1.67$, and for oscillatory cases, $\omega^*$ values are chosen such that for a channel with a cross-section of $100 \mu m$ and water as the working fluid at room temperature, the frequency ranges between 2Hz and 200Hz, which is mostly referred to in the literature \cite{dincau2020pulsatile}. The validation of our numerical framework, as well as grid and domain independence studies, are discussed in detail in our previous work \cite{lafzi2021dynamics}.
Figure \ref{flow_rates} illustrates the dimensionless mass flow rate over dimensionless time. It can be seen that while the steady regime has the largest constant flow rate in a single direction, the average of oscillatory flow rates in each half of a periodic cycle decreases by increasing the frequency \cite{lafzi2020inertial, lafzi2021dynamics}. Although the average of a sinusoidal function in half of a period is constant regardless of its oscillation frequency ($\frac{1}{\frac{\pi}{\omega}}\int_{0}^\frac{\pi}{\omega} sin({{\omega}t})dt=\frac{2}{\pi}$), the lower maximum absolute value of the flow rate at higher frequencies is the chief reason for the observed phenomenon.
\begin{figure}[h]
\centering
\includegraphics[width=5in]{flowrates_Ca=167.png}
\caption{Flow rate versus time at $Ca=1.67$ and $Re=10$}
\label{flow_rates}
\end{figure}
Figure \ref{area_Ca=1.67} visualizes the dimensionless time-dependent frontal projected area of the droplet (parameter $A_d$ in equation \ref{drag force}) as it migrates toward its lateral equilibrium position traveling both upper and lower-release trajectories. The first thing we note is that in the transient stage before focusing, the drop has a higher average projected area while traveling the upper trajectory (fig. \ref{area_upper}) compared to the one in the lower trajectory (fig. \ref{area_lower}) in each of the flow regimes correspondingly. This is because the drop experiences more shear and deforms easier when traveling the upper trajectory. Moreover, in each subfigure, the average of $A^*$ is lower at a higher frequency since the average deformation parameter decreases by increasing the frequency \cite{lafzi2021dynamics}. It is important to note that the minimum projected area of the drop in the steady flow is its initial value when the drop is still undeformed and has a spherical shape; in the oscillatory cases, this minimum value occurs when the direction of the flow changes in each periodic cycle. Also, as expected, the drop at higher $Ca$ deforms more and has a higher projected area. This is why $Ca=1.67$ is used for visualization here among all the other cases in the present study.
\begin{figure}
\centering
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{area_lower_Ca=167.png}
\caption{}
\label{area_lower}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{area_upper_Ca=167.png}
\caption{}
\label{area_upper}
\end{subfigure}
\caption{Transient cross-stream frontal projected area of the drop when it is released from a (a) lower initial location, and an (d) upper initial location}
\label{area_Ca=1.67}
\end{figure}
Figure \ref{all_steady_total_lift} demonstrates the dimensionless total lift coefficient as a function of the dimensionless distance of the drop from the channel center in the steady flows and at different $Ca$. Similar to \cite{chen2014inertial}, all of the lift coefficients in this work are obtained by dividing the derived lift force, according to the introduced methodology in the previous section, by a factor of $\frac{\pi}{8}\rho{V^2_{avg,s}}(2a)^2$, in which $V_{avg,s}$ is the average of flow velocity across the channel cross-section in the steady flow. As expected, we observe that each lift curve has a stable equilibrium point at the corresponding drop focal point. Furthermore, at each $Ca$, the maximum positive total lift occurs when the drop migration velocity is also maximum. This maximum value is the highest at the lowest $Ca$. In addition, the maximum negative total lift is at the initial location of the upper trajectory, and its absolute value is the highest for higher $Ca$ except for $Ca=1.67$. This is because as we go further up from the channel center and the drop focal point, the deformation lift becomes the dominant force. According to equation \ref{deformation force}, this force is larger at higher $Ca$. Also, the negative lift sign in this region is due to the direction of the deformation-induced force, which is toward the center. The drop at $Ca=1.67$ is released from an initial location closer to the center compared to other cases because it has the highest deformability among all. When it was released from the same location as that of the others, it experienced an extremely large deformation that led to its break up. Therefore, the selected initial point for $Ca=1.67$ is the furthest possible one from the center that results in the largest possible deformation of the drop throughout its upper trajectory without its break up. Consequently, since the drop in this case starts to travel from a closer distance from the center, it has a lower maximum negative total lift compared to $Ca=1$ and $Ca=0.5$ (please refer to eq. \ref{deformation force} that shows the dependence of the deformation force on the drop distance from the center).
\begin{figure}[h]
\centering
\includegraphics[width=4.3in]{all_steady_total_lift.png}
\caption{Total lift coefficient for the steady flows at different $Ca$}
\label{all_steady_total_lift}
\end{figure}
Since the principal hypothesis underlying equation \ref{deformation force} is that the wall effect is negligible due to the large distance of the drop from it \cite{zhang2016fundamentals, ho1974inertial, jahromi2019improved}, we can assume that the shear-gradient force is the dominant component of the inertial lift in our study. After subtracting the calculated deformation lift based on this equation from the obtained total lift (fig. \ref{all_steady_total_lift}), we can derive the inertial lift, as shown in fig. \ref{all_steady_inertial_lift}. We see that the inertial lift coefficient increases as we increase the $Ca$ number. This could make sense as the more deformed shape of the drop can help further increase the difference between the relative velocities of the fluid with respect to the drop on the channel wall and center sides, which is the chief reason for the shear-gradient force existence \cite{zhang2016fundamentals}. According to eq. \ref{deformation force}, the deformation force is a linear function of the drop distance from the center and is larger for higher values of $Ca$. Because of this trivial conclusion, a plot of this force is not depicted here.
\begin{figure}[h]
\centering
\includegraphics[width=4.3in]{all_steady_inertial_lift.png}
\caption{Inertial lift coefficient for the steady flows at different $Ca$}
\label{all_steady_inertial_lift}
\end{figure}
Total lift curves acting on the drop in steady and different oscillatory flows at a few $Ca$ numbers are expressed in fig. \ref{all_total_lift}. In each subfigure, the higher the drop migration velocity, the larger are both the amplitude of oscillations and the distance between two corresponding points (e.g. maximum or minimum in the oscillatory cycle) on two consecutive periodic cycles. Hence, similar to steady regimes, the maximum absolute values of oscillatory lift coefficients occur when the drop migration velocities are maximum as well. Similarly, the lift oscillations around the drop focal point and near the lower initial point are lower because the drop migration velocities are minimum at those locations.
\begin{figure}
\centering
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_total_lift_Ca=167.png}
\caption{}
\label{all_total_lift_Ca=167}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_total_lift_Ca=1.png}
\caption{}
\label{all_total_lift_Ca=1}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_total_lift_Ca=05.png}
\caption{}
\label{all_total_lift_Ca=0.5}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_total_lift_Ca=033.png}
\caption{}
\label{all_total_lift_Ca=0.33}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_total_lift_Ca=025.png}
\caption{}
\label{all_total_lift_Ca=0.25}
\end{subfigure}
~
\caption{Total lift coefficients for steady and oscillatory flows with different frequencies at (a) $Ca=1.67$, (b) $Ca=1$, (c) $Ca=0.5$, (d) $Ca=0.33$, and (e) $Ca=0.25$}
\label{all_total_lift}
\end{figure}
The moving averages of the total lift coefficients in fig. \ref{all_total_lift} are plotted in fig. \ref{all_averaged_lift}. The selected $d^*_{avg}$ while computing the average of lift in each corresponding oscillatory cycle is chosen to be the middle value of $d^*$ in that period. Therefore, since oscillatory cycles with lower frequencies have longer periods, the averaged lift curves at lower frequencies cover a shorter length of $d^*_{avg}$ (a later beginning and a sooner ending). We first note that the obtained averaged lift curves for the oscillatory flows are not necessarily as smooth as that of the steady lift at the corresponding $Ca$. This observation becomes more pronounced as we increase $\omega^*$ or decrease $Ca$. Nevertheless, these fluctuations on the curves are negligible compared to those of the original oscillatory lifts (fig. \ref{all_total_lift}). Additionally, although the average of inertial and deformation-induced lift forces decrease separately by increasing $\omega^*$ \cite{lafzi2021dynamics}, the difference between them (fig. \ref{all_averaged_lift}) does not follow the same pattern. This confirms the existence of a drop focal point with an extremum distance from the channel center at an intermediate frequency, as elaborated in our previous work \cite{lafzi2021dynamics}. Despite this, the average of lift is the largest in the steady flow and the smallest in the oscillatory flow with the highest frequency at each $Ca$ in our study.
\begin{figure}
\centering
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_averaged_lift_Ca=167.png}
\caption{}
\label{all_averaged_lift_Ca=167}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_averaged_lift_Ca=1.png}
\caption{}
\label{all_averaged_lift_Ca=1}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_averaged_lift_Ca=05.png}
\caption{}
\label{all_averaged_lift_Ca=0.5}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_averaged_lift_Ca=033.png}
\caption{}
\label{all_averaged_lift_Ca=0.33}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{all_averaged_lift_Ca=025.png}
\caption{}
\label{all_averaged_lift_Ca=0.25}
\end{subfigure}
~
\caption{Averaged total lift coefficients for steady and oscillatory flows with different frequencies at (a) $Ca=1.67$, (b) $Ca=1$, (c) $Ca=0.5$, (d) $Ca=0.33$, and (e) $Ca=0.25$}
\label{all_averaged_lift}
\end{figure}
By taking another close look at fig. \ref{all_total_lift}, we realize that each of the oscillatory total lift coefficients can be fitted using an expression that comprises of a base curve, which can be best fitted by a 4th order polynomial, combined with the absolute value of some sinusoidal oscillations. Both the amplitude of oscillations and oscillatory periods can be controlled by the drop migration velocity and have direct relationship with it. In other words, the proposed expression can have the following form:
\begin{equation}\label{regression_curve}
f_{tot}\approx\left\{m+nv(t)\left|cos\left(\frac{c}{av(t)}d(t)+b\right)\right|\right\}\left\{gd^4(t)+hd^3(t)+kd^2(t)+ld(t)+q\right\}
\end{equation}
Where $d(t)$ and $v(t)$ are the time-dependent drop distance from the channel center and its migration velocity, respectively, and $m$, $n$, $a$, $b$, $c$, $g$, $h$, $k$, $l$, and $q$ are the constants to be determined while performing the optimization. The constant $m$ is just placed for achieving higher accuracy for the fits, and it ends up to be almost zero compared to other constants in the expression. The resultant curves are plotted in fig. \ref{all_lift_regression_Ca=167} against their corresponding data (fig. \ref{all_total_lift_Ca=167}) for $Ca=1.67$ and $\omega^*=0.01$, $\omega^*=0.1$, $\omega^*=0.5$, and $\omega^*=1$ with $R^2$ scores of 0.99, 0.99, 0.99, and 0.97, respectively. Similar curves having the same proposed expression with $R^2$ scores of 0.97 or higher and capable of capturing all the infinitesimal details are obtained for other $Ca$ numbers as well.
\begin{figure}
\centering
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{lift_regression_Ca=167_w=0.01.png}
\caption{}
\label{lift_regression_Ca=167_w=0.01}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{lift_regression_Ca=167_w=0.1.png}
\caption{}
\label{lift_regression_Ca=167_w=0.1}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{lift_regression_Ca=167_w=0.5.png}
\caption{}
\label{lift_regression_Ca=167_w=0.5}
\end{subfigure}
~
\begin{subfigure}[ht]{.488\textwidth}
\centering
\includegraphics[height=.225\textheight]{lift_regression_Ca=167_w=1.png}
\caption{}
\label{lift_regression_Ca=167_w=1}
\end{subfigure}
\caption{The fitted total lift coefficients to the ground-truth data using the introduced non-linear regression at $Ca=1.67$ and (a) $\omega^*=0.01$, (b) $\omega^*=0.1$, (c) $\omega^*=0.5$, and (d) $\omega^*=1$}
\label{all_lift_regression_Ca=167}
\end{figure}
To further extend the total lift prediction to more general cases within a continuous range of $Ca$ and $\omega^*$, we consider the steady and averaged oscillatory lifts (fig. \ref{all_averaged_lift}) for regression with a 4th order polynomial here. In other words, the expression in the first bracket of eq. \ref{regression_curve} is replaced with a value of 1. Using the analytical set of equations \ref{equations_for_coeffients_determination}, we can derive the unknown coefficients $a$, $b$, $c$, $g$, and $h$ of the polynomial analytically. In these equations, the subscripts for $d$ and $f$ denote their locations. For instance, the subscript max represents where the magnitude of total (or averaged) lift is maximum and its first derivative is zero. The obtained polynomials with this approach have $R^2$ scores of around 0.9 or higher for the cases presented in fig. \ref{all_averaged_lift}.
\begin{equation}\label{equations_for_coeffients_determination}
\begin{cases}
ad_{first}^4+bd_{first}^3+cd_{first}^2+gd_{first}+h=f_{first},\\
ad_{max}^4+bd_{max}^3+cd_{max}^2+gd_{max}+h=f_{max},\\
4ad_{max}^3+3bd_{max}^2+2cd_{max}+g=0,\\
ad_{eq}^4+bd_{eq}^3+cd_{eq}^2+gd_{eq}+h=0,\\
ad_{last}^4+bd_{last}^3+cd_{last}^2+gd_{last}+h=f_{last},
\end{cases}
\end{equation}
Parameters $d_{first}$, $d_{max}$, $d_{eq}$, $d_{last}$, $f_{first}$, $f_{max}$, and $f_{last}$ are already available for the cases in fig. \ref{all_averaged_lift} to solve the system of equations \ref{equations_for_coeffients_determination} for them. However, we use the multi-fidelity Gaussian processes (MFGP) method to predict these unknown parameters for any given double inputs of $0.25\leq Ca\leq1.67$ and $0\leq\omega^*\leq1$. MFGP is a Bayesian stochastic approach that does a casual inference on a set of high and low-fidelity datasets, and it is extremely effective if there are strong correlations between them \cite{perdikaris2017nonlinear}. This method is described in detail in our previous work, and it is carried out to predict the distance of the drop equilibrium position from the channel center $(d_{eq})$ with $R^2$ of 0.99 and root mean squared error (RMSE) of 0.01 in that work \cite{lafzi2021dynamics}. Here, we refer to the data for the cases in fig. \ref{all_averaged_lift} except for $Ca=1$ as our high-fidelity data. We generate similar data for all the cases in that figure, but with a grid of $128\times 76 \times 76$ in the $x$, $y$, and $z$ directions, respectively, and having 13038 triangular elements for the discretization of the drop. We consider this data as our low-fidelity dataset. Therefore, we have a total of 25 low and 20 high-fidelity data points, which satisfies the required nested structure to apply MFGP on the data \cite{perdikaris2017nonlinear}. We randomly allocate 5 data points of the entire high-fidelity dataset as our test set since the high-fidelity response is our main target. We train the algorithm on the remaining 40 training data points and evaluate its performance on the test set. We repeat this procedure 30 times and compute the average of evaluation metrics so that the selection of test sets does not significantly affect the overall algorithm performance.
Table \ref{MFGP evaluation} presents the average of $R^2$ and RMSE on our 6 remaining unknown parameters after completing the aforementioned steps. We can see that the trained algorithm is capable of predicting the intended parameters with very high accuracies. Especially, the accurate prediction of $f_{max}$ and $f_{last}$ is useful for determining the maximum and minimum values of averaged total lift for any given input in the range, respectively. The slightly less accurate prediction for $d_{max}$ (i.e. where the maximum averaged lift occurs) is because of the present randomness in its values among different cases. This is unlike the consistent pattern that exists for other parameters for a combination of $\omega^*$ values across different $Ca$ numbers. The similar lower accurate prediction for $d_{last}$ is due to the lack of data points between $Ca=1$ and $Ca=1.67$ since all the cases with $Ca\leq1$ have the same upper release initial location. However, the $R^2$ score of around 0.8 for this prediction is still high, and it can help us determine the furthest starting point from the center and the widest traveling region of a droplet with $1\leq Ca \leq1.67$ in the microchannel so that it undergoes the largest possible deformation without breaking up.
\begin{table}[h]
\centering
\begin{tabular}{|P{4.12cm}|P{4.12cm}|P{4.12cm}|}
\hline
\textbf{Parameter} & \boldsymbol{$R^2$} & \textbf{RMSE} \\ \hline
$d_{max}$ & 0.79 & 0.0140\\ \hline
$d_{last}$ & 0.78 & 0.0213 \\ \hline
$d_{first}$ & 0.97 & 0.0007 \\ \hline
$f_{last}$ & 0.92 & 0.0177 \\ \hline
$f_{max}$ & 0.99 & 0.0015 \\ \hline
$f_{first}$ & 0.99 & 0.0003 \\
\hline
\end{tabular}
\caption{MFGP averaged performance metrics on 30 randomly chosen test sets for the parameters required for the determination of the analytical averaged total lift polynomial coefficients}
\label{MFGP evaluation}
\end{table}
\section{Conclusions}
The dynamics of particles and biological cells in microchannels has caught many researchers' attention because of several biomicrofluidic applications it has. The underlying physics owes its behavior mainly to the presence of different lift forces in such channels. Hence, many scientists have dedicated their time to calculate or measure these forces. However, most of these works have focused on analyzing the lift forces acting on solid and non-deformable particles and studied the effects of parameters such as particles' size, Reynolds number, etc on them. Consequently, such analysis on deformable droplets or bubbles and studying the effects of varying parameters like Capillary number is almost missing in the literature. In this work, we have extended such analysis to the case of a single deformable droplet in the channel. We have calculated the main components of the lift force based on a unique methodology that merely depends on the drop trajectory. To do so, first, the drop migration velocity and its frontal projected area as it travels its lateral trajectory have been computed to calculate the drag force in the wall-normal direction accurately. After applying Newton's second law on the drop, the total lift profile is obtained over a region where the drop has a distance higher than its diameter from the wall. It has been observed that the total lift has a higher maximum at a lower Capillary, and its minimum decreases as we increase the $Ca$. The inertial and deformation-induced lift forces both increase by increasing the $Ca$ number. Moreover, since the oscillatory flows within the microchannel were previously shown to enable working with sub-micron biological particles as well as introducing new focal points for them, we have also included these flow regimes in our analysis and investigated the effects of oscillation frequency on the lift in addition to the Capillary number. We have seen that for all cases, the total lift and for oscillatory ones, the amplitude of oscillations are both higher when the drop migration velocity is higher. At each $Ca$, the steady lift and moving averages of oscillatory ones at different $\omega^*$ have also been compared. It has been shown that the steady lift has the largest magnitude, and the average of oscillatory one with the highest frequency in this study has the smallest strength. However, there is not a constant decreasing pattern in the average of lift by increasing the frequency, which is why the drop focuses furthest from the channel center at an intermediate $\omega^*$. Additionally, an accurate mathematical expression has been proposed that captures the detailed total oscillatory lift curves at various $\omega^*$ with $R^2$ scores of 0.97 or higher. Finally, the multi-fidelity Gaussian processes has been used to accurately predict the 7 unknown parameters required to define a simple 4th order polynomial to fit the steady and averaged oscillatory lifts with $R^2$ scores of about 0.9 or higher for any given $Ca$ and $\omega^*$ within the ranges of $0.25\leq Ca\leq1.67$ and $0\leq\omega^*\leq1$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,423 |
You are here: Home / Uncategorized / Don't Try Divorce Alone!
Divorce is an incredibly difficult journey – no bones about it; you may be wracked with grief, stress, guilt and fear. Trying to navigate your divorce and co-parenting all alone is like rock climbing without having someone on the other end of your rope to catch you if you fall. Most definitely NOT advisable!
When we are scared and lonely, we tend to turn to our loved ones and friends for support. We vent to them. We cry on their shoulders. We count on them and often want them to agree with us and seek their advice and guidance.
Click on the link HERE to watch a video on why that's not such a great idea! | {
"redpajama_set_name": "RedPajamaC4"
} | 668 |
Slowing down the tempo of a fast moving tune can be a very helpful technique for learning the tune. Here are two software products designed for that purpose.
Most Digital Audio Workstations include a feature to slow down music without altering the pitch. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,562 |
\section{Introduction}\label{s1}
Integrability has proved to be a very powerful tool in quantum field theory in $1+1$ dimensions. It allows for the exact determination of the S-matrix of a theory based on the symmetries of the underlying QFT. Often the symmetries that arise are modified by the fact that quantum operators have non-trivial exchange properties as a consequence of working in one spatial dimension. Hence in many situations symmetry groups are deformed into quantum groups, which are particular deformations of the universal enveloping algebra of the original Lie algebra.
Of its many applications one of the most striking is to the world-sheet theory of the string moving in space times appearing in the AdS/CFT correspondence. The classic example is the case of the string in AdS$_5 \times S^5$ \cite{Beisert:2010jr,Arutyunov:2009ga}. One of the most practical approaches to quantizing this world-sheet theory has been the use of the BMN light-cone gauge. However, the integrable QFT that arises with this gauge fixing is not relativistically invariant---the energy and momentum do not satisfy the usual relativistic dispersion relation. This lack of relativistic invariance presents considerable complications to the application of the integrable tool box of exact S-matrix theory and the Bethe Ansatz.
We now know that the light-cone gauge-fixed theory, with its coupling constant $g$ (the 't ~Hooft coupling of the dual gauge theory), lies in a larger class of S-matrix theories with a new coupling $k$ \cite{Hoare:2011wr,Hoare:2012fc,Arutyunov:2012zt,Arutyunov:2012ai}, based on the R-matrix associated to the $q$ deformation of the light-cone symmetry algebra \cite{Beisert:2008tw,Beisert:2010kk,Beisert:2011wq,deLeeuw:2011jr}. The string world-sheet theory is obtained in the limit $k\to\infty$ with fixed $g$. Another interesting limit is obtained by taking $g\to\infty$ with fixed $k$, in which case the dispersion relation becomes relativistic and the S-matrix is identified with the S-matrix of a generalized sine-Gordon theory whose classical equation-of-motion is identical to the Pohlmeyer reduction of the equations-of-motion on the string world-sheet \cite{Pohlmeyer:1975nb,Grigoriev:2007bu,Mikhailov:2007xr,Grigoriev:2008jq,Miramontes:2008wt,Hollowood:2009tw,Hollowood:2009sc,Hollowood:2010dt,Hollowood:2011fq}. This generalized sine-Gordon theory involves a WZW model in the bosonic sector and this suggests that $k$ should be a positive integer \cite{Witten:1983ar}. The theories with general $(g,k)$ and their particular limits are illustrated in Figure \ref{f1}. We will argue that a special r\^ole is played by the theories with integer $k$ since these are the ones with a spectrum that naturally truncates. It is very likely that consistent theories only exist with integer $k$.
\FIG{
\begin{tikzpicture} [scale=1.2]
\draw[thick] (0,0) -- (0,5) -- (5,5) -- (5,0) -- (0,0);
\node at (-0.4,0) (q1) {$0$};
\node at (-0.4,5) (q2) {$\infty$};
\node at (-0.4,2.5) (q3) {$g$};
\node at (4.5,5.4) (q1) {$3$};
\node at (3.5,5.4) (q2) {$4$};
\node at (0,5.4) (q2) {$\infty$};
\node at (2.5,5.4) (q1) {$5$};
\node at (1.7,5.4) (q2) {$6$};
\node at (1.1,5.4) (q2) {$7$};
\node at (2.5,5.9) (a1) {$k$};
\node[rotate=90] at (-2,2.5) (a1) {world-sheet theory in $\text{AdS}_5\times S^5$};
\node[rotate=90] at (-1.5,2.5) (a1) {'t~Hooft coupling $\propto g^2$ ($k\to\infty$)};
\node at (2.5,7) (a1) {generalized sine-Gordon theory};
\node at (2.5,6.5) (a1) {relativistic, SUSY, WZW level $=k$ ($g\to\infty$)};
\draw[red] (0.05,5) -- (0.05,0);
\draw[red] (1.1,5) -- (1.1,0);
\draw[red] (1.7,5) -- (1.7,0);
\draw[red] (2.5,5) -- (2.5,0);
\draw[red] (3.5,5) -- (3.5,0);
\draw[red] (4.5,5) -- (4.5,0);
\filldraw[red] (0.3,2.5) circle (0.03cm);
\filldraw[red] (0.55,2.5) circle (0.03cm);
\filldraw[red] (0.8,2.5) circle (0.03cm);
\filldraw[red] (0.3,4.5) circle (0.03cm);
\filldraw[red] (0.55,4.5) circle (0.03cm);
\filldraw[red] (0.8,4.5) circle (0.03cm);
\filldraw[red] (0.3,0.5) circle (0.03cm);
\filldraw[red] (0.55,0.5) circle (0.03cm);
\filldraw[red] (0.8,0.5) circle (0.03cm);
\filldraw[black] (0.05,5) circle (0.05cm);
\filldraw[black] (1.1,5) circle (0.05cm);
\filldraw[black] (1.7,5) circle (0.05cm);
\filldraw[black] (2.5,5) circle (0.05cm);
\filldraw[black] (3.5,5) circle (0.05cm);
\filldraw[black] (4.5,5) circle (0.05cm);
\end{tikzpicture}
\caption{\small The red lines represent the $q=e^{i\pi/k}$-deformed world-sheet theories for fixed integer $k>2$ and varying $g$. The relativistic generalized sine-Gordon theories are obtained in the limit $g\to\infty$, for fixed integer $k$, while the original world-sheet theory is obtained in the limit $k\to\infty$, fixed $g$.}
\label{f1}
}
A general S-matrix for the basis states of the theory with general $(g,k)$ was written down in \cite{Hoare:2011wr} and initial investigations into its properties have been carried out in \cite{Hoare:2012fc,Arutyunov:2012zt,Arutyunov:2012ai}. It was pointed out in \cite{Arutyunov:2012zt} that the S-matrix theory is not manifestly unitary since its elements are not Hermitian analytic. Ordinarily, Hermitian analyticity, along with the quantum group braiding unitarity, is enough to imply QFT unitarity so a lack of Hermitian analyticity is a serious problem. In \cite{Miramontes:1999gd} it was observed that Hermitian analyticity (and therefore also QFT unitarity) is a basis-dependent identity and hence there is the possibility that there exists a new basis which makes it manifest. On the other hand, there is what seems like a separate issue concerned with how one deals with the representation theory of the quantum group when $k$ is an integer and the deformation parameter $q=e^{i\pi/k}$ is a root of unity. The latter issue is well known in the context of integrable lattice models and requires changing from the vertex picture where the states, in the case of an $\text{SU}(2)$ symmetry, transform in the doublet, to the Interaction-Round-a-Face (IRF), also know as the Solid-On-Solid (SOS), picture, where states correspond to kinks between vacua labelled by highest weight states, that is arbitrary spins: see Figure \ref{f2}.
\FIG{
\begin{tikzpicture}[scale=0.8]
\begin{scope}[scale=0.8]
\filldraw[black] (0,0) circle (4pt);
\draw (-1,-1) -- (1,1);
\draw (-1,1) -- (1,-1);
\node at (-1.5,-1.5) (a1) {$i$};
\node at (1.5,-1.5) (a1) {$j$};
\node at (-1.5,1.5) (a1) {$k$};
\node at (1.5,1.5) (a1) {$l$};
\draw[double,<->] (2.5,0) -- (4.5,0);
\end{scope}
\begin{scope}[xshift=6cm,scale=0.8]
\draw (-1,-1) -- (-1,1) -- (1,1) -- (1,-1) -- (-1,-1);
\node at (-1.5,0) (a1) {$a$};
\node at (0,-1.5) (a2) {$b$};
\node at (1.5,0) (a3) {$c$};
\node at (0,1.5) (a4) {$d$};
\end{scope}
\end{tikzpicture}
\caption{\small The vertex and IRF labels for the Boltzmann weights of an integrable lattice model. In the vertex picture the labels $i,j,k,l\in\{\pm\frac12\}$ are the weights of the spin $\frac12$ representation of $\text{SU}(2)$. In the IRF picture the labels $a,b,c,d\in\{0,\frac12,1,\frac32,\ldots\}$ are spins of arbitrary irreducible representations of $\text{SU}(2)$ with $|a-b|=|b-c|=|a-d|=|d-c|=\frac12$.}
\label{f2}
}
This vertex-to-IRF transformation was used in the context of S-matrix theory by Bernard and LeClair in the example of the sine-Gordon soliton S-matrix in \cite{LeClair:1989wy,Bernard:1990cw,Bernard:1990ys}. What is remarkable, and as far as we know unrecognised in the old quantum group S-matrix literature, is that the IRF form of the S-matrix is manifestly Hermitian analytic and consequently satisfies QFT unitarity.\footnote{Some relevant references for this old literature are
\cite{deVega:1990av,Hollowood:1992sy,Hollowood:1993fj,Delius:1995he,Gandenberger:1997pk,MacKay:1990mp}.} Whilst the $\text{SU}(2)$ example of Bernard and LeClair is simple enough that the original vertex form of the S-matrix can be rendered Hermitian analytic by a suitable conjugation, this does not work for $\text{SU}(N)$ generalizations and was an outstanding puzzle. Bernard and LeClair went on to argue that the IRF, or SOS, form for the S-matrix was the ideal basis for dealing with the complications of the representation theory of the $U_q(\mathfrak{su}(2))$ quantum group when $q$ is a root of unity which are summarized in Appendix~\ref{a5}. In fact, the IRF form of the S-matrix naturally projects out the ``bad" representations in the Hilbert space to leave a perfectly consistent S-matrix. In statistical mechanics language this restriction leads to the Restricted-Solid-On-Solid RSOS lattice models. In this work we will argue that the vertex-to-IRF transformation is exactly what is needed to make QFT unitarity manifest for the general $(g,k)$ theories and, at the same time, the IRF picture is perfectly adapted to implementing the RSOS-like restriction on the Hilbert space when $k$ is an integer. These reduced theories are likely to be the only consistent S-matrix theories with an finite spectrum.
Further evidence that the kink picture is the most natural basis comes from studying various limits of the S-matrix in the IRF picture. Firstly, we find that if we take the limit in which $k \rightarrow \infty$ (up to some minor subtleties) we recover the by now well-known light-cone string S-matrix \cite{Beisert:2005tm,Klose:2006zd}. This should be expected as this corresponds to the limit in which $q$, the quantum deformation parameter, goes to unity. More unexpected is what happens when we take the limit $g \rightarrow \infty$, which is meant to correspond to the Pohlmeyer-reduced theory. Expanding around large $k$ (again up to some subtleties) we find that the tree-level S-matrix should satisfy a modified classical Yang-Baxter equation. Furthermore this term in the expansion agrees with the perturbative computation of \cite{Hoare:2009fs}, explaining the apparent anomaly in this calculation.
The kink picture also arises naturally in the context of the Pohlmeyer-reduced theory, which is a generalized sine-Gordon theory whose Lagrangian action includes a WZW term in the bosonic sector. Its definition requires careful treatment due to the existence of field configurations (solitons) with non-trivial boundary conditions~\cite{Hollowood:2011fq}. Namely, it has to be defined on a world-sheet with boundary, and its consistency imposes quantization conditions on the boundary conditions themselves~\cite{Alekseev:1998mc,Gawedzki:2001ye,Elitzur:2001qd}, in addition to the well known quantization of the coupling constant~\cite{Witten:1983ar}.
The resulting picture is that soliton solutions are kinks that interpolate between a finite set of vacua labelled by highest weight states~\cite{Hollowood:2013oca}.
Before we proceed it is worthwhile stating the S-matrix axioms of relevance in the context of a non-relativistic theory. Due to integrability an $n$-body S-matrix element factorizes into 2-body building blocks for which the separate rapidities are preserved. However the S-matrix is a function of the rapidities separately $S(\theta_1,\theta_2)$ and not just of $\theta_1-\theta_2$ as in a relativistic theory. The rapidity is a familiar variable in $1+1$-dimensions determining the velocity via $v=\tanh\theta$, however the relativistic relations $E=m\cosh\theta$ and $p=m\sinh\theta$ are {\em not\/} satisfied.
The S-matrix elements generally satisfy two important identities that are known as crossing symmetry and ``braiding unitarity". It is important that the latter is not the same as ``QFT unitarity", which is the familiar requirement that the S-matrix is a unitary matrix for physical (real) rapidities. Assuming that particles come in multiplets whose states are labelled by $i$, crossing symmetry implies
\EQ{
S_{ij}^{kl}(\theta_1,\theta_2)={\cal C}_{kk'}S_{k'i}^{lj'}(i\pi+\theta_2,\theta_1){\cal C}^{-1}_{j'j}\ ,
\label{cross}
}
where ${\cal C}$ is the charge conjugation matrix. The braiding unitarity relation takes the form
\EQ{
\sum_{kl}S_{ij}^{kl}(\theta_1,\theta_2)S_{kl}^{mn}(\theta_2,\theta_1)=\delta_{im}\delta_{jn}\ ;
\label{bru}
}
however, this only implies QFT unitarity, that is
\EQ{
\sum_{kl}S_{ij}^{kl}(\theta_1,\theta_2)S_{mn}^{kl}(\theta_1,\theta_2)^*=\delta_{im}\delta_{jn}\ ,\qquad (\theta_i\text{ real})\ ,
\label{qft}
}
when the S-matrix satisfies Hermitian analyticity:
\EQ{
S_{ij}^{kl}(\theta_1^*,\theta_2^*)^*=S_{kl}^{ij}(\theta_2,\theta_1)\ .
\label{ha}
}
The other S-matrix axioms govern the structure of bound states and the bootstrap equations. For completeness we have included a discussion of the bootstrap programme in Appendix \ref{a4}. Overall, non-relativistic integrable S-matrix theory enjoys most of the properties of relativistic integrable S-matrix theory as we summarize in the check-list below:
\begin{center}
\begin{tabular}{lcc}
\toprule
&Relativistic & Non-relativistic \\
\midrule[\heavyrulewidth]
Factorization & {\Huge\Checkedbox} &{\Huge\Checkedbox}\\
Function of rapidity difference & {\Huge\Checkedbox} & {\Huge\Crossedbox}\\
Meromorphic function of rapidity & {\Huge\Checkedbox} & {\Huge\Crossedbox}\\
Hermitian analyticity & {\Huge\Checkedbox} & {\Huge\Checkedbox}\\
QFT Unitarity & {\Huge\Checkedbox} &{\Huge\Checkedbox}\\
Crossing & {\Huge\Checkedbox} & {\Huge\Checkedbox}\\
Bound-state poles &{\Huge\Checkedbox}&{\Huge\Checkedbox}\\
Bootstrap&{\Huge\Checkedbox}&{\Huge\Checkedbox}\\
\bottomrule
\end{tabular}
\end{center}
In the non-relativistic case the only difference is that S-matrix elements like $S(\theta_1,\theta_2)$ are not functions of the rapidity difference since there is no boost invariance and the S-matrix is not a meromorphic function on the complex rapidity plane; on the contrary, there are branch cuts~\cite{Hoare:2011wr}. However, all other properties hold just as in the relativistic case.
\section{Lessons from the Restricted Sine-Gordon Theory}\label{s2}
S-matrices for relativistic integrable quantum field theories are built out of solutions to the Yang-Baxter equation, for which quantum groups provide an algebraic framework. The simplest solution is related to the quantum group deformation of the affine (loop) Lie algebra $\mathfrak{su}(2)^{(1)}$. The basis states $\ket{\phi_m}$ transform in the spin $\frac12$ representation with $m=\pm\frac12$. If $V_{j}$ is the spin $j$ representation space then the two-body S-matrix, from which the complete S-matrix is constructed by factorization, is a map (known as an intertwiner)
\EQ{
S(\theta):\qquad V_{\frac12}(\theta_1)\otimes V_{\frac12}(\theta_2)\longrightarrow V_{\frac12}(\theta_2)\otimes V_{\frac12}(\theta_1)\ .
}
Here, we have indicated the rapidity of the states by $\theta_i$, and $\theta=\theta_1-\theta_2$. The S-matrix takes the form
\EQ{
S(\theta)=v(\theta)\check R(x(\theta))\ ,
}
where $x(\theta)=e^{\lambda\theta}$, $\check R(x)$ is the ``$R$-matrix" of the affine quantum group and $v(\theta)$ is a scalar factor, in current parlance the ``dressing phase", required to ensure all the necessary S-matrix axioms are satisfied. To make connection with~\eqref{cross}--\eqref{ha}, we define the S-matrix elements so that
\EQ{
\ket{\phi_i(\theta_1)}\otimes\ket{\phi_j(\theta_2)}\longrightarrow S_{ij}^{kl}(\theta)\, \ket{\phi_k(\theta_2)}\otimes \ket{\phi_l(\theta_1)}\,.
}
The non-trivial elements of the S-matrix are
\EQ{
\ket{\phi_{\pm\frac12}(\theta_1)\phi_{\pm\frac12}(\theta_2)}&\longrightarrow v(\theta)(qx-q^{-1}x^{-1})\,\ket{\phi_{\pm\frac12}(\theta_2)\phi_{\pm\frac12}(\theta_1)}\ ,\\
\ket{\phi_{\pm\frac12}(\theta_1)\phi_{\mp\frac12}(\theta_2)}&\longrightarrow v(\theta)(x-x^{-1})\,\ket{\phi_{\mp\frac12}(\theta_2)\phi_{\pm\frac12}(\theta_1)}\\ &~~~~~~~~~~~~~~~~~+
v(\theta)x^{\pm1}(q-q^{-1})\,\ket{\phi_{\pm\frac12}(\theta_2)\phi_{\mp\frac12}(\theta_1)}\ .
\label{wea}
}
There are consequently three basic processes; ``identical particle", ``transmission" and ``reflection" (although two of the latter):
\EQ{
S_I(\theta)&=\begin{tikzpicture}[baseline=-0.65ex,scale=0.4]
\filldraw[black] (0,0) circle (4pt);
\draw[->] (-0.6,-0.6) -- (0.6,0.6);
\draw[<-] (-0.6,0.6) -- (0.6,-0.6);
\node at (-1.3,1.3) (a1) {$\pm\frac12$};
\node at (1.3,-1.3) (a2) {$\pm\frac12$};
\node at (1.3,1.3) (a3) {$\pm\frac12$};
\node at (-1.3,-1.3) (a4) {$\pm\frac12$};
\end{tikzpicture}(\theta)=v(\theta)(xq-q^{-1}x^{-1})\ ,\\
S_T(\theta)&=\begin{tikzpicture}[baseline=-0.65ex,scale=0.5]
\filldraw[black] (0,0) circle (4pt);
\draw[->] (-0.6,-0.6) -- (0.6,0.6);
\draw[<-] (-0.6,0.6) -- (0.6,-0.6);
\node at (-1.3,1.3) (a1) {$\mp\frac12$};
\node at (1.3,-1.3) (a2) {$\mp\frac12$};
\node at (1.3,1.3) (a3) {$\pm\frac12$};
\node at (-1.3,-1.3) (a4) {$\pm\frac12$};
\end{tikzpicture}(\theta)=v(\theta)(x-x^{-1})\ ,\\
S_R^\pm(\theta)&=\begin{tikzpicture}[baseline=-0.65ex,scale=0.5]
\filldraw[black] (0,0) circle (4pt);
\draw[->] (-0.6,-0.6) -- (0.6,0.6);
\draw[<-] (-0.6,0.6) -- (0.6,-0.6);
\node at (-1.3,1.3) (a1) {$\pm\frac12$};
\node at (1.3,-1.3) (a2) {$\mp\frac12$};
\node at (1.3,1.3) (a3) {$\mp\frac12$};
\node at (-1.3,-1.3) (a4) {$\pm\frac12$};
\end{tikzpicture}(\theta)=v(\theta)x^{\pm1}(q-q^{-1})\ .
\label{wea2}
}
There are two theories whose S-matrices are built out of the $R$-matrix of the $\mathfrak{su}(2)$ quantum group. The first is associated to solitons of the sine-Gordon theory for which \cite{Jimbo:1985zk,Jimbo:1985vd,Jimbo:1987ra,Jimbo:1989qm,Bernard:1990cw,Bernard:1990ys}
\EQ{
x=e^{\frac{\theta}{k}}\ ,\qquad q=-e^{-i\pi/k}\ ,
\label{de1}
}
and the second to certain symmetric space sine-Gordon (SSSG) theories. In particular, we have in mind
the $S^5$ SSSG theory whose underlying quantum group symmetry
is $\mathfrak {so}(4) \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ and, accordingly, the S-matrix should factorize into two copies of the $R$-matrix of the $\mathfrak{su}(2)$ quantum group up to an overall phase.\footnote{The same $R$-matrix also describes the ${\mathbb C}P^3$ SSSG theory whose underlying symmetry is $\mathfrak{u}(2) \cong \mathfrak{su}(2) \oplus\mathfrak{u}(1)$\cite{Hollowood:2010rv}.} In this case, we have
\EQ{
x=e^{\frac{k+1}{k+2}\theta}\ ,\qquad q=e^{\frac{i\pi}{k+2}}\ .
\label{de2}
}
In both cases \eqref{de1} and \eqref{de2}
\EQ{
x(\theta)=-q^{-1}x(i\pi-\theta)^{-1}\ ,
}
which implies
\EQ{
S_I(\theta)=S_T(i\pi-\theta)\ ,\qquad S_R^+(\theta)=-q^{-1}S_R^-(i\pi-\theta)\ ,
}
as long as the dressing phase satisfies $v(\theta)=v(i\pi-\theta)$.
Crossing symmetry is then ensured by defining charge conjugation as
\EQ{
{\cal C}\ket{\phi_{\pm\frac12}(\theta)}=\pm i q^{\pm1/2}\ket{\phi_{\mp\frac12}(\theta)}\ .
\label{phicross}
}
The S-matrix satisfies the braiding relation
\EQ{
\sum_{kl}S_{ij}^{kl}(\theta)S_{kl}^{mn}(-\theta)=v(\theta)v(-\theta)(qx-q^{-1}x^{-1})(qx^{-1}-q^{-1}x)\delta_{im}\delta_{jn}\ .
}
Therefore, as long as the dressing phase satisfies
\EQ{
v(\theta)v(-\theta)=\frac1{(qx-q^{-1}x^{-1})(qx^{-1}-q^{-1}x)}\ ,
\label{phasecross}}
the S-matrix satisfies the braiding unitarity relation \eqref{bru}. However, this is not equivalent to QFT unitarity because as it stands the S-matrix written in this ``particle''-like basis does not satisfy Hermitian analyticity \eqref{ha} \cite{Miramontes:1999gd}. Whilst, given that the dressing factor satisfies $v(\theta^*)^*=-v(-\theta)$, the identical particle and transmission elements satisfy the required identity
\EQ{
S_I(\theta^*)^*=S_I(-\theta)\ ,\qquad S_T(\theta^*)^*=S_T(-\theta)\ ,
}
the reflection amplitudes are non-compliant because they satisfy
\EQ{
S_R^\pm(\theta^*)^*=S_R^\mp(-\theta)\ ,
}
rather than $S_R^\pm(\theta^*)^*=S_R^\pm(-\theta)$,
clearly violating \eqref{ha}.
In the case of $\mathfrak{su}(2)$, Hermitian analyticity can be restored by a simple rapidity-dependent transformation on the states of the form \cite{Jimbo:1985zk,Jimbo:1985vd,Jimbo:1987ra,Jimbo:1989qm,Bernard:1990cw,Bernard:1990ys}
\EQ{
\ket{\phi_{\pm\frac12}(\theta)}\longrightarrow x(\theta)^{\pm1/2}\ket{\phi_{\pm\frac12}(\theta)}\ .
}
This transformation removes the factors of $x^{\pm1}$ from the reflection amplitudes and restores Hermitian analyticity.\footnote{To ensure crossing symmetry charge conjugation needs to be modified so that ${\cal C} \ket{\phi_{\pm \frac12}} = \ket{\phi_{\mp \frac12}}$, in agreement with the original construction of \cite{Zamolodchikov:1978xm,Dorey:1996gd}
.}
It has an algebraic interpretation of moving from the homogeneous to the principal gradation of the affine algebra $\mathfrak{su}(2)^{(1)}$.
The new S-matrix is precisely the S-matrix of the solitons of the sine-Gordon theory. In is interesting to note that the same kind of transformation is not sufficient to restore Hermitian analyticity for the $\mathfrak{su}(n)$ generalization of the S-matrix and this deficiency of quantum group S-matrices was never resolved in the literature.
However, there is another way to restore Hermitian analyticity that does not involve changing the gradation and can be generalized to higher rank algebras \cite{Ahn:1990gn,Hollowood:1992sy}. This is the vertex-to-IRF transformation \cite{LeClair:1989wy,Bernard:1990cw,Bernard:1990ys}.
In the context of the sine-Gordon theory, the
transformation is a mathematical procedure that takes the S-matrix of one theory (the sine-Gordon theory) and produces the S-matrix of a new one (the restricted sine-Gordon theory). At this stage it is worth recalling that the 1-particle states in the sine-Gordon theory are labelled by $m=\pm\frac12$, which is a $U(1)$ topological charge, while in the restricted sine-Gordon theory the 1-particle states are labelled by a pair of spins $(j,j')$ with $|j-j'|=\frac12$ and $j,j'\in\{0,\frac12,\ldots,j_{\text{max}}\}$.
The vertex-to-IRF transformation involves two steps. The first one is simply a change of basis in the Hilbert space of multi-particle states
$\ket{\phi_{m_1}(\theta_1)\phi_{m_2}(\theta_2)\cdots\phi_{m_N}(\theta_N)}$ which transform in the tensor product representation ${V_{\frac12}}^{\otimes N}$ of the quantum group.
This is the ``vertex" basis. The new basis corresponds to decomposing the multi-particle states into irreducible representations. The group theory here is analogous to the decomposition of representations of $\mathfrak{su}(2)$ (at least when $q$ is generic). In the new basis, the $N$-soliton states that transform in the representation of total spin $J$, say $\ket{{\cal J},M}$, are labelled by the chain of the spins in the decomposition, ${\cal J}=(j_1\equiv J,j_2,\ldots,j_N\equiv\frac12)$ with $|j_{i-1}-j_i|=\frac12$, and by the $j_z$ quantum number $-J\leq M\leq+J$. In order to describe the change of basis we will need the $q$-deformed Clebsch-Gordan ($q$-CG) coefficients which we define by
\EQ{
\ket{J,M}=\sum_{m_1,m_2}\left[\ARR{J & j_1 & j_2\\
M & m_1 & m_2}\right]_q\ket{j_1,m_1}\otimes\ket{j_2,m_2}\ ,
\label{qCGor}
}
where the sum is taken with $M=m_1+m_2$ fixed. The $q$-CG coefficients that we need are given in Appendix \ref{A1}.
In terms of them, the change of basis takes the form
\EQ{
\ket{{\cal J},M;\{\theta_i\}}&=\sum_{\{m_i=\pm\frac12\}}\left[\ARR{j_1 & \tfrac12 & j_2\\
M & m_1 & M_2}\right]_q
\left[\ARR{j_2 & \frac12 & j_3\\
M_2 & m_2 & M_3}\right]_q\cdots\\[5pt]
&\hspace{1.5cm}
\cdots
\left[\ARR{j_{N-1} & \tfrac12 & \frac12\\
M_{N-1} & m_{N-1} & m_N}\right]_q \ket{\phi_{m_1}(\theta_1)\phi_{m_2}(\theta_2)\cdots\phi_{m_N}(\theta_N)}\ ,
\label{VtI}
}
where $M_{i}=M_{i+1}+m_i$, $M_1\equiv M$, $M_N\equiv m_N$, and the sum is over all the $\{m_i=\pm\frac12\}$ subject to $M=\sum_{i=1}^Nm_i$ being fixed.
The new basis can be interpreted in terms of the states
\EQ{
\ket{\Phi_{j_1j_2}(\theta)}^{M_1}{}_{M_2}=\sum_{m=\pm\frac12}\left[\ARR{j_1 & \tfrac12 & j_2\\
M_1 & m & M_2}\right]_q\ket{\phi_m (\theta)}\,,
\label{newop}
}
so that
\EQ{
\ket{(j_1,j_2,\ldots,j_N),M_1;\{\theta_i\}}&=
\ket{\Phi_{j_1j_2}(\theta_1)\Phi_{j_2j_3}(\theta_2)\cdots \Phi_{j_N0}(\theta_N)}^{M_1}{}_0\,,
\label{mpsL}
}
which involves an implicit sum over $M_2,M_3,\ldots, M_{N}$, and where the product satisfies the adjacency conditions $|j_i-j_{i+1}|=j_N=\frac12$.
A crucial part of the vertex-to-IRF transformation is the observation that, in this basis, the two-body S-matrix elements are given by
\EQ{
&\ket{\Phi_{j_{i-1}\, j_i}(\theta_{i-1})\,\Phi_{j_{i}\, j_{i+1}}(\theta_{i})}^{M_{i-1}}{}_{M_{i+1}} \\[5pt]
&
\qquad\qquad\longrightarrow
\sum_{j_i'}\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.4,0) (a1) {$j_{i-1}$};
\node at (0,-1.1) (a2) {$j_i$};
\node at (1.4,0) (a3) {$j_{i+1}$};
\node at (0,1.2) (a4) {$j_i'$};
\end{tikzpicture}(\theta)\,\,
\ket{\Phi_{j_{i-1}\, j'_i}(\theta_{i})\Phi_{j'_{i}\, j_{i+1}}(\theta_{i-1})}^{M_{i-1}}{}_{M_{i+1}} \,,
\label{smatL}
}
which is illustrated in Appendix \ref{A1}. Here, the box denotes a function of the rapidity difference $\theta=\theta_{i-1}-\theta_i$ and of the spins $j_{i-1}$, $j_i$, $j_{i+1}$ and $j_i'$ given by~\eqref{gtw}.
In other words, the two-body S-matrix elements are diagonal in the $j_z$ quantum numbers and completely independent of them.
This fact is a consequence of the quantum group invariance of the $R$-matrix, but at an explicit level it will be important later to notice that it depends on the specific identity
\EQ{
S_R^\pm(x)
+q^{\mp1}S_T(x)=S_I(x)\ ,
\label{kid}
}
that is
\EQ{
x^{\pm1}(q-q^{-1})+q^{\mp1}(x-x^{-1})=qx-q^{-1}x^{-1}\ .
\label{sid}
}
Eq.~\eqref{smatL} motivates the second step of the vertex-to-IRF transformation, which simply amounts to
being blind to the $j_z$ quantum numbers
\EQ{
\ket{\Phi_{j_1\, j_2}(\theta)}^{M_1}{}_{M_2} \longrightarrow \ket{K_{j_1\, j_2}(\theta)}\,.
}
The multi-particle states~\eqref{mpsL} corresponding to different values of $M_1$ are then identified with a single state
\EQ{
\ket{K_{j_1j_2}(\theta_1)K_{j_2j_3}(\theta_2)\cdots K_{j_N 0}(\theta_N)}\,.
\label{mkL}
}
This is the tensor product of $N$ one-particle states $\ket{K_{j j'}(\theta)}$ labelled by two spins with $|j-j'|=\frac12$ which are scalar with respect to the quantum group.
They can be naturally interpreted as describing a set of kinks in a theory with a degenerated set of vacua labelled by $j\in\{0,\tfrac12,1,\tfrac32,\ldots\}$ and, thus, associated to the highest weight representations of $U_q(\mathfrak{su}(2))$. Within this interpretation, $\ket{K_{j j'}(\theta)}$ corresponds to a kink between neighboring vacua, and~\eqref{mkL} to an $N$-kink state
between the vacua labelled by 0 and $j_1$.
In this IRF picture, it is natural to generalize the multi-kink states~\eqref{mkL} by allowing them to be built on an arbitrary vacuum on the right. The more general states are then
\EQ{
\ket{K_{j_1j_2}(\theta_1)K_{j_2j_3}(\theta_2)\cdots K_{j_N j_{N+1}}(\theta_N)}\,,
\label{mkL2}
}
which is an $N$-kink state
between the vacua labelled by $j_{N+1}$ and $j_1$.
In the context of~\eqref{VtI} and~\eqref{mpsL}, the vacuum labelled by $j_1$ has a
moduli space corresponding to the whole irreducible representation $V_{j_1}$, with basis $\ket{j_1,M_1}$ for $-j_1\leq M_1\leq +j_1$.
In a QFT in $1+1$ dimensions a continuous symmetry cannot be spontaneously broken and operationally this means that potential vacuum moduli spaces should actually be integrated over in the functional integral. In the present situation, since S-matrix elements do not depend on the points in the moduli spaces and, in particular, on the quantum number $M_1$, performing the integrals simply amounts to being blind to the quantum number $M_1$, which makes the vertex-to-IRF transformation natural.
The vertex-to-IRF transformation is particularly relevant in the case when $k$ is a positive integer and, hence,
$q$ is a root of unity. Then, the number of irreducible representations $V_j$ is bounded by $k$ and, moreover, the tensor product of two of them decomposes as a sum of both irreducible and {reducible but indecomposable} representations.\footnote{The main features of the representation theory of $U_q(\mathfrak{su}(2))$ are summarized in Appendix~\ref{a5}.}
For $q=e^{i\pi/k}$, the tensor product can be consistently restricted to the irreducible representations $V_j$ with $j=0,\tfrac12,1,\ldots, \tfrac{k}2-1$, so that it becomes~\eqref{ttp}. Correspondingly,
the S-matrix theory naturally preserves the sub-sector of states formed from kinks associated to the vacua labelled by the subset $j\in\{0,\frac12,\ldots,\frac{k}2-1\}$. This restriction corresponds to taking the restricted-SOS, or RSOS, models of statistical physics. In the context of the sine-Gordon theory the restricted model is known as the restricted $k/(k+1)$ sine-Gordon theory which involves $2(k-2)$ different elementary kinks.
According to~\eqref{smatL}, the two-body S-matrix elements in the IRF picture correspond to the pairwise processes
\EQ{
\ket{\cdots K_{j_{i-1}j_i}(\theta_{i-1})K_{j_ij_{i+1}}(\theta_i)\cdots}\longrightarrow
\ket{\cdots K_{j_{i-1}j'_i}(\theta_i)K_{j'_ij_{i+1}}(\theta_{i-1})\cdots}\ .
}
As we show in Appendix \ref{A1}, the explicit non-trivial elements of the S-matrix are
\EQ{
&\ket{K_{j\pm1,j\pm\frac12}(\theta_1)K_{j\pm\frac12,j}(\theta_2)}\longrightarrow S_I(\theta)\ket{K_{j\pm1,j\pm\frac12}(\theta_2)K_{j\pm\frac12,j}(\theta_1)}\ ,\\[5pt]
&\ket{K_{j,j\pm\frac12}(\theta_1)K_{j\pm\frac12,j}(\theta_2)}\longrightarrow\frac{\sqrt{[2j][2j+2]}}{[2j+1]}S_T(\theta)\ket{K_{j,j\mp\frac12}(\theta_1)K_{j\mp\frac12,j}(\theta_2)}\\[5pt]
&\hspace{131pt}+\frac{q^{2j+1}S_R^\mp(\theta)-q^{-2j-1}S_R^\pm(\theta)}{q^{2j+1}-q^{-2j-1}}
\ket{K_{j,j\pm\frac12}(\theta_1)K_{j\pm\frac12,j}(\theta_2)}\ .
\label{web}
}
In this formula, we have used the $q$-number
\EQ{
[n]=\frac{q^n-q^{-n}}{q-q^{-1}}\ .
\label{qnb}
}
and $\theta=\theta_1-\theta_2$.
We can summarize the complete S-matrix in the following way:\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$b$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$d$};
\end{tikzpicture}(\theta)
:\quad K_{ab}(\theta_1)+K_{bc}(\theta_2)\longrightarrow K_{ad}(\theta_2)+K_{dc}(\theta_1)\ ,
}
where the kinks $K_{ab}(\theta)$ are required to satisfy the adjacency condition $|a-b|=\frac12$. The S-matrix can then be written compactly as
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$b$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$d$};
\end{tikzpicture}(\theta)=S_I(\theta)\delta_{bd}+e^{i\pi(a+c-b-d)}S_T(\theta)\sqrt{\frac{[2b+1][2d+1]}{[2a+1][2c+1]}}\, \delta_{ac}\ .
\label{gtw}
}
Using this notation, it is worth noticing that the independence of the S-matrix element~\eqref{smatL} on the $j_z$ quantum numbers corresponds to the identity
\EQ{
&\sum_{\alpha,\beta} \left[\ARR{j_{i-1} & \tfrac12 & j_i\\
M_{i-1} & \alpha & M_i}\right]_q
\left[\ARR{j_i & \tfrac12 & j_{i+1}\\
M_i & \beta & M_{i+1}}\right]_q \, S_{\alpha\beta}^{\beta'\alpha'}(\theta)\\[5pt]
&\qquad
=\sum_{j'_i}
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$j_{i-1}$};
\node at (0,-1) (a2) {$j_i$};
\node at (1.3,0) (a3) {$j_{i+1}$};
\node at (0,1) (a4) {$j'_i$};
\end{tikzpicture}(\theta)
\left[\ARR{j_{i-1} & \tfrac12 & j'_i\\
M_{i-1} & \beta' & M_i'}\right]_q
\left[\ARR{j'_i & \tfrac12 & j_{i+1}\\
M_i' & \alpha' & M_{i+1}}\right]_q\ .
}
\noindent{\bf Crossing symmetry}
\nopagebreak
The S-matrix satisfies the following crossing symmetry relation
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$b$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$d$};
\end{tikzpicture}(\theta)=e^{i\pi(a+c-b-d)}\sqrt{\frac{[2b+1][2d+1]}{[2a+1][2c+1]}}\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$d$};
\node at (0,-1) (a2) {$a$};
\node at (1,0) (a3) {$b$};
\node at (0,1) (a4) {$c$};
\end{tikzpicture}(i\pi-\theta)\ ,
}
which is the correct crossing equation with charge conjugation defined as
\EQ{
{\cal C}\ket{K_{ab}(\theta)}=e^{i\pi(b-a)}\sqrt{\frac{[2a+1]}{[2b+1]}}\, \ket{K_{ba}(\theta)} \ .
}
This is consistent with the charge conjugation of $\ket{\phi_{\pm\frac12}(\theta)}$ \eqref{phicross} and the vertex-to-IRF transformation \eqref{VtI} as shown in Appendix \ref{A1}.
\noindent{\bf Hermitian analyticity}
\nopagebreak
The resulting S-matrix now satisfies the kink version of the Hermitian analyticity
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$b$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$d$};
\end{tikzpicture}(\theta^*)^*=
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$d$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$b$};
\end{tikzpicture}(-\theta)\ .
\label{kha}
}
Note that it is important that in each case the expressions inside the square roots in \eqref{web} and \eqref{gtw} are real and positive. In the kink basis the braiding unitarity relation takes the form
\EQ{
\sum_e
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$b$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$e$};
\end{tikzpicture}(\theta)
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$e$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$d$};
\end{tikzpicture}(-\theta)
=\delta_{bd}\ .
\label{bui}
}
Putting \eqref{kha} and \eqref{bui} together we find the kink version of the QFT unitarity condition \eqref{qft}
\EQ{
\sum_e
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$b$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$e$};
\end{tikzpicture}(\theta)
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1,0) (a1) {$a$};
\node at (0,-1) (a2) {$d$};
\node at (1,0) (a3) {$c$};
\node at (0,1) (a4) {$e$};
\end{tikzpicture}(\theta)^*
=\delta_{bd}\ ,\qquad (\theta\text{ real})\ .
\label{unit}
}
\noindent{\bf Yang-Baxter Equation}\nopagebreak
Finally, we briefly describe the Yang-Baxter equation in the IRF picture. As in the vertex picture it is easiest to represent it graphically, shown in Figure \ref{figybe}. The labels $a,b,c,d,e,f$ denote fixed vacua, while the internal vacuum $g$ should be summed over (in direct analogy with the sum of the indices labelling the internal lines in the usual Yang-Baxter equation). It can be checked that the kink S-matrix satisfies this version of the Yang-Baxter equation.\footnote{An alternative diagrammatical representation is given by the hexagon picture \cite{Baxter}, gotten from Figure \ref{figybe} by considering the dual graphs so that the vacua now label vertices.}
\begin{figure}
\begin{center}
\begin{tikzpicture} [scale=0.8,line width=1.5pt,inner sep=2mm,
place/.style={circle,draw=blue!50,fill=blue!20,thick}]
\begin{pgfonlayer}{top layer}
\node at (1.5,1.5) (pm1) {$S_{12}$};
\node at (1.8,3.4) (pm2) {$S_{23}$};
\node at (3.5,2.5) (m3) {$S_{13}$};
\node at (9.5,2.5) (m4) {$S_{13}$};
\node at (11.2,1.4) (m5) {$S_{23}$};
\node at (11.5,3.5) (m6) {$S_{12}$};
\end{pgfonlayer}
\begin{pgfonlayer}{foreground layer}
\node at (1.5,1.5) [place] (sm1) {\phantom{i}};
\node at (1.8,3.4) [place] (sm2) {\phantom{i}};
\node at (3.5,2.5) [place] (sm3) {\phantom{i}};
\node at (9.5,2.5) [place] (sm4) {\phantom{i}};
\node at (11.2,1.4) [place] (sm5) {\phantom{i}};
\node at (11.5,3.5) [place] (sm6) {\phantom{i}};
\end{pgfonlayer}
\node at (6,2.5) {$=$};
\node at (11,0) (i1) {};
\draw[->] (i1) -- (11.7,5);
\node at (12.5,0.5) (i2) {};
\draw[->] (i2) -- (8.25,3.45);
\node at (8.25,1.8) (i3) {};
\draw[->] (i3) -- (13,4.25) ;
\node at (5,1.7) (i4) {};
\draw[->] (i4) -- (0.4,4.2);
\node at (0.1,0.8) (i5) {};
\draw[->] (i5) -- (5,3.25);
\node at (1.25,0) (i6) {};
\draw[->] (i6) -- (2,4.8);
\node at (-1,2.5) {$\sum_g$};
\node at (7.2,2.5) {$\sum_g$};
\node at (0.6,2.5) {$a$};
\node at (2.3,2.5) {$g$};
\node at (4.6,2.5) {$d$};
\node at (8.6,2.5) {$a$};
\node at (10.7,2.5) {$g$};
\node at (12.2,2.5) {$d$};
\node at (0.8,0.5) {$b$};
\node at (11.5,0.5) {$c$};
\node at (2.7,1.3) {$c$};
\node at (9.8,1.3) {$b$};
\node at (1.3,4.5) {$f$};
\node at (12.2,4.5) {$e$};
\node at (3.0,3.7) {$e$};
\node at (10.2,3.7) {$f$};
\end{tikzpicture}
\caption{\small
The Yang-Baxter equation in the IRF picture. $a,b,c,d,e,f$ label fixed vacua, while the internal vacuum $g$ should be summed over.}
\label{figybe}
\end{center}
\end{figure}
In principle one could check the Yang-Baxter equation in the IRF picture for a large class of external vacua. However, the kink S-matrix \eqref{web} can be thought of as a $4 \times 4$ matrix, depending not only on the difference of rapidities, but also say on the right vacuum (labelled as $c$ in \eqref{gtw}). Relabelling the external vacuum $d$ as $j$, Figure \ref{figybe} can then be understood as a deformed version of the usual Yang-Baxter equation (see Figure \ref{f2a}), known as the dynamical Yang-Baxter equation \cite{Felder:1994be}
\EQ{
S_{12}(\theta_1,\theta_2,j+\tfrac12\hat h_3) &S_{13}(\theta_1,\theta_3,j)S_{23}(\theta_2,\theta_3,j+\tfrac12\hat h_1)=
\\& S_{23}(\theta_2,\theta_3,j)S_{13}(\theta_1,\theta_3,j+\tfrac12\hat h_2)S_{12}(\theta_1,\theta_2,j)\ .
\label{dybe}
}
The operator $\hat h_i$ acts on the kink with rapidity $\theta_i$ as follows:
\EQ{
\hat h_i \, \ket{K_{ab}(\theta_i)} = 2(a-b)\ket{K_{ab}(\theta_i)}\ .
}
Note that $\hat h$ always acts on the kink that is not involved in the relevant scattering process---in practice it accounts for the fact that the same two-particle S-matrix can have different right vacua in different terms of the Yang-Baxter equation.
It is of particular interest that the kink S-matrix satisfies a dynamical Yang-Baxter equation as the latter's semi-classical expansion leads to a deformed version of the standard classical Yang-Baxter equation \cite{Felder:1994be}. This appears to partially explain the apparent anomaly noticed in the perturbative computation of various symmetric space sine-Gordon model S-matrices \cite{Hoare:2010fb,Hoare:2011fj}, discussed in detail in section \ref{slim}.
\section{The \texorpdfstring{$\boldsymbol q$}{q}-Deformed World-Sheet S-matrix}\label{s3}
The S-matrix is constructed using a product of two copies of the fundamental $R$-matrix of the quantum group deformation of the triply extended superalgebra $\mathfrak h=\mathfrak{psu}(2|2)\ltimes{\mathbb R}^3$ in \cite{Beisert:2008tw}, with the central extensions identified. Concentrating on a single $R$-matrix factor each particle multiplet is $4$ dimensional with two bosonic and two fermionic states, denoted here as $\{|\phi_m\rangle,|\psi_m\rangle\}$, where $m=\pm\frac12$ are the $j_z$ quantum numbers for the two $\mathfrak{su}(2)$ bosonic subalgebras of $\mathfrak{psu}(2|2)$. In general the theory is non-relativistic and therefore the kinematics is rather exotic as described in detail in Appendix \ref{A2}. States can be labelled by their rapidity which determines the velocity as usual by $v=\tanh\theta$. However, there is a maximum rapidity and for each $\theta$ there are two distinct physical states with different energy and momentum. In the following we will leave the choice of rapidity branch implicit.
The two-body S-matrix has non-vanishing elements
\EQ{
& \ket{\phi_m\phi_m} \longrightarrow A\ket{\phi_m\phi_m}\ ,\qquad \ket{\psi_m\psi_m} \longrightarrow D\ket{\psi_m\psi_m}\ ,\\
& \ket{\phi_{\pm\frac12}\phi_{\mp\frac12}} \longrightarrow \frac1{[2]}\Big((A-B)\ket{\phi_{\mp\frac12}\phi_{\pm\frac12}}+
(q^{\pm1} A+q^{\mp1}B)\ket{\phi_{\pm\frac12}\phi_{\mp\frac12}}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\;\;
+q^{\mp1}C\ket{\psi_{\pm\frac12}\psi_{\mp\frac12}}-C\ket{\psi_{\mp\frac12}\psi_{\pm\frac12}}\Big)\ ,\\
& \ket{\psi_{\pm\frac12}\psi_{\mp\frac12}} \longrightarrow \frac1{[2]}\Big((D-E)\ket{\psi_{\mp\frac12}\psi_{\pm\frac12}}
+(q^{\pm1}D+q^{\mp1}E)\ket{\psi_{\pm\frac12}\psi_{\mp\frac12}}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;
+q^{\mp1}C\ket{\phi_{\pm\frac12}\phi_{\mp\frac12}}-C\ket{\phi_{\mp\frac12}\phi_{\pm\frac12}}\Big)\ ,\\
& \ket{\phi_m\psi_n} \longrightarrow G\ket{\psi_n\phi_m}+H\ket{\phi_m\psi_n}\ ,\qquad
\ket{\psi_m\phi_n} \longrightarrow H\ket{\psi_m\phi_n}+L\ket{\phi_n\psi_m}\ .
\label{jjs}
}
The functions $A=A(\theta_1,\theta_2)$, etc., are defined in \cite{Hoare:2011wr} (taken from the original reference \cite{Beisert:2008tw}). As the theory is generally not relativistically invariant the S-matrix is not a function of the difference $\theta_1-\theta_2$. Below we write the functions, including the dressing phase $v(\theta_1,\theta_2)$, in terms of the quantities $x^\pm_i=x^\pm(\theta_i)$ defined in Appendix \ref{A2}
\EQ{
A&=v\frac{U_1V_1}{U_2V_2}\cdot\frac{x_2^+-x_1^-}{x_2^--x_1^+}\ ,\qquad \qquad \,
B=A\Big(1-(1+q^{-2})\cdot\frac{x_2^+-x_1^+}{x_2^+-x_1^-}\cdot\frac{x_2^--\frac1{x_1^+}}{x_2^--\frac1{x_1^-}}\Big)\ ,\\
C&=iv(1+q^{-2})\Big(\frac{U_1V_1}{U_2V_2}\Big)^{3/2}\cdot\frac{1-\frac{x_2^+}{x_1^+}}{x_2^--\frac1{x_1^-}}\cdot\frac{\sqrt{(x_1^+-x_1^-)(x_2^+-x_2^-)}}{x_2^--x_1^+}\ ,\\
D&=-v\ ,\qquad \qquad \qquad \qquad\qquad \,
E=D\Big(1-\frac{1+q^{-2}}{U_2^2V_2^2}\cdot\frac{x_2^+-x_1^+}{x_2^--x_1^+}\cdot
\frac{x_2^+-\frac1{x_1^-}}{x_2^--\frac1{x_1^-}}\Big)\ , \\
G&=vq^{-1/2}\frac1{U_2V_2}\cdot\frac{x_2^+-x_1^+}{x_2^--x_1^+}\ ,\qquad
L=vq^{1/2}U_1V_1\cdot\frac{x_2^--x_1^-}{x_2^--x_1^+}\ ,\\
H&=v\sqrt{\frac{U_1V_1}{U_2V_2}}\cdot\frac{\sqrt{(x_1^+-x_1^-)(x_2^+-x_2^-)}}{x_2^--x_1^+}\ .
\label{eqnsb}
}
The dressing phase was constructed in \cite{Hoare:2011wr}. Here we are implicitly considering the $q$-deformation of the usual magnon dressing phase (denoted $\sigma$ in \cite{Hoare:2011wr}). The single-particle quantities $U(\theta)$ and $V(\theta)$ are defined by
\EQ{
U^2=q^{-1}\frac{x^++\xi}{x^-+\xi}=q\frac{\frac1{x^-}+\xi}{\frac1{x^+}+\xi}\ ,\qquad
V^2=q^{-1}\frac{\xi x^++1}{\xi x^-+1}=q\frac{\frac\xi{x^-}+1}{\frac\xi{x^+}+1}\ ,
\label{jww}
}
where
\EQ{
q=\exp(i\pi/k)\ ,\qquad\xi=\frac{2g\sin(\pi/k)}{\sqrt{1+4g^2\sin^2(\pi/k)}}\ .
}
Physical states with real rapidity satisfy the reality condition $(x^\pm)^*=x^\mp$ which corresponds to real rapidity $\theta$. If the rapidity is continued into the complex plane the functions $A,B$, etc., satisfies a reality condition of the form
\EQ{
A(\theta_1,\theta_2)^*=A(\theta_2^*,\theta_1^*)\ .
}
Using these reality conditions, one can verify that the S-matrix is almost Hermitian analytic but, as in the $\mathfrak{su}(2)$ case described in section \ref{s2}, this is spoiled by the reflection amplitudes involving four bosons or four fermions:
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.5]
\filldraw[black] (0,0) circle (4pt);
\draw[->] (-0.6,-0.6) -- (0.6,0.6);
\draw[<-] (-0.6,0.6) -- (0.6,-0.6);
\node at (-1.3,1.3) (a1) {$\phi_{\pm\frac12}$};
\node at (1.3,-1.3) (a2) {$\phi_{\mp\frac12}$};
\node at (1.3,1.3) (a3) {$\phi_{\mp\frac12}$};
\node at (-1.3,-1.3) (a4) {$\phi_{\pm\frac12}$};
\end{tikzpicture}=\frac1{[2]}\big(q^{\pm1} A+q^{\mp1}B\big)\ ,\qquad
\begin{tikzpicture}[baseline=-0.65ex,scale=0.5]
\filldraw[black] (0,0) circle (4pt);
\draw[->] (-0.6,-0.6) -- (0.6,0.6);
\draw[<-] (-0.6,0.6) -- (0.6,-0.6);
\node at (-1.3,1.3) (a1) {$\psi_{\pm\frac12}$};
\node at (1.3,-1.3) (a2) {$\psi_{\mp\frac12}$};
\node at (1.3,1.3) (a3) {$\psi_{\mp\frac12}$};
\node at (-1.3,-1.3) (a4) {$\psi_{\pm\frac12}$};
\end{tikzpicture}=\frac1{[2]}\big(q^{\pm1}D+q^{\mp1}E\big)\ .
}
The lessons from the $\mathfrak{su}(2)$ quantum group example in section \ref{s2} suggests that Hermitian analyticity should be manifest in a kink basis obtained by making the vertex-to-IRF transformation on the bosonic $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ subalgebra of $\mathfrak{psu}(2|2)$. In the $\mathfrak{su}(2)$ example a key identity which ensures the overall consistency of the transformation is \eqref{kid} and one can verify that this also holds in the present case, in both the bosonic and fermionic sector. In each sector, involving four bosonic or four fermionic particles, we have the same structure of the S-matrix as in \eqref{wea}--\eqref{wea2} with
\EQ{
&S_I=A\ ,\qquad S_T=\frac1{[2]}(A-B)\ ,\qquad S_R^\pm=\frac1{[2]}(q^{\pm1}A+q^{\mp1}B)\ ,
\\
&\widetilde S_I=D\ ,\qquad \widetilde S_T=\frac1{[2]}(D-E)\ ,\qquad \widetilde S_R^\pm=\frac1{[2]}(q^{\pm1}D+q^{\mp1}E)\ ,
}
In both case the crucial identity \eqref{kid} holds. Of course it had to be so because of the $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ quantum group invariance of the S-matrix.
The above identity is important because it means that we can use the same formulae for the S-matrix used in the $\mathfrak{su}(2)$ case given in Appendix \ref{A1} with the appropriate values of $S_I$, $S_T$ and $S_R^\pm$. In the kink picture, the vacua are labelled by a pair of spins $(j,l)$, one for each $\mathfrak{su}(2)$, and kinks of the form $K_{j,j\pm\frac12}^{l,l}$ are bosonic while $K_{j,j}^{l,l\pm\frac12}$ are fermionic.\footnote{For the world-sheet S-matrix, states come in a tensor product of two copies of the $\mathfrak{psu}(2|2)$ $R$-matrix. Therefore the vacua are labelled by four spins associated to the bosonic $\mathfrak{su}(2)^{\oplus4}$.} Making the vertex-to-IRF transformation as in section \ref{s2} gives the kink S-matrix
\EQ{
\ket{K^{l,l}_{j\pm 1,j\pm\frac12}K^{l,l}_{j\pm\frac12,j}} \longrightarrow & \ A\ket{K^{l,l}_{j\pm 1,j\pm\frac12}K^{l,l}_{j\pm\frac12,j}}\ ,\\
\ket{K_{j,j}^{l\pm1,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}} \longrightarrow & \ D\ket{K_{j,j}^{l\pm1,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}}\ ,\\
\ket{K^{l,l}_{j,j\pm\frac12}K^{l,l}_{j\pm\frac12,j}} \longrightarrow & \ \frac{[2j+1\mp1]A+[2j+1\pm1]B}{[2][2j+1]}\ket{K^{l,l}_{j,j\pm\frac12}K^{l,l}_{j\pm\frac12,j}}\\ &+
\frac{\sqrt{[2j+2][2j]}}{[2][2j+1]}(A-B)\ket{K^{l,l}_{j,j\mp\frac12}K^{l,l}_{j\mp\frac12,j}}\\
&+\sqrt{\frac{[2j+1\pm1][2l+1\pm1]}{[2j+1][2l+1]}}\frac{C}{[2]}\ket{K_{j,j}^{l,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}}\\ &-\sqrt{\frac{[2j+1\pm1][2l+1\mp1]}{[2j+1][2l+1]}}\frac{C}{[2]}\ket{K_{j,j}^{l,l\mp\frac12}K_{j,j}^{l\mp\frac12,l}}\ ,\\
\ket{K_{j,j}^{l,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}} \longrightarrow & \ \frac{[2l+1\mp1]D+[2l+1\pm1]E}{[2][2l+1]}\ket{K_{j,j}^{l,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}}\\ &+
\frac{\sqrt{[2l+2][2l]}}{[2][2l+1]}(D-E)\ket{K_{j,j}^{l,l\mp\frac12}K_{j,j}^{l\mp\frac12,l}}
\ ,\\
&+\sqrt{\frac{[2j+1\pm1][2l+1\pm1]}{[2j+1][2l+1]}}\frac{C}{[2]}\ket{K_{j,j\pm\frac12}^{l,l}K_{j\pm\frac12,j}^{l,l}}\\ &-\sqrt{\frac{[2j+1\mp1][2l+1\pm1]}{[2j+1][2l+1]}}\frac{C}{[2]}\ket{K_{j,j\mp\frac12}^{l,l}K_{j\mp\frac12,j}^{l,l}}\ ,\\
\ket{K_{j\pm\frac12,j}^{l\mp\frac12,l\mp\frac12}K_{j,j}^{l\mp\frac12,l}} \longrightarrow & \
G\ket{K_{j\pm\frac12,j\pm\frac12}^{l\mp\frac12,l}K_{j\pm\frac12,j}^{l,l}}
+H\ket{K_{j\pm\frac12,j}^{l\mp\frac12,l\mp\frac12}K_{j,j}^{l\mp\frac12,l}}\ ,\\
\ket{K_{j\mp\frac12,j\mp\frac12}^{l\pm\frac12,l}K_{j\mp\frac12,j}^{l,l}} \longrightarrow & \
H\ket{K_{j\mp\frac12,j\mp\frac12}^{l\pm\frac12,l}K_{j\mp\frac12,j}^{l,l}}
+L\ket{K_{j\mp\frac12,j}^{l\pm\frac12,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}}\ ,
\label{web2}
}
These S-matrix elements can be summarized as follows. Firstly kinks $K_{ab}^{uv}(\theta)$ must have either $|a-b|=\frac12$ and $u=v$, or $a=b$ and $|u-v|=\frac12$---the former being a boson and the latter a fermion. We introduce the notation
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$b,v$};
\node at (1.3,0) (a3) {$c,w$};
\node at (0,1) (a4) {$d,y$};
\end{tikzpicture}(\theta_1,\theta_2)\ ,
\label{dek}
}
for the process $K_{ab}^{uv}(\theta_1)+K_{bc}^{vw}(\theta_2)\longrightarrow
K_{ad}^{uy}(\theta_1)+K_{dc}^{yw}(\theta_2)$.
The processes involving $BB\to BB$ can then be written in the compact form
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$b,u$};
\node at (1.3,0) (a3) {$c,u$};
\node at (0,1) (a4) {$d,u$};
\end{tikzpicture}=S_I\delta_{bd}+e^{i\pi(a+c-b-d)}S_T\sqrt{\frac{[2b+1][2d+1]}{[2a+1][2c+1]}}\delta_{ac}\ ,
\label{gtw2}
}
while those involving $FF\to FF$ are given by
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$a,v$};
\node at (1.3,0) (a3) {$a,w$};
\node at (0,1) (a4) {$a,y$};
\end{tikzpicture}=\widetilde S_I\delta_{vy}+e^{i\pi(u+w-y-v)}\widetilde S_T\sqrt{\frac{[2v+1][2y+1]}{[2u+1][2w+1]}}\delta_{uw}\ .
\label{gtw3}
}
For those involving $BB\to FF$ or $FF\to BB$ we have
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$b,u$};
\node at (1.3,0) (a3) {$a,u$};
\node at (0,1) (a4) {$a,v$};
\end{tikzpicture}=\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$a,v$};
\node at (1.3,0) (a3) {$a,u$};
\node at (0,1) (a4) {$b,u$};
\end{tikzpicture}=
-e^{i\pi(a-b+u-v)}\frac C{[2]}\sqrt{\frac{[2b+1][2v+1]}{[2a+1][2u+1]}}\ .
\label{gtw4}
}
Finally there are the processes involving $BF\to FB+BF$ and $FB\to BF+FB$, respectively,
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$b,u$};
\node at (1.3,0) (a3) {$b,v$};
\node at (0,1) (a4) {$d,y$};
\end{tikzpicture}=G\delta_{ad}\delta_{vy}+H\delta_{bd}\delta_{uy}\ ,\qquad
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$a,v$};
\node at (1.3,0) (a3) {$b,v$};
\node at (0,1) (a4) {$d,y$};
\end{tikzpicture}=H\delta_{ad}\delta_{vy}+L\delta_{bd}\delta_{uy}\ .
\label{gtw5}
}
The functions involved in the S-matrix satisfy the crossing symmetry relations
\EQ{
&S_I(\theta_1,\theta_2)=S_T(i\pi+\theta_2,\theta_1)\ ,\qquad
\widetilde S_I(\theta_1,\theta_2)=\widetilde S_T(i\pi+\theta_2,\theta_1)\ ,\\ &
C(\theta_1,\theta_2)=[2]H(i\pi+\theta_2,\theta_1)\ ,\qquad G(\theta_1,\theta_2)=
L(i\pi+\theta_2,\theta_1)\ .
}
Using these identities,
the kink S-matrix in \eqref{dek} is seen to be crossing symmetric if
we define charge conjugation as\footnote{Recall that in this superalgebra case we should use the supertranspose as oppose to the transpose. This contributes a factor of $e^{2i\pi(v+y-u-w)}$ to the crossing equation.}
\EQ{
{\cal C}\ket{K_{ab}^{uv}(\theta)}= e^{i\pi(b-a+v-u)}\sqrt{\frac{[2a+1][2u+1]}{[2b+1][2v+1]}} \ket{K_{ba}^{vu}(\theta)} \ .
}
One can check that the S-matrix satisfies Hermitian analyticity in the kink picture
\EQ{
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$b,v$};
\node at (1.3,0) (a3) {$c,w$};
\node at (0,1) (a4) {$d,y$};
\end{tikzpicture}(\theta_1^*,\theta_2^*)^*
=
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$d,y$};
\node at (1.3,0) (a3) {$c,w$};
\node at (0,1) (a4) {$b,v$};
\end{tikzpicture}(\theta_2,\theta_1)\ .
\label{bui1}}
In the kink basis, the braiding unitarity relation takes the form
\EQ{
\sum_{e,z}
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$b,v$};
\node at (1.3,0) (a3) {$c,w$};
\node at (0,1) (a4) {$e,z$};
\end{tikzpicture}(\theta_1,\theta_2)
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$e,z$};
\node at (1.3,0) (a3) {$c,w$};
\node at (0,1) (a4) {$d,y$};
\end{tikzpicture}(\theta_2,\theta_1)
=\delta_{bd}\delta_{vy}\ .
\label{bui2}
}
Putting \eqref{bui1} and \eqref{bui2} together then gives the kink version of the QFT unitarity condition \eqref{unit}
\EQ{
\sum_{e,z}
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$b,v$};
\node at (1.3,0) (a3) {$c,w$};
\node at (0,1) (a4) {$e,z$};
\end{tikzpicture}(\theta_1,\theta_2)
\begin{tikzpicture}[baseline=-0.65ex,scale=0.6]
\draw (-0.6,-0.6) -- (-0.6,0.6) -- (0.6,0.6) -- (0.6,-0.6) -- (-0.6,-0.6);
\node at (-1.3,0) (a1) {$a,u$};
\node at (0,-1) (a2) {$d,y$};
\node at (1.3,0) (a3) {$c,w$};
\node at (0,1) (a4) {$e,z$};
\end{tikzpicture}(\theta_1,\theta_2)^*
=\delta_{bd}\delta_{vy}\ ,\qquad(\theta_i \text{ real})\ .
\label{bui3}
}
Finally, it can be checked that \eqref{web2} satisfies both the Yang-Baxter equation in the IRF picture and also the dynamical Yang-Baxter equation given by the obvious generalization of the $\mathfrak{su}(2)$ case---see Figure \ref{figybe} and equation \eqref{dybe} respectively.
\section{Limits of the Kink S-Matrix\label{slim}}
In this section we discuss two limits of the kink S-matrix. In the first we expect to recover the string S-matrix, while the second is conjectured to be related to the Pohlmeyer-reduced $\text{AdS}_5 \times S^5$ superstring.
{
Taking the string limit ($k \rightarrow \infty$) of the vertex S-matrix \eqref{jjs} we recover the S-matrix of the light-cone gauge-fixed string theory \cite{Beisert:2008tw}---in this limit the vertex S-matrix is QFT unitary \cite{Arutyunov:2009ga}. However, taking this limit in the kink S-matrix we find additional $(j,l)$-dependent factors originating from the $q$-numbers in equation \eqref{web2}, and to recover the string S-matrix, we also need to take the $\mathfrak{su}(2)$ spins $j$ and $l$ to infinity.
The limit $j,l\to\infty$ can be justified as follows.
Recall that, in the kink picture, $(j,l)$ label the vacua so that, for $q=e^{i\pi/k}$,
\EQ{
j,l\in\Big\{0,\frac12,1,\ldots, \frac{k}2-1\Big\}
}
and, in the $k\to\infty$ limit, the kinks interpolate between an infinite number of vacua. Naively, therefore, this limit distinguishes between the vacua $j,l=0$ and $j,l=\frac{k}2-1\to\infty$, but it is worth noticing that the S-matrix elements~\eqref{gtw} and~\eqref{gtw2}--\eqref{gtw5} are invariant if we change all the labels corresponding to the vacua according to
$j\to \tfrac{k}2-1-j$ and $l\to \tfrac{k}2-1-l$. This suggests to take the $k\to\infty$ limit keeping the symmetry between $j,l=0$ and $j,l=\frac{k}2-1\to\infty$. In order to do that, we redefine the spins
\begin{equation}\label{rdef}
j = \frac{k-2}4 + \tilde \jmath \ , \qquad
l = \frac{k-2}4 + \tilde l \ ,
\end{equation}
and then take the $k\to\infty$ limit keeping $\tilde \jmath$ and $\tilde l$ fixed. This implies $j,l\to\infty$, and the dependence on $\tilde \jmath$
and $\tilde l$ drops out leaving the S-matrix of the light-cone gauge-fixed string theory. Notice that this limit gives rise to an infinite number of vacua labelled by $\tilde \jmath, \tilde l$ which play the role of topological charges.
}
The relativistic limit ($g \rightarrow \infty$) in the vertex picture is known to give a relativistic S-matrix related to the Pohlmeyer reduction of the $\text{AdS}_5 \times S^5$ superstring \cite{Hoare:2011fj,Hoare:2011nd}. However, the large $k$ expansion of the vertex S-matrix does not agree with the perturbative computation of \cite{Hoare:2009fs,Hoare:2011fj}. This might be expected as the perturbative S-matrix does not satisfy the Yang-Baxter equation. In particular, the tree-level result does not satisfy the classical Yang-Baxter equation, which follows from the quantum one \eqref{ybe} assuming the S-matrix has the form ${\cal P} + \tfrac 1k \, {\cal T}$, where ${\cal P}$ is the permutation operator. Furthermore, the perturbative S-matrix is unitary while the vertex S-matrix is not.
In sections \ref{s2} and \ref{s3}, it was shown that starting from the vertex S-matrix one can use the vertex-to-IRF transformation to move to the kink picture, in which both QFT unitarity and the Yang-Baxter equation are satisfied. Therefore, it is natural to ask whether the large $k$ expansion of the relativistic limit of the kink S-matrix \eqref{web2} has any relation to the perturbative S-matrix of the Pohlmeyer-reduced theory. For reference we give the relativistic limit of the functions parametrizing the S-matrix:\footnote{
To facilitate comparison with the perturbative computation \cite{Hoare:2011fj} we have extracted a factor from the phase
\EQ{\tilde v = v \, \sinh \frac\theta 2 \operatorname{csch}\left(\frac\theta 2 + \frac{i\pi}{2k}\right) \ .}
The dressing phase in the relativistic limit is given in \cite{Hoare:2011fj} and in integral form in \cite{Hoare:2011nd}.}
\EQ{
& A(\theta) = \tilde v \, \operatorname{csch}\frac\theta2 \sinh\left(\frac\theta 2 - \frac{i\pi}{2k}\right) \ ,\quad \ \ \,
D(\theta) = - \, \tilde v \, \operatorname{csch}\frac\theta2 \sinh\left(\frac\theta 2 + \frac{i\pi}{2k}\right) \ , \\
& B(\theta) = - 2 \, i \, \tilde v \, \operatorname{csch}\theta \left(\sin\frac{\pi }{2 k}-i \cosh\left(\frac{\theta }{2}+\frac{3 i \pi }{2 k}\right)\sinh\frac{\theta }{2}\right) \ , \\
& E(\theta) = - 2 \, i \, \tilde v \, \operatorname{csch}\theta \left(\sin\frac{\pi }{2 k}+i \cosh\left(\frac{\theta }{2}-\frac{3 i \pi }{2 k}\right)\sinh\frac{\theta }{2}\right) \ , \\
& C(\theta) = - \, 2 \,\tilde v \cos\frac{\pi }{k} \sin\frac{\pi }{2 k} \operatorname{sech}\frac{\theta }{2} \ , \quad \ \ \
H(\theta) = - \, i\, \tilde v\, \sin\frac{\pi }{2 k}\operatorname{csch} \frac \theta 2\ , \vphantom{\left(\frac\theta 2 \right)} \\
& G(\theta) = L(\theta) = \tilde v \ . \vphantom{\left(\frac\theta 2 \right)}
}
The S-matrix now depends only on the difference of rapidities, $\theta = \theta_1 - \theta_2$, as required by Lorentz symmetry. The relation between $x^\pm$ and the rapidity for arbitrary $(g,k)$ is discussed in Appendix \ref{A2}.
In addition to depending on $k$ and $g$ the kink S-matrix also depends on the $\mathfrak{su}(2)$ spins $j$ and $l$, which should be taken large for a good semi-classical interpretation (see Appendix \ref{A3}). However, the $(j,l) \rightarrow \infty$ limit is not well-defined for finite $k$.\footnote{As discussed in section \ref{s5}, for a consistent physical theory $k$ is required to be an integer while $j$ and $l$ should take values in the finite set $\{0,\tfrac12,\ldots,\tfrac k2-1\}$.} Our approach is therefore to first redefine the spins using \eqref{rdef} and then expand around large $k$. To the one-loop order we find the following S-matrix
\EQ{
\ket{K^{l,l}_{j\pm1,j\pm\frac12}K^{l,l}_{j\pm\frac12,j}} \longrightarrow
& \ \tilde v \, \big(1-\frac{i\pi}{2k}\coth\frac\theta2-\frac{\pi^2}{8k^2}\big)
\ket{K^{l,l}_{j\pm1,j\pm\frac12}K^{l,l}_{j\pm\frac12,j}} \ , \\
\ket{K_{j,j}^{l\pm1,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}} \longrightarrow
& \ \tilde v \, \big(-1-\frac{i\pi}{2k}\coth\frac\theta2+\frac{\pi^2}{8k^2}\big)
\ket{K_{j,j}^{l\pm1,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}} \ , \\
\ket{K^{l,l}_{j,j\pm\frac12}K^{l,l}_{j\pm\frac12,j}} \longrightarrow
& - \tilde v \, \big(\frac{i\pi}{k}\coth\theta \mp \frac{2\pi^2 \tilde \jmath}{k^2}-\frac{\pi^2}{2k^2}\big)
\ket{K^{l,l}_{j,j\pm\frac12}K^{l,l}_{j\pm\frac12,j}} \\
& + \tilde v \, \big(1+\frac{i\pi}{2k}\tanh\frac\theta 2-\frac{5\pi^2}{8k^2}\big)
\ket{K^{l,l}_{j,j\mp\frac12}K^{l,l}_{j\mp\frac12,j}} \\
& - \tilde v \, \big(\frac{\pi}{2k}\operatorname{sech}\theta\big)
\ket{K_{j,j}^{l,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}} \\
& + \tilde v \, \big(\frac{\pi}{2k}\operatorname{sech}\theta\big)
\ket{K_{j,j}^{l,l\mp\frac12}K_{j,j}^{l\mp\frac12,l}} \ , \\
\ket{K_{j,j}^{l,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}} \longrightarrow
& - \tilde v \, \big(\frac{i\pi}{k}\coth\theta \pm \frac{2\pi^2\tilde l}{k^2}+\frac{\pi^2}{2k^2}\big)
\ket{K_{j,j}^{l,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}} \\
& - \tilde v \, \big(1-\frac{i\pi}{2k}\tanh\frac\theta2-\frac{5\pi^2}{8k^2}\big)
\ket{K_{j,j}^{l,l\mp\frac12}K_{j,j}^{l\mp\frac12,l}} \ , \\
& - \tilde v \, \big(\frac{\pi}{2k}\operatorname{sech}\theta\big)
\ket{K_{j,j\pm\frac12}^{l,l}K_{j\pm\frac12,j}^{l,l}} \\
& + \tilde v \, \big(\frac{\pi}{2k}\operatorname{sech}\theta\big)
\ket{K_{j,j\mp\frac12}^{l,l}K_{j\mp\frac12,j}^{l,l}} \ , \\
\ket{K_{j\pm\frac12,j}^{l\mp\frac12,l\mp\frac12}K_{j,j}^{l\mp\frac12,l}} \longrightarrow
& \ \tilde v \,
\ket{K_{j\pm\frac12,j\pm\frac12}^{l\mp\frac12,l}K_{j\pm\frac12,j}^{l,l}}
- \tilde v \, \big(\frac{i\pi}{2k}\operatorname{csch}\frac\theta 2\big)
\ket{K_{j\pm\frac12,j}^{l\mp\frac12,l\mp\frac12}K_{j,j}^{l\mp\frac12,l}} \ , \\
\ket{K_{j\mp\frac12,j\mp\frac12}^{l\pm\frac12,l}K_{j\mp\frac12,j}^{l,l}} \longrightarrow
& - \tilde v \, \big(\frac{i\pi}{2k}\operatorname{csch}\frac\theta 2\big)\ket{K_{j\mp\frac12,j\mp\frac12}^{l\pm\frac12,l}K_{j\mp\frac12,j}^{l,l}}
+ \tilde v \,
\ket{K_{j\mp\frac12,j}^{l\pm\frac12,l\pm\frac12}K_{j,j}^{l\pm\frac12,l}} \ .
\label{webex}}
Comparing with \cite{Hoare:2009fs,Hoare:2011fj} we see that at the tree level the expansion of the kink S-matrix matches the perturbative computation of the Pohlmeyer-reduced theory S-matrix.\footnote{Note that here we have redefined the fermions by a factor of $e^{i\pi/4}$ compared to \cite{Hoare:2009fs,Hoare:2011fj}.} Furthermore, setting $\tilde \jmath = \tilde l = 0$ in the one-loop terms we recover the real part of the perturbative computation. The exact agreement however breaks down at one-loop. This may be expected as it is only for large $k$ that the perturbative states are a good approximation for the kink states \cite{Hollowood:2011fq}.
While agreement with the perturbative S-matrix is no longer true at one-loop, the presence of the $\mathfrak{su}(2)$ spins in the one-loop amplitudes provides us with the explanation of the Yang-Baxter ``anomaly'' of the tree-level S-matrix \cite{Hoare:2009fs,Hoare:2010fb,Hoare:2011fj}. The identity that the tree-level S-matrix should satisfy is modified from the classical Yang-Baxter equation by a contribution originating from the shifts in vacua in the dynamical Yang-Baxter equation. This is the usual construction whereby one recovers the modified classical Yang-Baxter equation from the semi-classical expansion of the dynamical Yang-Baxter equation \cite{Felder:1994be}.
\noindent
\textbf{A simpler example}\nopagebreak
For completeness we also discuss the analogous construction for the $\mathfrak{su}(2)$ quantum group and the $S^5$ symmetric space sine-Gordon theory. In this case the Lie algebra underlying the symmetry of the theory is $\mathfrak {so}(4) \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ and the S-matrix should factorize accordingly into two copies of the $R$-matrix associated to the $\mathfrak{su}(2)$ quantum group of section \ref{s2}. The map between the parameters $x$ and $q$ and $\theta$ and $k$ is given in \eqref{de2}. It is therefore useful to define
\EQ{
\lambda = \frac{k+1}{k+2} \ , \qquad \omega = \frac{\pi}{k+2} \ ,}
so that
\EQ{
x = e^{\lambda \theta} \ , \qquad q = e^{i\omega}\ .}
The perturbative S-matrix for this theory was computed in \cite{Hoare:2010fb} to the one-loop order, or equivalently ${\cal O}(\omega^2)$. Pulling out an overall factor\footnote{This expression is motivated by various observations in \cite{Hollowood:2010rv} and Appendix G of \cite{Hoare:2011fj}. Furthermore it satisfies the braiding unitarity and crossing relations \eqref{phasecross} and $v(\theta) = v(i\pi-\theta)$. Note that the square root disappears in the tensor product.}
\EQ{
v(\theta) = & \frac{1}{2\sinh\big(\lambda \theta + i\omega\big)}
\sqrt{\frac{\sinh\big(\frac{\lambda \theta}{2} + \frac{i\omega}{2}\big)\cosh\big(\frac{\lambda \theta}{2} - \frac{i\omega}{2}\big)}{\sinh\big(\frac{\lambda \theta}{2} - \frac{i\omega}{2}\big) \cosh\big(\frac{\lambda \theta}{2} + \frac{i\omega}{2}\big)}} \ \frac{\cosh\big(\frac{\lambda \theta}{2} + \frac{i\omega}{4}\big)}{\cosh\big(\frac{\lambda \theta}{2} - \frac{i\omega}{4}\big)}
\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \Big(1 - \frac{i\omega^2}{\pi} \lambda \theta \coth \lambda \theta \operatorname{csch} \lambda \theta + {\cal O}(k^{-3}) \Big)\ ,}
the perturbative result is given by
\EQ{
\mathbb S\ket{\phi_m(\theta_1) \phi_n(\theta_2)} = v(\theta)\Big[&\Big(2\sinh \lambda \theta + \frac{i\omega^2}{\pi} \big(1+(i \pi -\lambda \theta \big) \sinh \lambda \theta\big) \Big) \delta_m^q \delta_n^p \\ & + \Big(2 i \omega \cosh \lambda \theta +\frac{2 i \omega^2}{\pi } \lambda \theta \sinh\lambda \theta \Big)\delta_m^p \delta_n^q\Big]\ket{\phi_p(\theta_2)\phi_q(\theta_1)}\ .
\label{compare1}}
The expansion of the S-matrix associated to the $\mathfrak{su}(2)$ quantum group in the kink basis \eqref{web} to the same order is given by
\EQ{
&\ket{K_{j\pm1,j\pm\frac12}(\theta_1)K_{j\pm\frac12,j}(\theta_2)}\longrightarrow
\\ & \hspace{15pt} v(\theta) \Big(2\sinh \lambda \theta +2 i \omega \cosh \lambda \theta - \omega^2 \sinh \lambda \theta \Big)\ket{K_{j\pm1,j\pm\frac12}(\theta_2)K_{j\pm\frac12,j}(\theta_1)}\ ,\\
&\ket{K_{j,j\pm\frac12}(\theta_1)K_{j\pm\frac12,j}(\theta_2)}\longrightarrow
\\ & \hspace{15pt} v(\theta) \Big(2\sinh \lambda \theta -\omega^2\sinh \lambda \theta \Big) \ket{K_{j,j\mp\frac12}(\theta_2)K_{j\mp\frac12,j}(\theta_1)}
\\ &\hspace{40pt}+v(\theta) \Big(2 i \omega \cosh \lambda \theta \pm 4 \omega^2 \tilde \jmath \sinh\lambda \theta \Big) \ket{K_{j,j\pm\frac12}(\theta_2)K_{j\pm\frac12,j}(\theta_1)}\ .
\label{compare2}
}
Here we have first redefined the $\mathfrak{su}(2)$ spin
\begin{equation}\label{rrdef}
j = \frac{k}4 + \tilde \jmath \ ,
\end{equation}
which is equivalent to \eqref{rdef} taking account of the shift in $k$, and then expanded around large $k$. Comparing \eqref{compare1} and \eqref{compare2} we again see that we recover the perturbative result at the tree level. Furthermore, if we set $\tilde \jmath = 0$, we find agreement with the real part of the perturbative computation at the one-loop level. However, in the imaginary part there is disagreement. Again, this may be expected as it is only for large $k$ that the perturbative states are a good approximation for the kink states \cite{Hollowood:2011fq}.
The disagreement stems from the presence of ${\cal O}(\tilde \jmath)$ terms in the one-loop reflection amplitudes. More precisely, they imply the crossing and unitarity relations that the one-loop amplitudes should satisfy are different to those that the perturbative result satisfies. Consequently, for the perturbative computation to match the expansion of the kink S-matrix beyond the tree level it would need to be modified to include the $\mathfrak{su}(2)$ spin $j$.
\section{The Restricted Theories}\label{s5}
When $q$ is generic it is known that representations of quantum groups are simple deformations of representations of the undeformed group. {However, when $q$ is a root of unity, the case pertinent to our discussion, the representation theory of quantum groups is more subtle{, and we have summarized its main features for the case of $SU(2)$ in Appendix~\ref{a5}}. For $q=e^{i\pi/k}$, there are a set of irreducible ``good" representations of dimension $<k$, and the idea is to define a restricted theory by removing from the Hilbert space the remaining ``bad" representations. The ``good" representations are of Type A and denoted $V_j^{(+1)}$ in Appendix \ref{a5}, with $j\leq k/2-1$. The ``bad" representations consists of a finite set of reducible but indecomposable representations of dimension $2k$, in addition to the irreducible representation $V_{j}^{(+1)}$ with $j=(k-1)/2$ of dimension~$k$.}\footnote{ {Notice that the q-CG coefficients~\eqref{qCG} blow up for $j=(k-1)/2$.}}
It is then possible to define a restricted representation theory which only includes the ``good" representations and the
way to do this in practice is to use the vertex-to-IRF change of basis to go to the kink basis and simply insist that the vacua lie in the finite set $\{0,\frac12,1,\ldots,\frac k2-1\}$. This consistently implements the restriction.
What is particularly nice about this restriction is that it meshes perfectly with the spectrum of bound states of the theory and the bootstrap procedure that determines the S-matrix elements of the bound states. With the vacua restricted to the set $\{0,\frac12,1,\ldots,\frac k2-1\}$ it is clear that states in the original vertex picture are restricted. For instance, the Hilbert space cannot contain the states $\ket{\phi_m(\theta_1)\phi_m(\theta_2)\cdots\phi_m(\theta_N)}$ with $N>k-2$. The bound {states} transform in the short representations $\langle a-1,0\rangle$ (the magnons) or $\langle0,a-1\rangle$ (the solitons) of $U_q(\mathfrak h)$. {Taking the magnon bound states,
they contain states in the representation\footnote{The $\mathfrak{su}(2)$ labels here are twice the spin and the representation theory of $U_q(\mathfrak h)$ is discussed in \cite{Beisert:2008tw,Hoare:2011nd}.} $(a,0)\oplus(a-1,1)\oplus(a-2,0)$ of the subalgebra $U_q(\mathfrak{su}(2))\times U_q(\mathfrak{su}(2))$ and, clearly, only states with $a\leq k$ can appear in the spectrum of the restricted model.} Note also that the bound states near the top of the tower, those with $a=k$ and $k-1$, have a modified content. More specifically, in the truncated representation theory, the representation $a=k$, that is $\langle k-1,0\rangle$, consists of the $U_q(\mathfrak{su}(2))\times U_q(\mathfrak{su}(2))$ representation $(k-2,0)$ only, while for $a=k-1$ we have $\langle k-2,0\rangle=(k-2,1)\oplus(k-3,0)$. The fact that the tower of states is restricted to $a=1,2,\ldots,k$ also meshes perfectly with the dispersion relation for these states (\eqref{dismag} in Appendix \ref{A2}) which has a built-in periodicity $a\to a+2k$ and symmetry $a\to 2k-a$:
\EQ{
\sin^2\Big(\frac{\xi E}{2g}\Big)-\xi^2\sin^2\Big(\frac p{2g}\Big)=
(1-\xi^2)\sin^2\Big(\frac{\pi a}{2k}\Big)\ .
\label{dismag2}
}
Although we will not investigate the representation theory of the quantum supergroup $U_q(\mathfrak h)$ when $q$ is a root of unity in any detail here, one can infer what happens from thinking about the bosonic sub-algebra $U_q(\mathfrak{su}(2))\times U_q(\mathfrak{su}(2))$. In Appendix \ref{a5} we review the way that the representations of $U_q(\mathfrak{su}(2))$ satisfy a truncated Clebsch-Gordon decomposition. The relevant representations are $V_j^{(+1)}$, which we label as $(2j)$ above, {with $j\leq k/2-1$,} whose tensor product decomposition takes the form \eqref{ttp}. This implies that the particular short representations $\langle a-1,0\rangle$ of $U_q(\mathfrak h)$ that describe the magnons have the truncated tensor product, when $q=e^{i\pi/k}$, of the form
\EQ{
\langle a_1-1,0\rangle\, \widetilde{\otimes}\, \langle a_2-1,0\rangle = \bigoplus_{m=|a_1-a_2|}^{\text{min}(a_1+a_2-2,2k-a_1-a_2-2)}\, \{m,0\}\ ,
\label{ttp4}
}
where $\{m,0\}$ is a long representation. There is a similar expression for the solitons involving representations $\langle0,a-1\rangle$.
This decomposition meshes with the bootstrap
procedure of S-matrix theory (reviewed in Appendix \ref{a4}). The S-matrix for $a_1$ scattering with $a_2$ (either magnon or soliton) has two direct channel poles shown in Figure \ref{f8}. The pole I is physical if $a_1+a_2\leq k$ and the term $\{a_1+a_2-2,0\}$ becomes reducible (but indecomposable) implying that the bound state transforms in the short representation $\langle a_1+a_2-1,0\rangle$ as shown in Figure \ref{f8}. What is particularly noteworthy here is that the kinematical condition $a_1+a_2\leq k$ dovetails perfectly with the truncation of the Clebsch-Gordon decomposition. At the latter pole II, the term $\{|a_1-a_2|,0\}$ becomes reducible (but indecomposable) implying that the bound state transforms in the short representation $\langle |a_1-a_2|-1,0\rangle$ as shown in Figure \ref{f8} for $a_1\geq a_2$. There is a similar discussion for the solitons.
\section{Discussion}
In this work we have argued that the $q$-deformation of the string world-sheet S-matrix in $\text{AdS}_5\times S^5$ is described by an IRF, or RSOS, type S-matrix. The original ``vertex" form of the S-matrix is just a starting point for the vertex-to-IRF transformation. Unlike its vertex cousin, the new IRF S-matrix is manifestly unitary. It also satisfies all the S-matrix axioms familiar from a relativistic theory, albeit with a more complicated analytic structure. The bootstrap equations are discussed in Appendix \ref{a4}, including some strong checks that they mesh with the representation theory.
We found that in the relativistic limit, $g \rightarrow \infty$, the perturbative tree-level S-matrix of the Pohlmeyer-reduced theory is recovered in a particular semi-classical expansion. The details of this expansion clarify why the tree-level result satisfies a deformed classical Yang-Baxter equation. While we found agreement at the tree level, at one-loop there is still a discrepancy. This might be expected as the perturbative computations with which we are comparing assumed trivial boundary conditions. This is in contrast with the excitations whose scattering is described by the kink S-matrix. To find agreement at higher orders one would then need to incorporate the non-trivial boundary conditions and introduce the $\mathfrak{su}(2)$ spins $j$ and $l$ into the perturbative computation. This remains an open problem, however, progress in this direction has been made in \cite{Hollowood:2013oca}, in which an action was constructed with the required properties.
In the other limit of interest, $q \rightarrow 1$, we recover the string S-matrix. Again this limit is subtle and should be taken in such a way that preserves the symmetry between $j,l = 0$ and $j,l = \tfrac{k}{2}-1$, the end-points of the range of vacua. This can be done by just performing a shift in $j,l$ such the range is symmetric about $0$, see Eq.~\eqref{rdef}. The shifted vacuum labels $\tilde{\jmath}$ and $\tilde{l}$ then drop out of the S-matrix in the $q \to 1$ limit. It is an interesting open question as to whether the shifted vacuum labels have a physical meaning in the string limit or are just an artefact of the non-trivial vacuum structure of the interpolating theory. This could be the case if, for example, the vacua become degenerate as $q \rightarrow 1$.
Strong evidence that the quantum group restriction is the correct procedure to apply to the S-matrix in the present context could be obtained by studying the Thermodynamic Bethe Ansatz in the relativistic $g\to\infty$ limit.\footnote{We would like to thank the referee of this paper for suggesting this as a worthwhile analysis.} If the central charge of the UV theory could be extracted then this could be compared to the central charge of the UV CFT; namely, the $G/H$ gauge WZW model. Whilst this analysis has yet to be done for the string theory case, in the simpler context of the purely bosonic symmetric space ${\mathbb C}P^2$, where
$G/H=\text{SU}(2)/U(1)$, the calculation of the central charge from the TBA has been performed \cite{Hollowood:2010rv} and precise agreement was found. This is additional circumstantial evidence that the quantum group restriction is the correct paradigm.
Finally, one of the key messages of this work is that the interpolating theory has a non-trivial vacuum structure with kinks playing the r\^ole of one-particle states. Therefore this should be respected by any fundamental off-shell (Lagrangian or otherwise) origin for the interpolating theory. In fact, for the Pohlmeyer reduced theory, whose Lagrangian description is known, the non-trivial vacuum structure arises {as} a consequence of the presence of a WZW term in the bosonic sector of the theory~\cite{Hollowood:2013oca}.
\section*{Acknowledgements}
\noindent
BH is supported by the Emmy Noether Programme ``Gauge fields from Strings'' funded by the German Research Foundation (DFG). He would like to thank Gleb Arutyunov, Wellington Galleas, Arkady Tseytlin and Stijn van Tongeren for useful discussions.
\noindent
TJH is supported in part by the STFC grant ST/G000506/1 and further acknowledges support
from the European Science Foundation (ESF) for the activity entitled ``Holographic Methods for Strongly Coupled Systems".
\noindent
JLM is supported in part by MINECO (FPA2011-22594), the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), and FEDER. He would like to thank Manuel Asorey and Joaqu\'\i n S\'anchez Guill\'en for useful discussions.
\vspace{1cm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,834 |
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Emeline was born and raised in Windsor, CA. Their roots are from Michoacán, Mexico. They earned their Associates Degree in Social and Behavioral Science, and Sociology. They have been working in nonprofit specializing in the youth community for over 7 years. They are looking forward to continuing working with the latinx/bi-cultural community with Humanidad.
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Maxine Hall
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Heather Harshbarger
Christina Zapata
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I am the Clinical Director of Humanidad Therapy and Education Services where I provide clinical supervision to our trainees and associates who are gaining hours towards becoming therapists like myself. I also work with adults, couples, families and children doing therapy that is both bilingual and bicultural. Having migrant parents, I have experienced the challenges, complexities and opportunities that come with being a first generation Latina and living in and bridging two cultures. I am also a licensed MFT therapist and have been an active member of Sonoma County's mental health community, providing psychotherapy in community mental health agencies, schools and court-affiliated services and as well as in my own bilingual private practice. I received an Associates of Art from Santa Rosa Junior College, Bachelors of Science in both Psychology and Spanish from Sonoma State University and then completed a Masters of Science in Counseling Marriage and Family at San Francisco State University. It was during that time in working as an intern to become licensed MFT that I became inspired by my mentor Dr. Maria Hess and her work as a Humanistic depth-oriented therapist and teacher. In collaboration with Dr. Hess, Claudia Cendejas, MA, and I, we co-founded Humanidad Therapy and Education Services. The first non-profit mental health counseling agency in Sonoma County that primarily serves the Latinx community, offering bilingual and bicultural sensitive services to hard to reach communities. Dr. Maria Hess lead the way for me in my interest of expanding multicultural awareness and delving deeper into issues that impact diverse populations. We developed the therapeutic program of Convivencia that is currently being assessed to train future therapists and social workers into the field of mental health on a culturally based practice that supports the needs of the Latinx immigrant community. I am dedicated to serving the most vulnerable populations and helping to bring awareness to the barriers that limit access and the stigma of mental health services.
Maxine Hall, MFT has been a Marriage and Family Therapist in Private Practice since 1999. She began supervising Trainees and Interns in 2006 and became a CAMFT Certified Supervisor in 2011. It is with great pleasure that she is involved and supervises Trainees and Associates at Humanidad.
Heather Harshbarger is a Licensed Marriage and Family Therapist, bilingual in English and Spanish, with over a decade of experience working with children, families, and adults. Heather is originally from the east coast, but moved to California after completing graduate school to start her career in mental health. Heather also carries a specialization in Early Childhood Mental Health and is pursuing endorsement with the state of California as an Early Childhood Mental Health Specialist. Outside of work, Heather is a busy parent alongside her partner to their toddler daughter. She enjoys spending time with her family in their garden and out in nature.
Christina (LCSW, CDEP) is a licensed social worker carrying out program and clinical duties at Humanidad. She is EMDR certified, a therapeutic intervention used to relieve psychological stress and is the program manager for the Community Convivencia & Group Convivencia research project. Christina's experience prior to clinical training includes organized labor, migrant education, and advocacy for policy change.
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Bianca Avina is a bilingual, bicultural, Licensed Marriage and Family Therapist (LMFT). She is a first-generation college student to obtain her Master's in counseling psychology from the University of San Francisco. Bianca works with individual clients of all ages including adults, adolescents, and children. Bianca is also a facilitator for our Convivencias, facilitates youth groups, and conducts mental health lectures in the community.
Dolores "Dee" Edwinson immigrated to the United States from El Salvador at the age of five. After graduating from high school, she continued her studies, married, and had three daughters. She eventually earned a Bachelor's degree in Liberal Studies, with an emphasis in Multicultural Studies and then earned a Bilingual Teaching Credential from Sonoma State University and taught elementary school for 17 years. She earned a Master's degree in Psychology from California Southern University in 2015, and joined Humanidad in 2016. As an Associate Marriage Therapist (AMFT), she enjoys working with couples, individuals, families, children, and adolescents. Dee enjoys reading, chorus singing, taking long walks, and visiting her three grown daughters who live in Sacramento, El Dorado, and Sonoma Counties. She lives with her husband in Sonoma County. (AMFT #132971)
Cody has been with Humanidad as an Associate Marriage and Family Therapist for over three years. She works with clients of all ages, and has a particular focus on multicultural competency in her training that supports her work with all intersecting identities. She is committed to helping others transcend their experiences of pain and loss in order to have, be, and do more with their lives than they may have thought they could. She holds undergraduate degrees in Psychology and Social and Behavioral Science, with a Masters in Community Mental Health from Sonoma State University. As a member of the Humanidad lecture team, she lectures and provides trainings at the Santa Rosa Junior College, Lawrence Cook Middle School, and many community locations. *Her clients report feeling seen, valued, and validated in their work with her, and as a result of their dedication to their healing, have achieved their goals and gone on to find fulfillment and self love. She sees it as a privilege to be asked to join her clients in the therapeutic process, and looks forward to doing this work for many years to come.
Since 2016, Teresa Perez-Martinez has been seeking to make difference in the life of clients by improving their self-confidence and helping them through life problems. Her inspiration to enter the field began after witnessing the lack of mental health options in the LatinX community. Teresa graduated in 2016 from Argosy University with a Master of Arts degree in Clinical Mental Health Counseling. Teresa has worked with children, adolescents, couples, adults, and families. Teresa continues to work in various settings by providing support to bilingual families also she is working in psycho-educational groups.
Nancy Herrera is a Licensed Clinical Social Worker. Ms. Herrera received he bachelor's degree in Human Services from Judson University in Elgin, Illinois and a Master's in Social Work from Aurora University in Aurora, Illinois. Growing up in an agrarian community among farmworkers in the central valley of California, Ms. Herrera developed a passion for helping vulnerable people. Ms. Herrera has worked in the social work field in various capacities, such as hospice, grief, homelessness, domestic violence, school counselor, and co-facilitator of NAMI's Family-to-Family group. Ms. Herrera also served on an advisory board to implement a Clinical Mental Health Counseling program at Judson University. Ms. Herrera looks forward to serving, adolescents and adults in the Latinx community at Humanidad Therapy and Education Services.
Noemi Degante Blancas AMFT128554 is a bilingual, bicultural, Associate Marriage and Family Therapist (AMFT). She graduated with a masters in counseling psychology from the University of San Francisco. Noemi works with individual clients of all ages including adults, adolescents, and children; and is currently working specifically with middle school students at Cesar Chavez Language Academy. Noemi has facilitated youth groups, Convivencias and has given presentations in the community on mental health.
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Valerie Pacheco is an Associate Professional Clinical Counselor born and raised in Los Angeles, CA. Ms. Pacheco received her BA in Psychology and Spanish, and an MA in Counseling with an emphasis in community mental health from Loyola Marymount University. Ms. Pacheco continues to embody her alma mater's mission of the education of the whole person, service of faith, and the promotion of justice by assisting the LatinX community. Ms. Pacheco has worked with children, adolescents, adults, couples, and families in a variety of settings including schools, community based organizations, and in private practice. Since coming to Sonoma county, Ms. Pacheco continues to work with various settings by providing support to bilingual and monolingual families and individuals through Sonoma County Behavioral Health, Child Protective Services, and Redwood Children's Clinic. Ms. Pacheco is now part of the Humanidad family and is pursuing more training in Trauma Focus work.
Jordan is a Licensed Marriage and Family Therapist. He works with individuals, couples and families.
We're hiring! …for fore information go here: https://srosahtes.org/careers/ | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,883 |
\section{A results for actions on the circle} \label{s-circle}
To conclude this paper we consider the problem of studying actions of locally moving groups on the circle. This question turns out to be much simpler turns out to be much simpler then the question studied so far, essentially due to the compactness of $\mathbb{S}^1$. In fact, we can actually prove a result for actions on $\mathbb{S}^1$ of a group of homeomorphisms $G\subseteq\homeo(X)$ where $X$ is an arbitrary locally compact space, provided the action of $G$ on $X$ satisfies suitable dynamical conditions.
Given a group $G$ of homeomorphisms of a space $X$ and an open subset $U\subset X$, we let $G_U$ be the pointwise fixator of $X\setminus U$.
Similarly to Definition \ref{d.lm}, we say that a subgroup $G\subseteq \homeo(X)$ is \emph{micro-supported} if for every non-empty subset $U\subset X$, the subgroup $G_U$ is non-trivial. The action of $G$ on $X$ is \emph{extremely proximal} if for every compact subset $K\subsetneq X$ there exists $y\in X$ such that for every open neighborhood $V$ of $y$, there exists $g\in G$ with $g(K)\subset V$.
When $G\subset \homeo(X)$ is a micro-supported group, we denote by $M_G\subset G$ the subgroup of $G$ generated by the subgroups $[G_U, G_U]$ where $U$ varies over relatively compact non-dense open subsets of $X$. Note that $M_G$ is non-trivial and normal in $G$.
In fact, standard arguments similar to the proof of Proposition \ref{p-micro-normal} imply the following (see e.g.\ \cite[Proposition 4.6]{LB-lattice-embeddings}).
\begin{prop}
Let $X$ be a locally compact Hausdorff space, and $G\subset\homeo(X)$ be a micro-supported group acting minimally and strongly proximally on $X$. Then every non-trivial normal subgroup of $G$ contains $M_G$. In particular if $G=M_G$, then $G$ is simple. \end{prop}
Thus when $G\neq M_G$ then the group $G/M_G$ is the largest non-trivial quotient of $G$.
For $x\in X$ we also denote by $G^0_x$ the subgroup of elements that fix pointwise a neighborhood of $x$, and call it the \emph{germ-stabilizer} of $x$.
Further we say that $G$ has the \emph{independence property for pairs of germs} if for every distinct $x_1, x_2\in X$ and for every elements $g_1, g_2\in G$ such that $g_1(x_1)\neq g_2(x_2)$, there exists $g\in G$ and open neighborhoods $U_i\ni x_i$ such that $g$ coincides with $g_i$ in restriction to $U_i$ for $i\in \{1, 2\}$.
\begin{thm} \label{t-circle} Let $X$ be a locally compact Hausdorff space, and let $G\subseteq \homeo(X)$ be a minimal micro-supported subgroup of homeomorphisms of $X$ satisfying the following conditions:
\begin{enumerate}[label=\roman*)]
\item the action of $G$ on $X$ is extremely proximal;
\item for every $x\in X$ the germ-stabilizer $ G^0_x$ acts minimally on $X\setminus\{x\}$.
\end{enumerate}
Assume that $\varphi\colon G\to \homeo_+(\mathbb S^1)$ is a faithful minimal action.
Then there exists a continuous surjective map $\pi \colon \mathbb S^1 \to X$ which is $G$-equivariant with respect to the $\varphi$-action on $\mathbb S^1$ and the natural action on $X$. Moreover, if the action of $G$ on $X$ has the independence property for pairs of germs, then $X$ is homeomorphic to $\mathbb S^1$ and the map $\pi\colon \mathbb S^1 \to X$ is a covering map.
\end{thm}
Before giving the proof we give some comment on the statement.
\begin{rem}
The condition that $\varphi$ be minimal is not so restrictive, since every group action on $\mathbb S^1$ either has a finite orbit (and thus is semi-conjugate to an action factoring through a finite cyclic group), or is semi-conjugate to a minimal action. The condition that $\varphi$ is faithful cannot be avoided, since in this generality little can be said about actions of the largest quotient $G/M_G$ (which could even be a non-abelian free group, see \cite[Proposition 6.11]{LB-lattice-embeddings}). However if $G=M_G$, then theorem implies that every non-trivial action $\varphi\colon G\to \homeo(\mathbb S^1)$ factors onto its standard action on $X$.
\end{rem}
\begin{rem}
It is likely that the assumptions on $G$ in Theorem \ref{t-circle} may be relaxed or modified, and we did not attempt to identify the optimal ones. In particular we do not know whether the assumption that the action of $G$ has independent germs is needed in the last statement. However, note that even with this assumption, the map $\pi$ may be a non-trivial self-cover of $\mathbb S^1$ (thus it is not necessarily a semi-conjugacy to the standard action). Indeed the groups of homeomorphisms of the circle constructed by Hyde, Lodha, and the third named author in \cite[Section 3]{HLR} satisfy all assumptions in Theorem \ref{t-circle} and their action lifts to an action on the universal cover $\mathbb{R}\to \mathbb S^1$, thus also lifts to an action under all self-coverings of $\mathbb S^1$
\end{rem}
The proof of Theorem \ref{t-circle} follows a similar approach the proofs of its special cases that appeared in \cite[Theorem 4.17]{LBMB-sub} for Thompson's group $T$ and in \cite[Theorem D]{MatteBonTriestino} for the groups $\mathsf{T}(\varphi)$ of piecewise linear homeomorphisms of suspension flows defined there. The main difference is that some arguments there make crucial use on specific properties of those groups, such as the absence of free subgroups in the group of piecewise linear homeomorphisms of an interval, while here we get rid of these arguments using Proposition \ref{p-centralizer-fix}, and this allows for a generalization to a much broader class of groups.
\begin{proof}[Proof of Theorem \ref{t-circle}]
Assume that $\varphi\colon G\to \homeo_0(\mathbb S^1)$ is a faithful minimal action. First of all, note that there is no loss of generality in supposing that $\varphi$ is extremely proximal. Namely since $G$ is non-abelian, then by Theorem \ref{t-Margulis} every minimal faithful action of $\varphi\colon G\to \homeo_0(\mathbb{S}^1)$ has a finite centralizer $C_\varphi$ and the action $\varphi$ descends via the quotient map $\mathbb{S}^1\to \mathbb{S}^1/C_\varphi\cong \mathbb{S}^1$ to an extremely proximal action $\varphi_{ep}$. Thus by replacing $\varphi$ with $\varphi_{ep}$ we can assume that $\varphi$ is extremely proximal to begin with.
Given $x\in X$ we will denote by $K_x:=(G^0_x)_c$ the subgroup of all elements whose support is a relatively compact subset of $X\setminus \{x\}$, i.e.\ the union of the groups $G_U$, where $U$ varies over open subsets with $U\Subset X\setminus\{x\}$. Note that $K_x$ is a normal subgroup of $G^0_x$.
By extreme proximality and minimality, for every $U\Subset X$ there exists $g\in G$ such that $g(\overline{U})\cap \overline{U}=\varnothing$, so that $G_U$ is conjugate to a subgroup of its centralizer. Thus by Proposition \ref{p-centralizer-fix} we have $\operatorname{\mathsf{Fix}}^\varphi([G_U, G_U])\neq \varnothing$. By compactness of $\mathbb S^1$, we deduce that for every $x\in X$ we have
\[\operatorname{\mathsf{Fix}}^\varphi([K_x, K_x])=\bigcap_{ U\Subset X,\,x\in U}\operatorname{\mathsf{Fix}}^\varphi([G_U, G_U])\neq \varnothing.\]
Note also that $\operatorname{\mathsf{Fix}}^\varphi([K_x, K_x])\neq \mathbb S^1$, otherwise the subgroup $[K_x, K_x]$ would act trivially, contradicting that the action is faithful. Now fix $x\in X$ and consider the subset $C:=\operatorname{\mathsf{Fix}}^\varphi([K_x, K_x])$, which is $G_x^0$-invariant by normality of $[K_x, K_x]$ in $G_x^0$. Since we are assuming that $\varphi$ is extremely proximal, we can find a sequence $(g_n)$ in $G$ such that $g_n.C$ tends to a point $\xi \in \mathbb S^1$ in the Hausdorff topology. Upon extracting a subnet from $(g_n)$, we can suppose that $g_n(x)$ tends to a limit in the Alexandroff compactification $\hat{X}:=X\cup \{\infty_X\}$. Suppose that this limit is $\infty_X$, that is, that $(g_n(x))$ escapes every compact subset of $X$. Then every element $g\in G_c$ belongs to $G^0_{g_n(x)}$ for $n$ large enough, so that $\varphi(g)$ preserves $g_n.C$ for every $n$ large enough, and thus fixes the point $\xi$. Since $g\in G_c$ is arbitrary, we deduce that $\varphi(G_c)$ fixes $\xi$, and by normality of $G_c$ in $G$ and minimality of $\varphi$ we deduce that $\varphi(G_c)=\{\mathsf{id}\}$, contradicting that the action is faithful. Thus the limit of $(g_n(x))$ is a point $y\in X$. The same argument as above then shows that $\varphi(G^{0}_y)$ fixes $\xi$.
Once the existence of such points $y\in X$ and $\xi\in\mathbb S^1$ has been proven, the rest of the proof is essentially the same as in \cite{LBMB-sub} or \cite{MatteBonTriestino}, but we outline a self-contained argument for completeness. We claim that for every $\zeta\in \mathbb S^1$ there exists a unique point $\pi(\zeta)\in X$ such that $\varphi(G^{0}_{\pi(\zeta)})$ fixes $\zeta$, and that the map $\pi\colon \mathbb S^1\to X$ defined in this way is continuous.
Let us first show that if such a point exists, it must be unique. Namely assume that $x_1, x_2\in X$ are distinct points such that $\varphi(G^0_{x_i})$ fixes $\zeta$ for $i\in \{1, 2\}$, so that the $\varphi$-image of $H:=\langle G_{x_1}^0, G_{x_2}^0\rangle$ fixes $y$. Let $U\subset X$ be any non-dense open subset of $X$. After the assumption of minimality of the action of $G_{x_1}^0$ on $X\setminus \{x_1\}$, we can find $g\in G_{x_1}^0$ such that $g(x_2)\notin \overline{U}$, so that $g^{-1}G_Ug=G_{g^{-1}(U)} \subset G_{x_2}^0\subset H$. Since we also have $g\in H$, we obtain that $G_U\subset H$. Thus $H$ contains the non-trivial normal subgroup $N$ of $G$ generated by $G_U$ where $U$ varies over all non-dense open subsets of $X$. Then $\varphi(N)$ fixes $\zeta$, which is a contradiction using again minimality and normality of $N$.
To show existence and continuity, one first checks the following fact ($\ast$): if $(z_i)$ is a net of points in $X$ converging to a limit $z\in \hat{X}:=X\cup \{\infty_X\}$, and if $(\zeta_i)$ is a net of points in $\mathbb S^1$ converging to some limit $\zeta\in \mathbb S^1$ such that $\varphi(G_{z_i}^0)$ fixes $\zeta_i$ for every $i$, then $z\in X$ and $\varphi(G^0_z)$ fixes $\zeta$. The proof of ($\ast$) follows similar arguments as above; more precisely, the fact that $z$ belongs to $X$ is shown by arguing as above, to prove that otherwise the group $\varphi(G_c)$ would fix $\zeta$, and the same argument also shows that $\varphi(G^0_z)$ fixes $\zeta$.
Now let $\zeta\in \mathbb S^1$ be arbitrary, and choose $y\in X$ and $\xi\in \mathbb S^1$ such that $\varphi(G^0_y)$ fixes $\xi$ (whose existence has already been proven). By minimality of $\varphi$ we can find a net $(g_i)$ in $G$ such that $\zeta_i:=g_i.\zeta$ converges to $\zeta$, and upon extracting a subnet we can suppose that $y_i:=g_i(y)$ converges to some $z\in \hat{X}$. Then by ($\ast$) we have $z\in X$ and $\varphi(G^0_z)$ fixes $\zeta$. This shows the existence of a point $\pi(\zeta):=z$ as desired, and ($\ast$) also implies the continuity of the map $\pi \colon \mathbb S^1 \to X$ defined in this way. The map is clearly $G$-equivariant, so its image $\pi(\mathbb S^1)$ is a compact $G$-invariant subset of $X$, so by minimality of the standard action of $G$ on $X$, the map $\pi$ must be surjective.
\smallskip
Suppose now that $G$ has the independence property for pairs of germs and let us show that the map $\pi$ must be injective (and thus a homeomorphism). As a preliminary observation, note that for every $x\in X$ the fiber $\pi^{-1}(x)$ must have empty interior, since otherwise by $G$-equivariance, the open subset $\Int(\pi^{-1}(x))$ would be a wandering domain in $\mathbb S^1$, contradicting minimality.
Assume by contradiction that there exist $\xi_1\neq \xi_2$ in $\mathbb S^1$ such that $\pi(\xi_1)=\pi(\xi_2)=:x$, and let $I:=(\xi_1, \xi_2)$ be the arc between them (with respect to the clockwise orientation of $\mathbb S^1$). Since $I\not\subset \pi^{-1}(x)$, we can choose $\zeta\in I$ with $z:=\pi(\zeta)\neq x$. By minimality of $\varphi$, there exists $g\in G$ such that $g.\zeta\notin I$.
Assume first that $g(x)\neq z$. Using the independence property for pairs of germs, we can find $h\in G$ which coincides with the identity on some neighborhood of $z$, and with $g^{-1}$ on some neighborhood of $x$. Then $h^{-1}\in G^0_z$, so $h^{-1}.\zeta=\zeta$. On the other hand $gh\in G^0_x$, so that $\varphi(gh)$ fixes $\xi_1$ and $\xi_2$ and preserves the arc $I$. Writing $g=(gh)h^{-1}$, we see that $g.\zeta=gh.\zeta\in I$, contradicting the choice of $g$.
Assume now that $g(x)=z$. In this case choose an open subarc $J\subset I$ containing $\zeta$ such that $g.J\cap I=\varnothing$. Since $\pi^{-1}(z)$ has empty interior, we can find a point $\zeta'\in J$ such that the point $z':=\pi(\zeta')$ is different from $z$. Then we have $g(y)\neq z'$, so we can repeat the previous reasoning using the points $\zeta', z'$ instead of $\zeta, z$. This provides the desired contradiction and shows that the map $\pi$ is injective. \qedhere
\end{proof}
This implies a rigidity result for groups that are given by a locally moving action on the circle (in the sense that for every proper interval $I\subset \mathbb S^1$, the rigid stabilizer acts on $I$ without fixed points).
\begin{cor}\label{c-circle-circle}
For $X=\mathbb S^1$, let $G\subseteq \homeo_0(X)$ be locally moving. Then for every faithful minimal action $\varphi\colon G\to \homeo_0(\mathbb S^1)$, there exists a continuous surjective equivariant map $\pi\colon \mathbb S^1\to X$. Moreover, if the standard action of $G$ on $X$ has the independence property for pairs of germs, then $\pi$ is a covering map.
\end{cor}
\begin{ex}
Let us explain how Corollary \ref{c-circle-circle} recovers the result of Matsumoto that every non-trivial action of $G=\homeo_0(\mathbb S^1)$ on $\mathbb S^1$ is conjugate to its standard action. Note that $G$ clearly satisfies all assumptions of Corollary \ref{c-circle-circle} and has the independence property for pairs of germs. Since $G$ has elements of finite order, it cannot act non-trivially on the circle with a fixed point, and since moreover it is simple it cannot act with a finite orbit, so that every non-trivial action $\varphi\colon G\to \homeo_0(\mathbb S^1)$ must be semi-conjugate to the minimal action $\varphi_{min}$ (which is automatically faithful). Corollary \ref{c-circle-circle} then shows that $\varphi_{min}$ is the lift of its natural action via a self-cover, and again it is not difficult to see (using elements conjugate to rotations) that this is possible only if $\varphi_{min}$ is conjugate to the standard action, and in particular all its orbits are uncountable. If $\varphi$ had an exceptional minimal set $\Lambda \subsetneq \mathbb S^1$, then every connected component of the complement would be mapped to a point with a countable orbit, so that this is not possible. We deduce that $\varphi$ is minimal, thus conjugate to $\varphi_{min}$ and thus to the standard action.
\end{ex}
\begin{ex}
In a similar fashion, Corollary \ref{c-circle-circle} recovers the result of Ghys \cite{Ghys} that every action of Thompson's group $T$ on the circle is semi-conjugate to its standard action. Indeed $T$ satisfies all assumptions in Corollary \ref{c-circle-circle}. As above, one uses its simplicity to show that every action $\varphi\colon T\to \homeo_0(\mathbb S^1)$ is semi-conjugate to a faithful minimal action, and thus by the corollary, to the lift of the standard action through a self-cover $\pi \colon \mathbb S^1\to \mathbb S^1$, and then argue that this is possible only if $\pi$ is a homeomorphism.
\end{ex}
By considering other kind of spaces, Theorem \ref{t-circle} can be used to construct groups that cannot admit interesting actions on the circle.
\begin{cor} \label{c-circle-fixed}
Let $X$ be a locally compact Hausdorff space which is not a continuous image of $\mathbb S^1$ (e.g.\ if $X$ is not compact, or not path-connected), and let $G\subseteq \homeo(X)$ be a subgroup as in Theorem \ref{t-circle}. Then $G$ has no faithful minimal action on $\mathbb S^1$. In particular, if $G$ is simple, every action $\varphi\colon G\to \homeo_0(\mathbb S^1)$ has a fixed point.
\end{cor}
\begin{ex}Examples of groups to which Corollary \ref{c-circle-fixed} are the groups of piecewise linear homeomorphisms of flows $\mathsf{T}(\varphi)$ from \cite{MatteBonTriestino}. For every homeomorphism $\varphi$ of the Cantor set $X$, the group $\mathsf{T}(\varphi)$ is a group of homeomorphisms of the mapping torus $Y^\varphi$ of $(X, \varphi)$ defined analogously to Thompson's group $T$ (see \cite{MatteBonTriestino} for details). When $\varphi$ is a minimal homeomorphism, the group $\mathsf{T}(\varphi)$ is simple \cite[Theorem B]{MatteBonTriestino} and its action on $Y^\varphi$ satisfies all assumptions in Theorem \ref{t-circle}. Since however the space $Y^\varphi$ is not path-connected, Corollary \ref{c-circle-fixed} recovers the fact proven in \cite[Theorem D]{MatteBonTriestino} that every action of the group $\mathsf{T}(\varphi)$ on the circle has a fixed point. Moreover, it allows to extend the conclusion to many groups defined similarly but not by PL homeomorphisms, for instance any simple overgroup of $\mathsf{T}(\varphi)$ in $\homeo(Y^\varphi)$ (see Darbinyan and Steenbock \cite{darbinyan2020embeddings} for a vast family of such groups). \end{ex}
\section{A structure theorem for exotic actions of a class of locally moving groups}
\label{sec.locandfoc}
\subsection{The class $\mathcal F${}}
In this section we will focus on a class of locally moving groups, and prove a more precise structure theorem on their exotic continuous actions on the line. Such class, named $\mathcal F${}, is defined in terms of a finite generation condition on the rigid stabilizers.
\begin{dfn}\label{def.classF} For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be a subgroup. We say that $G$ is in the \emph{class $\mathcal F${}\;}if it satisfies the following conditions:
\begin{enumerate}[label=(\arabic*)]
\item $G$ is locally moving,
\item for every $x\in X$ the group $G_{(a, x)}$ is contained in a finitely generated subgroup of $G_+$, and the group $G_{(x, b)}$ is contained in a finitely generated subgroup of $G_-$.
\end{enumerate}
\end{dfn}
Various examples of groups in the class $\mathcal F${}\;arise as groups of piecewise linear and projective homomorphisms: Thompson's group $F$ and all Thompson--Brown--Stein groups $F_{n_1,\ldots, n_k}$, many other Bieri--Strebel groups (see Section \ref{s-few-actions}) and the groups of Lodha--Moore \cite{LodhaMoore}. The following proposition shows that this class goes far beyond groups of piecewise linear or projective homeomorphisms.
\begin{prop}\label{prop.superF}
For $X=(a,b)$, let $H\subset \homeo_0(X)$ be a countable subgroup. Then there exists a finitely generated subgroup $G\subset \homeo_0(X)$ which contains $H$, has an element without fixed points, and belongs to the class $\mathcal F${}\;(hence to the class $\mathcal{F}_0${}\;defined later).
\end{prop}
\begin{proof} We can assume without loss of generality that $H$ is finitely generated, since every countable subgroup of $\homeo_0(X)$ is contained in a finitely generated one \cite{LeRouxMann}. Assume also for simplicity that $X=(0, 1)$. Let $b_\ell\colon (0, 1)\to (0, {3/4})$ and $b_r\colon (0, 1)\to (1/4, 1)$ be the homeomorphisms defined respectively by
\[b_\ell(x)=\left\{\begin{array}{lr}x & x\in (0, 1/2),\\[.5em] \frac{1}{2}x+\frac{1}{4} & x\in [1/2, 1),
\end{array} \right. \quad b_r(x)=\left\{\begin{array}{lr}\frac{1}{2}x +\frac{1}{4}& x\in (0, 1/2),\\[.5em] x & x\in [1/2, 1).
\end{array} \right. \]
Set also $b_0=b_\ell b_r$.
That is, $b_0$ is the homothety of slope $1/2$ centered at the point $1/2$, while each of $b_\ell$ and $b_r$ fixes half of the interval $(0, 1)$ and coincides with $b_0$ on the other half. See Figure \ref{fig.maps_classF}.
\begin{figure}[ht]
\includegraphics[scale=1]{classF-1.pdf}
\caption{The maps $b_r$ (red) and $b_\ell$ (blue).}\label{fig.maps_classF}
\end{figure}
Set $H_0=b_0Hb_0^{-1}$, $H_\ell=b_\ell H b_\ell^{-1}$, and $H_r=b_rHb_r^{-1}$. Note that each of these groups is a group of homeomorphisms of a subinterval of $(0, 1)$, and we see them as groups of homeomorphisms of $(0, 1)$ by extending them to the identity outside their support. Set $G=\langle H, H_\ell, H_r, H_0, F\rangle$, where $F$ is the standard copy of Thompson's group $F$ acting on $(0, 1)$.
As $F$ contains elements without fixed points, so does $G$.
Let us prove that $G$ is in the class $\mathcal F${}.
As $F$ and $H$ are finitely generated, so is $G$; moreover $G$ is locally moving since $F$ is so. Let us show that $G_{(0, 1/2)}$ is contained in a finitely generated subgroup of $G_+$. To this end, consider the group $\Gamma=b_\ell G b_\ell ^{-1}$. Then $\Gamma$ is a finitely generated subgroup of $\homeo_0((0, 1))$ supported in $(0, 3/4)$ (we again extend it as the identity on $(3/4, 1)$). Moreover, $\Gamma$ contains $G_{(0, 1/2)}$: indeed since $b_\ell$ acts trivially on $(0, 1/2)$, for $g\in G_{(0, 1/2)}$ we have $g=b_\ell gb_\ell^{-1}\in \Gamma$. Thus the desired conclusion follows if we show that $\Gamma$ is contained in $G$ (hence in $G_+$).
For this, note that $b_\ell F b_\ell^{-1}=F_{(0, 3/4)} \subset F \subset G$ and $b_\ell Hb_\ell^{-1}=H_\ell \subset G$; we also easily observe
\[
b_\ell H_rb_{\ell}^{-1}=b_\ell b_r H (b_\ell b_r)^{-1}=b_0H b_0^{-1}=H_0\subset G.
\]
Choose an element $f\in F$ which coincides with $b_\ell$ in restriction to $(0, 3/4)$. Then the conjugation action of $f$ on the subgroups $H_\ell, H_0$ coincides with the conjugation by $b_\ell$, thus $b_\ell H_\ell b_\ell^{-1}= f H_\ell f^{-1}$ and $b_\ell H_0 b_\ell^{-1}=f H_0 f^{-1}$ are also contained in $G$. Since
\[\Gamma=b_\ell \langle H, H_0, H_\ell, H_r, F\rangle b_\ell^{-1},\]
we conclude that that $\Gamma$ is contained in $G$. Thus $\Gamma \subset G_+$ is a finitely generated subgroup that contains $G_{(0, 1/2)}$. Since for every $x\in (0,1)$ the group $G_{(0, x)}$ is conjugate to a subgroup of $G_{(0, 1/2)}$, we have that $G_{(0, x)}$ is contained in a finitely generated subgroup of $G_+$ for every $x$. The case of the subgroups $G_{(x, 1)}$ is analogous. \qedhere
\end{proof}
\subsection{The main theorem for the class $\mathcal F${}}\label{ss.classFontheline}
The goal of this subsection is to show that all exotic actions of a group in the class $\mathcal F${}\;must be tightly related to the standard action, through the notion of horograding of $\mathbb{R}$-focal actions (see \S\ref{ss.horograding}).
\begin{thm}\label{t-C-trichotomy}
For $X=(a,b)$, let $G\subset \homeo_0(X)$ be a subgroup in the class $\mathcal F${}. Then every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ without fixed points is semi-conjugate to an action in one of the following families:
\begin{enumerate}[label=(\roman*)]
\item (\emph{Standard}) the natural action on $X$;
\item (\emph{Induced from a quotient}) an action induced from the largest quotient $G/[G_c, G_c]$
\item (\emph{$\mathbb{R}$-focal})\label{t-C-i-focal} a minimal $\mathbb{R}$-focal action, which can be horograded by the natural action of $G$ on $X$.
\end{enumerate}
\end{thm}
\begin{rem} In the $\mathbb{R}$-focal case, Proposition \ref{prop.dynamic-class-horo} allows to determine the dynamics of individual elements by looking at the standard action of $G$ on $X$. For this, consider the case where $\varphi$ is increasingly horograded by the natural action of $G$ on $X$ (the other case is analogous). Then, we have the following: \begin{itemize}
\item if $\operatorname{\mathsf{Fix}}(g)$ accumulates at $b$, then $\varphi(g)$ is totally bounded;
\item if $g(x)>x$ (respectively $g(x)<x$) in a neighborhood of $b$, then $\varphi$ is an expanding (respectively contracting) pseudohomothety and
\item if $g(x)>x$ (respectively $g(x)<x$) for every $x\in X$, then $\varphi(g)$ is an expanding (respectively contracting) homothety.
\end{itemize}
\end{rem}
\subsubsection{Sketch of the proof and preliminary observations}
We will divide the proof into steps, which individually yield some additional information. As $G$ is locally moving, by Theorem \ref{t-lm-trichotomy} it is enough to consider the case where $\varphi$ is an exotic action. By symmetry, we will assume that the subgroup $G_+$ is locally dominated by commuting elements (within $[G_c,G_c]$).
We first observe that we can assume that the action $\varphi$ is minimal (Lemma \ref{l-C-minimal}) and then describe the set of fixed points for the images of the rigid stabilizers $\varphi(G_{(a,x)})$ (Lemma \ref{p-C-fix}). This allows to introduce an invariant CF-cover, which will give that $\varphi$ is $\mathbb{R}$-focal (Proposition \ref{c-C-focal}). To determine the horograding action, we have to construct a focal action on a planar directed tree representing $\varphi$ (this is done throughout \S\ref{subs.construction}). We then put all of this together and prove Theorem \ref{t-C-trichotomy}. Let us start.
\begin{lem} \label{l-C-minimal}
For $X=(a,b)$, let $G\subset \homeo_0(X)$ be a subgroup in the class $\mathcal F${}. Consider a faithful exotic action $\varphi\colon G\to \homeo_0(\mathbb{R})$. Then $\varphi$ is semi-conjugate to a minimal action.
\end{lem}
\begin{proof}
We will assume that $G_+$ is locally dominated by commuting elements, the other case being analogous.
Then for every $x \in X$ the image $\varphi(N_{(x, b)})$ has no fixed points (see Proposition \ref{p-lm-trichotomy}). Since $G$ is in the class $\mathcal F${}, for every $x\in X$ there exists a finitely generated subgroup $\Gamma \subset G_-$ containing $N_{(x,b)}$,so that $\varphi(\Gamma)$ has no fixed points. Thus, there exists a compact interval $I\subset \mathbb{R}$ which intersects every $\varphi(\Gamma)$-closed invariant subset, and in particular every $\varphi(G)$-closed invariant subset. As in the proof of Proposition \ref{p-focal-semiconj}, it follows that $\varphi(G)$ has a unique minimal invariant set, and that $\varphi$ is semi-conjugate to a minimal action. \qedhere
\end{proof}
Thus without loss of generality, upon replacing $\varphi$ with a semi-conjugate action, we will assume that $\varphi$ is minimal. The key ingredient in the proof of Theorem \ref{t-C-trichotomy} is the following result, which follows again from the trichotomy for locally moving groups (Theorem \ref{t-lm-trichotomy}) together with the assumption that $G$ is in $\mathcal F${}.
\begin{lem}[Key lemma]\label{p-C-fix}
For $X=(a,b)$, let $G\subset \homeo_0(X)$ be a subgroup in the class $\mathcal F${}. Consider a minimal faithful exotic action $\varphi\colon G\to \homeo_0(\mathbb{R})$ satisfying that $G_+$ is locally dominated by commuting elements. Then for every $x\in X$ the subset $\operatorname{\mathsf{Fix}}^\varphi(G_{(a, x)})$ is non-empty, accumulates on both $\pm \infty$, and has empty interior. The family of closed subsets $\{\operatorname{\mathsf{Fix}}^\varphi(G_{(a, x)})\}_{x\in X}$ is decreasing in $x$, and we have
\[\lim_{x\to a} \operatorname{\mathsf{Fix}}^\varphi(G_{(a, x)})=\mathbb{R}\quad \text{and}\quad \lim_{x\to b} \operatorname{\mathsf{Fix}}^\varphi(G_{(a, x)}) =\varnothing,\]
where the limits are taken with respect to the Fell topology on closed subsets of $\mathbb{R}$.
\end{lem}
\begin{proof}
Since $G_+$ is locally dominated by commuting elements, all its finitely generated subgroups are totally bounded. Moreover, since for every $x\in X$ the subgroup $G_{(a, x)}$ is contained in a finitely generated subgroup of $G_+$, it is also totally bounded, i.e.\ the subset $C_x:=\operatorname{\mathsf{Fix}}^\varphi(G_{(a, x)})$ is non-empty and accumulates on both $\pm \infty$. Let us postpone the proof that it has empty interior. Observe that since $\{G_{(a, x)}\colon x\in X\}$ is an increasing family of subgroups, the family of subsets $\{C_x \colon x\in X\}$ is decreasing. In particular, the two limits in the statement exist and are given by
\[\ \lim_{x\to a} C_x=\overline{\bigcup_{x\in X} C_x } \quad \text{and}\quad \lim_{x\to b} C_x= \bigcap_{x\in X} C_x.\]
From this, since the family $\{C_x\}$ satisfies the equivariant property $C_{g(x)}=g.C_x$, it is clear that the two limits are closed $G$-invariant subsets, thus by minimality each of them is equal either to $\mathbb{R}$ or to $\varnothing$.
Since the limit on the left-hand side contains every subset $C_x$, it is non-empty, thus it must be equal to $\mathbb{R}$. As for the second, it is contained in each subset $C_x$, which is a strict subset of $\mathbb{R}$ (otherwise $\varphi(G_{(a, x)})$ would act trivially, contradicting faithfulness of the action), and therefore it must be equal to $\varnothing$.
Let us now show that every subset $C_x$ must have empty interior. Assume by contradiction that $J\subset C_x$ is a non-empty open interval. Fix also an arbitrary $y\in X$ and write $H=G_{(a, y)}$. Let $I$ be a connected component of $\suppphi(H)=\mathbb{R}\setminus C_y$. Since $\varphi$ is minimal, then it is proximal (Lemma \ref{l.exotic_proximal}), hence there exists $g\in G$ such that $g.J\supset I$. Thus $\varphi(G_{(a, g(x))})$ acts trivially on $I$, and since $G_{(a, x)} \cap H \neq \{\mathsf{id}\}$, we deduce that the action of $H$ on $I$ is not faithful. Since $H$ is a locally moving subgroup of $\homeo_{0}((a, y))$, by Proposition \ref{p-micro-normal} we see that the kernel of this action must contain $[H_c, H_c]$. Since $I$ was an arbitrary connected component of $\suppphi(H)$, this implies that $[H_c, H_c]$ acts trivially on all $\suppphi(H)$, and thus the $\varphi$-image of $[H_c, H_c]$ is trivial, contradicting the assumption that $\varphi$ is faithful. \qedhere
\end{proof}
\subsubsection{An invariant CF-cover}\label{ssc.CF_family} Lemma \ref{p-C-fix} allows to identify a natural invariant CF-family for the action $\varphi$, and to construct a planar directed tree. To this end, let us introduce the following notation.
\begin{dfn} \label{d-I-phi}
Let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be as in Lemma \ref{p-C-fix}. For every $x\in X$ and $\xi\in \mathbb{R}$, with $\xi\notin \operatorname{\mathsf{Fix}}^\varphi(G_{(a, x)})$, we denote by $\Iphi(x, \xi)$ the connected component of $\suppphi(G_{(a, x)})$ which contains $\xi$.
\end{dfn}
Note that the family of open subsets $\suppphi(G_{(a, x)})=\mathbb{R}\setminus \operatorname{\mathsf{Fix}}^\varphi(G_{(a, x)})$ is increasing with respect to $x\in X$. Thus if $\xi\in \suppphi(G_{(a, x)})$, then for every $y>x$ we have $\xi\in\suppphi(G_{(a, y)})$ and $\Iphi(x, \xi)\subset \Iphi(y, \xi)$ for every $y>x$.
Note also that Lemma \ref{p-C-fix} implies the subset $\suppphi(G_{(a, x)})$ is open and dense for every $x\in X$. Thus, by Baire's theorem, the intersection
\begin{equation} \label{e-residual} \Xi:= \bigcap_{x\in X} \suppphi(G_{(a, x)})\end{equation}
is a $G_\delta$ dense subset of $\mathbb{R}$, which is moreover invariant under the action of $G$. For $\xi\in \Xi$, the interval $\Iphi(x, \xi)$ is well-defined and non-empty for every $x\in X$. In the following $\xi$ will always denote a point in $\Xi$ unless mentioned otherwise.
With this notation, we have the following consequence of Lemma \ref{p-C-fix}.
\begin{prop}\label{c-C-focal}
With notation as above, the collection \[\mathcal{S}=\{\Iphi(x, \xi)\colon x\in X, \xi\in \Xi\}\] defines a $G$-invariant CF-cover. In particular, $\varphi$ is a minimal $\mathbb{R}$-focal action.
\end{prop}
\begin{proof}
It is clear from the definition of the intervals $\Iphi(x, \xi)$ that they verify the equivariance property $g.\Iphi(x, \xi)=\Iphi(g(x), g.\xi)$. Thus the family $\mathcal{S}$ is invariant, and covers $\mathbb{R}$ by the second limit in Lemma \ref{p-C-fix}. Note also that Lemma \ref{p-C-fix} implies that every interval in $\mathcal S$ is bounded. To see that it is cross-free, consider two intervals $I_1,I_2\subset \mathbb{R}$ of the form $I_1=\Iphi(x_1, \xi_1)$ and $I _2=\Iphi(x_2, \xi_2)$, with $x_1\leq x_2$.
Observe first that if $x_1=x_2$, then $I_1$ and $I_2$ are both connected components of $\suppphi(G_{(a, x_1)})$, so they are either equal or disjoint. Otherwise, assume that $I_1\cap I_2\neq \varnothing$. Note that $I_1\subset \Iphi(x_2, \xi_1)$, so that $I_2=\Iphi(x_2, \xi_1)$. Thus $I_1\subset I_2$. This shows that $\mathcal{S}$ is an invariant CF-cover. As $\varphi$ is minimal, Proposition \ref{prop.minimalimpliesfocal} implies that $\varphi$ is $\mathbb{R}$-focal.\qedhere
\end{proof}
\subsubsection{A planar directed tree}\label{subs.construction}
We now explain how to use the invariant CF-cover $\mathcal S$ to construct a planar directed tree with the horograding required for the conclusion of Theorem \ref{t-C-trichotomy}. For this, we need to examine some further properties of this family.
As already observed, for a fixed point $\xi\in \Xi$ the intervals $\{\Iphi(x, \xi)\colon x\in X\}$ are a totally order family, non-decreasing in the variable $x\in X$, but in general they do not vary continuously with respect to $x$ (in fact one could easily show that their endpoint \emph{cannot} vary continuously at every $x\in X$). Thus we also introduce the outer and inner limits
\begin{equation*} \label{e-iout}
\Iphiout(x, \xi):=\Int\left(\bigcap_{y>x} \Iphi(y, \xi)\right) \quad \text{and}\quad \Iphiinn(x, \xi)=\bigcup_{y<x} \Iphi(y, \xi).
\end{equation*}
Notice that the fact that being crossed is an open condition together with Proposition \ref{c-C-focal} imply that both $\{\Iphiout(x, \xi)\}$ and $\{\Iphiinn(x, \xi)\}$ are also invariant CF-covers.
We define the directed tree $\mathbb T$ as the family of intervals
\begin{equation*}
\label{e-C-def-tree}
\mathbb T:=\{\Iphiout(x, \xi)\colon x\in X, \xi\in \Xi\},
\end{equation*}
with the partial order $\triangleleft$ given by the inclusion of intervals. The group $G$ acts on $\mathbb T$, by the expression $g.\Iphiout(x, \xi)=\Iphiout(g(x), g.\xi)$, and this action clearly preserves the inclusion relations.
To show that this defines indeed a directed tree we first show the following lemma, which analyzes how these intervals vary as functions of $(x,\xi)$.
\begin{lem}[Injectivity in the variable $x$]\label{l-inj-x}
With notation as above, if $\Iphi(x_1, \xi_1)=\Iphi(x_2,\xi_2)$, then $x_1=x_2$. As a consequence, the same conclusion holds for the families of intervals $\Iphiout(x, \xi)$ and $\Iphiinn(x, \xi)$.
\end{lem}
\begin{proof}
Assume by contradiction that $x_1<x_2$. Then, since $G_{(a, x_2)}$ preserves the interval $\Iphi(x_2, \xi_2)$, for every $g\in G_{(a, x_2)}$ we have
\[\Iphi(x_2,\xi_2)=g.\Iphi(x_2,\xi_2)=g.\Iphi(x_1, \xi_1)=\Iphi(g(x_1), g.\xi_1).\]
By definition, the last interval is a connected component of $\suppphi(G_{(a, g(x_1))})$, so in particular the image $\varphi(G_{(a, g(x_1))})$ has no fixed point in $\Iphi(x_2, \xi_2)$ for every $g\in G_{(a, x_2)}$. However since $G_{(a, x_2)}$ acts without fixed points on $(a, x_2)$ we can choose $g$ such that $g(x_1)$ is arbitrarily close to $a$. This contradicts the fact that $\lim_{y\to a}\operatorname{\mathsf{Fix}}^\varphi(G_{(a, y)})=\mathbb{R}$, established by Lemma \ref{p-C-fix}.
\end{proof}
The previous lemma implies that the function
\begin{equation*}
\label{e-C-def-grading}
\dfcn{\pi}{\mathbb T}{X}{\Iphiout(x, \xi)}{x}
\end{equation*}
is well defined, and clearly $G$-equivariant.
\begin{lem}
With notation as above, the set $(\mathbb T, \triangleleft)$ is a directed tree and the map $\pi\colon \mathbb T\to X$ is an increasing horograding.
\end{lem}
\begin{proof}
Let us check that all conditions in Definition \ref{def d-trees} are satisfied. Given $I=\Iphiout(x, \xi)\in \mathbb T$, the set of intervals above $I$ is $\{\Iphiout(y, x)\colon y\ge x\}$, which by Lemma \ref{l-inj-x} is order-isomorphic to the interval $[x, b)$, showing \ref{i-tree}. To check \ref{i-directed} note that since intervals $\{\Iphi(x, \xi)\}$ define a CF-cover, any two elements in $\mathbb T$ have a common upper bound, and by definition of the outer approximations $\Iphiout$ the infimum of the set of such upper bounds is again a common upper bound (thus the smallest possible). Finally to see \ref{i-chains-R}, note that for a given $x\in X$ there are countably many intervals of the form $\Iphiout(x, \xi)$ when $\xi$ varies (since every two such intervals are either equal or disjoint). Thus by letting $X_0\subset X$ be any countable dense set, the collection $\Sigma=\{\Iphiout(x, \xi)\colon x\in X_0, \xi\in \Xi\}$ satisfies \ref{i-chains-R}. The fact that $\pi$ is an increasing horograding is clear from the construction. \qedhere
\end{proof}
Finally note that for a point $v=\Iphiout(x, \xi)\in \mathbb T$, the set $E^-_v$ of directions below $v$ is in one-to-one correspondence with the set of intervals $\{\Iphiinn(x, \xi')\colon \xi'\in \Iphiout(x, \xi)\}$. Since any two distinct intervals of this form are disjoint, this set of intervals inherits a natural order from the ordering of $\mathbb{R}$, which allows to define an order $<^v$ on $E_v^-$. The family of orders $\prec=\{<^v \colon v\in \mathbb T\}$ defined in this way is a planar order on $\mathbb T$ which is invariant for the action of $G$.
\begin{rem} We point out that the residual set $\Xi$ defined in \eqref{e-residual} is in natural correspondence with the $\pi$-complete boundary of $(\mathbb{T},\triangleleft)$ (see Definition \ref{dfn.pi_complete}). However, we do not elaborate on this since is not necessary for the rest of the discussion.
\end{rem}
\begin{proof}[Proof of Theorem \ref{t-C-trichotomy}]
After Theorem \ref{t-lm-trichotomy} we can suppose that $\varphi$ is an exotic action, and we will assume that the subgroup $G_+$ is locally dominated by commuting elements. Then Lemma \ref{l-C-minimal} and Proposition \ref{c-C-focal} together prove the first part in point \ref{t-C-i-focal} of Theorem \ref{t-C-trichotomy}, namely that, up to semi-conjugacy, $\varphi$ is $\mathbb{R}$-focal.
Consider then the action $\Phi\colon G\to \Aut(\mathbb T, \triangleleft, \prec)$ of $G$ on the planar directed tree defined above. Note that the fact that the action is proximal (Lemma \ref{l.exotic_proximal}) gives that the action is focal. The map $\pi\colon \mathbb T\to X$ is an increasing $G$-equivariant horograding, so we only need to show that the action $\varphi$ is conjugate the dynamical realization of $\Phi$. To this end observe that every $\xi\in \Xi$ can be thought of as an end $\xi\in \partial_*\mathbb T$, namely the unique infimum of the maximal totally ordered subset $\{\Iphiout(x, \xi)\colon x\in X\}$. The order on the $G$-orbit of $\xi$ with respect to the action of $\varphi$ induced from $\mathbb{R}$ coincides with the order induced by the planar order on $\partial^*\mathbb T$. This shows that $\varphi$ is the dynamical realization of $\Phi:G\to \Aut(\mathbb T, \omega, \prec)$, as desired. \qedhere \end{proof}
\subsection{Non-orientation-preserving actions}
\label{ssec non-orientation-preserving actions}
Let us record here a simple consequence of the previous results, which might be worth point out. Namely if we allow group actions on intervals that are not necessarily orientation-preserving, then a stronger rigidity phenomenon occurs for groups in the class $\mathcal F${}, which rules out the existence of exotic actions altogether.
\begin{cor} \label{c-C-flip}
For $X=(a, b)$, let $G\subseteq\homeo(X)$ be a subgroup such that $G\nsubseteq \homeo_0(X)$ and $G\cap\homeo_0(X)$ is in the class $\mathcal F${}. Then every action $\varphi\colon G\to \homeo(\mathbb{R})$, which does not fix any point or a pair of points, is semi-conjugate either to the natural action of $G$ on $X$ or to a non-faithful action.
In particular every faithful minimal action $\varphi\colon G\to \homeo(\mathbb{R})$ is topologically conjugate to the natural action on $X$.
\end{cor}
Note that the proof does not require Theorem \ref{t-C-trichotomy}, but only Lemma \ref{p-C-fix} (actually only the first part of its statement).
\begin{proof}[Proof of Corollary \ref{c-C-flip}]
Set $H=G\cap \homeo_0(X)$, and let $\varphi\colon G\to \homeo(\mathbb{R})$ be an action without fixed points. Note that the assumption implies that the image of $H$ has no fixed points, since otherwise every point in $\operatorname{\mathsf{Fix}}^\varphi(H)$ has a $G$-orbit of order at most $2=|G/H|$. Assume first that the action $\varphi\restriction_H$ is exotic. Indeed, if this is not the case, then by Proposition \ref{p-lm-trichotomy} and Lemma \ref{p-C-fix} we can assume that $\operatorname{\mathsf{Fix}}^\varphi(H_{(a, x)})\neq \varnothing$ for every $x\in X$, while $\operatorname{\mathsf{Fix}}^\varphi(H_{(x, b)})=\varnothing$ for every $x\in X$ (or that the opposite condition holds). However, since $G$ contains elements that reverse the orientation of $X$, every subgroup of the form $H_{(a, x)}$ is conjugate inside $G$ to a subgroup of the form $H_{(y, b)}$, so this is not possible. Thus, $\varphi\restriction_H$ is semi-conjugate either to an action of $H/[H_c, H_c]$ or to the standard action of $H$ on $X$. The first case occurs if and only if $\operatorname{\mathsf{Fix}}^\varphi([H_c, H_c])\neq \varnothing$. Since $[H_c, H_c]$ is normal in the whole group $G$, this implies that in fact the action of the whole group $G$ is semi-conjugate to an action of $G/[H_c, H_c]$. If $\varphi\restriction_H$ is semi-conjugate to the standard action, then there exists a unique minimal $H$-invariant set $\Lambda\subset \mathbb{R}$, which is invariant under the whole group $G$, and we can obtain a minimal action $\psi\colon G\to \homeo(\mathbb{R})$ semi-conjugate to $\varphi$ by collapsing all connected components of $\mathbb{R}\setminus \Lambda$. Then by construction $\psi\restriction_H$ is conjugate the standard action of $H$ on $X$, so let $q\colon \mathbb{R}\to X$ be a topological conjugacy. It is therefore enough to check that the map $q$ is equivariant under the whole group $G$. This is easily done after observing that since $H$ is locally moving, different points in $X$ have distinct stabilizers in $H$, so both actions can be reconstructed from the conjugation action of $G$ on the stabilizers in $H$. Namely, for every $\xi\in \mathbb{R}$ we have $\operatorname{\mathsf{Stab}}^\varphi_H(\xi)=\St_H(q(\xi))$.
Thus conjugating by $g\in G$ we obtain
\[\operatorname{\mathsf{Stab}}^\varphi_H(g.\xi)=g\operatorname{\mathsf{Stab}}^\varphi_H(\xi)g^{-1}=g\St_H(q(\xi))g^{-1}=\St_H(g(q(\xi)))\]
Since on the other hand we have $\operatorname{\mathsf{Stab}}^\varphi_H(g.\xi)=\St_H(q(g.\xi))$, this implies that $q(g.\xi)=g(q(\xi))$ for every $g\in G$ and $\xi\in \mathbb{R}$, i.e.\ that the map $q$ is $G$-equivariant, as desired. \qedhere
\end{proof}
An example of a group satisfying the assumption of Corollary \ref{c-C-flip} is \emph{Thompson's group with flips} $F^{\pm}\subset \homeo((0,1))$, which is defined similarly to Thompson group $F$, but by allowing negative slopes in the definition.
\subsection{An attractor in the Deroin space and local rigidity}
\label{s-lr}
In this subsection we consider the following subclass $\mathcal{F}_0${}\;of the class $\mathcal F${}.
\begin{dfn}
Let $X=(a, b)$. We say that a group $G\subseteq \homeo_0(X)$ is in the class $\mathcal{F}_0${}\;if it is finitely generated, it belongs to $\mathcal F${}\;(Definition \ref{def.classF}) and there exists $f\in G$ without fixed points in $X$.
\end{dfn}
For a group $G$ in the class $\mathcal{F}_0${}\;we prove the following theorem, which corresponds to Theorem \ref{t-intro-space} from the introduction.
\begin{thm} \label{t-space}
Let $X$ be an open interval and let $G\subset \homeo_0(X)$ be a finitely generated group in the class $\mathcal{F}_0${}. Let $\mathcal{U}\subset \Homirr(G, \homeo_0(\mathbb{R}))$ be the set of irreducible actions that are not semi-conjugate to any action induced from the largest quotient $G/[G_c, G_c]$. Then $\mathcal{U}$ is open and admits a subset $\mathcal{S}\subset \mathcal{U}$ satisfying the following.
\begin{enumerate}[label=(\roman*)]
\item The set $\mathcal{S}$ is closed in $\mathcal{U}$ and relatively compact in $\Homirr(G, \homeo_0(\mathbb{R}))$. In particular it is a locally compact Polish space.
\item Every $\varphi\in \mathcal{S}$ is a minimal and faithful action.
\item Every $\varphi\in \mathcal{U}$ is positively semi-conjugate to a unique action $\bar{\varphi}\in \mathcal{S}$ and the map $\varphi\mapsto \bar{\varphi}$ is a continuous retraction from $\mathcal{U}$ to $\mathcal{S}$. In particular the quotient of $\mathcal{U}$ by the semi-conjugacy equivalence relation, with the quotient topology, is homeomorphic to $\mathcal{S}$.
\item The unique action $\psi\in \mathcal{S}$ which is positively semi-conjugate to the standard action of $G$ on $X$ is an isolated point in $\mathcal{S}$.
\end{enumerate}
\end{thm}
In order to prove Theorem \ref{t-space} we will proceed by analyzing the structure of the Deroin space $\Der_\mu(G)$ of normalized harmonic $G$-actions (see \S\ref{ssubsec.Deroin}).
Recall that the Deroin space carries a natural flow $\Phi$ which, given $\varphi\in \Der_\mu(G)$ and $t\in \mathbb{R}$, gives the action $\Phi^t(\varphi)$ obtained by conjugating $\varphi$ by the translation $x\mapsto x+t$ (whence the name \emph{translation flow}).
Throughout the subsection, we work in the following setting.
\begin{assumption} \label{a-lr} For $X=(a, b)$, let $G\subseteq\homeo_0(X)$ be a subgroup in the class $\mathcal{F}_0${}. We fix $f\in G$ such that $f(x)>x$ for every $x\in X$. We also fix once and for all a symmetric probability measure $\mu$ supported on a finite generating set of $G$.
\end{assumption}
\subsubsection{Description of the Deroin space} Let us introduce some further notation. Given an action $\varphi\in \Der_\mu(F)$, we denote by $\widehat{\varphi}$ its reversed action (the conjugate under the reflection $x\mapsto -x$). Then, by Theorem \ref{t-C-trichotomy} we have a decomposition of the Deroin space
\begin{equation} \label{e-lr-decomposition} \Der_\mu(G)=\mathcal{N}\sqcup \mathcal{I} \sqcup \widehat{\mathcal{I}} \sqcup \mathcal{P}, \end{equation}
defined by what follows.
\begin{itemize}
\item We let $\mathcal{N}\subset \Der_\mu(G)$ be the subset consisting of actions $\varphi$ which are not faithful, or equivalently such that $\varphi([G_c, G_c])=\{\mathsf{id}\}$.
\item We fix a representative $\iota\in \Der_\mu(G)$ positively conjugate to the standard action on $X$, and let $\mathcal{I}=\{\Phi^t(\iota)\colon t\in \mathbb{R}\}$ be its $\Phi$-orbit and $\widehat{\mathcal{I}}$ be the $\Phi$-orbit of $\widehat{\iota}$.
\item We let $\mathcal{P}=\mathcal{P_+}\sqcup \mathcal{P}_-$ be the subset of $\Der_\mu(G)$ of $\mathbb{R}$-focal actions, where $\mathcal{P}_+$ and $\mathcal{P}_-$ are the subsets of those which are increasingly (respectively, decreasing) horograded by the standard action on $X$.
\end{itemize}
Proposition \ref{prop.dynclasselements} implies that for every $\varphi\in \mathcal{P}$ the image $\varphi(f)$ is a homothety, which is expanding if $\varphi\in \mathcal{P}_+$ and contracting if $\varphi\in \mathcal{P}_-$. For $\varphi\in \mathcal{P}$, denote by $\xi_\varphi$ the unique fixed point of $\varphi(f)$ in $\mathbb{R}$. We say that $\varphi$ is $f$-\emph{centered} if $\xi_\varphi=0$. Let us denote by $\mathcal{P}^0\subset \mathcal{P}$ the subset of $f$-centered $\mathbb{R}$-focal actions, with $\mathcal{P}^0_\pm=\mathcal{P}^0\cap \mathcal{P}_{\pm}$.
Note that each of the subsets of $\Der_\mu(G)$ in \eqref{e-lr-decomposition} is $\Phi$-invariant. We aim to analyze the topology of these subsets and the dynamics of the flow $\Phi$ on them. A first obvious observation is that the set $\mathcal{N}$ is closed in $\Der_\mu(G)$. Our main goal is to show that $\mathcal{N}$ is a \emph{uniform attractor} for the flow $\Phi$, that is all $\Phi$-orbits in the complement of $\mathcal{N}$ spend a uniformly bounded amount of time away from any neighborhood of $\mathcal{N}$. This is formalized in Proposition \ref{p-lr-attractor} below. Whenever $U$ is a $\Phi$-invariant subset of $\Der_\mu(G)$ we say that $K\subset U$ is a \emph{cross section} of $\Phi$ inside $U$ if it intersects every $\Phi$-orbit contained in $U$ exactly once. With this notation we have the following.
\begin{lem}\label{l-lr-section} Under Assumption \ref{a-lr}, the following hold.
\begin{enumerate}[label=(\roman*)]
\item \label{i-P-open-section} Both subsets $\mathcal{P}_\pm$ are open and $\Phi$-invariant, and each subset $\mathcal{P}^0_\pm$ is a cross section of $\Phi$ inside $\mathcal{P}_\pm$.
\item \label{i-P-section-closure} The closures of the subsets $\mathcal{P}^0_\pm$ in $\Der_\mu(G)$ satisfy $\overline{\mathcal{P}^0_\pm}\subset \mathcal{P}^0_\pm\cup \mathcal{N}$.
\end{enumerate}
\end{lem}
\begin{proof}
It is clear that $\mathcal{P}_+$ and $\mathcal{P}_-$ are both $\Phi$-invariant. To show that $\mathcal{P}_+$ is open, fix $\varphi\in \mathcal{P}_+$. Since $\varphi(f)$ is an expanding homothety centered at $\xi_\varphi$, for every bounded open interval $I$ containing $\xi_\varphi$ we have $I\Subset \varphi(f)(I)$. Thus $\varphi$ has a neighborhood $\mathcal{U}$ in $\Der_\mu(G)$ such that every $\psi\in \mathcal{U}$ satisfies $I\Subset \psi(f)(I)$.
This gives that for $\psi\in \mathcal U$, the map $\psi(f)$ cannot be a contracting homothety, hence $\mathcal{U}\cap\mathcal{P}_-=\varnothing$. Similarly, this gives $\mathcal{U}\cap( \mathcal{I} \cup \widehat{\mathcal{I}})=\varnothing$,
as in the standard action the map $f$ has no fixed point. Moreover since $\mathcal{N}$ is closed, we can restrict $\mathcal{U}$ so that it is contained in $\mathcal{P}_+$, showing that $\mathcal{P}_+$ is open. An analogous argument shows that $\mathcal{P}_-$ is also open. Moreover for every $\varphi\in\mathcal{P}_\pm$ we have $\Phi^{t}(\varphi)\in \mathcal{P}^0_\pm$ if and only if $t=-\xi_\varphi$, showing that each $\mathcal{P}^0_\pm$ is a cross section for $\Phi$ inside $\mathcal{P}_\pm$.
Finally suppose that $\psi\in \overline{\mathcal{P}^0_+}$. Since $\varphi(f)$ fixes $0$ for every $\varphi\in \mathcal{P}^0_+$, so does $\psi(f)$, and in particular $\psi\notin \mathcal{I} \cup \widehat{\mathcal{I}}$.
Since $\mathcal{P}_-$ is open, the only possibility is that $\psi\in \mathcal{P}^0_+\cup \mathcal{N}$, so that $\overline{\mathcal{P}^0_+}\subset \mathcal{P}^0_+\cup \mathcal{N}$. Similarly $\overline{\mathcal{P}^0_-}\subset \mathcal{P}^0_-\cup \mathcal{N}$. \qedhere
\end{proof}
In particular it follows that the restriction of the flow $\Phi$ to the complement of $\mathcal{N}$ in $\Der_\mu(G)$ has a cross section, namely $\mathcal{P}^0\cup\{\iota, \widehat{\iota}\}$. The central point here is the following.
\begin{prop}[$\mathcal{N}$ is a uniform attractor]\label{p-lr-attractor}
Under Assumption \ref{a-lr}, let $\mathcal{U}$ be an open neighborhood of $\mathcal{N}$ in $\Der_\mu(G)$. Then there exists $t_0>0$ such that for every $t\in \mathbb{R}$ with $|t|> t_0$ and every $\varphi\in \mathcal{P}^0\cup \{\iota, \widehat{\iota}\}$ we have $\Phi^t(\varphi)\in \mathcal{U}$.
\end{prop}
Before proving Proposition \ref{p-lr-attractor}, let us observe that it implies Theorem \ref{t-space}.
\begin{proof}[Proof of Theorem \ref{t-space}]
Set $\mathcal{S}=\mathcal{P}^0\cup\{\iota, \widehat{\iota}\}$, and $\mathcal{V}=\mathcal{P}\cup \mathcal{I} \cup \widehat{\mathcal{I}}$. Note also that the set $\mathcal{U}$ in the statement of Theorem \ref{t-space} is precisely the preimage of $\mathcal{V}$ under the harmonic retraction $r\colon \Homirr(G, \homeo_0(\mathbb{R}))\to \Der_\mu(G)$ given by Theorem \ref{t.retraction_Deroin}. By Lemma \ref{l-lr-section} the set $\mathcal{S}$ is closed in $\mathcal{V}$ (and hence in $\mathcal{U}$), its closure is contained in $\Der_\mu(G)$ and thus is compact, and the point $\iota$ and $\hat{\iota}$ are isolated in $\mathcal{S}$.
Now the uniform convergence in Proposition \ref{p-lr-attractor} implies that the map $\sigma\colon \mathbb{R}\times \mathcal{S}\to \mathcal{V}$, given by $\sigma(t, \varphi)=\Phi^t(\varphi)$, is a homeomorphism from $\mathbb{R}\times \mathcal{S}$ to $\mathcal{V}$. It follows that there is a continuous retraction of $p\colon \mathcal{V}\to\mathcal{S}$ which preserves $\Phi$-orbits. Thus the map $p \circ r$ is a continuous retraction of $\mathcal{U}$ to $\mathcal{S}$ which preserves semi-conjugacy classes as desired.
\end{proof}
For the proof of Proposition \ref{p-lr-attractor} we need some preliminary lemmas. The case $\varphi\in \{\iota, \widehat{\iota}\}$ is actually straightforward and boils down to the following.
\begin{lem}\label{l-lr-standard-accumulation}
Under Assumption \ref{a-lr}, for $\varphi\in \{\iota, \widehat{\iota}\}$ all limit points of $\Phi^t(\varphi)$ as $t\to \pm \infty$ are contained in $\mathcal{N}$.
\end{lem}
\begin{proof}
For every finite subset $S\subset G_c$ the image $\varphi(S)$ is supported in a compact interval. Since $\varphi_t:=\Phi^t(\varphi)$ is the conjugate of $\varphi$ by a translation by $t$, it follows that when $|t|$ is sufficiently large, the image $\varphi_t(S)$ acts trivially on an arbitrarily large compact interval, so that $\psi(S)=\{\operatorname{\mathsf{id}}\}$ for every limit action $\psi$ of $\varphi_t$ as $|t|\to +\infty$. Since $S$ is arbitrary, we deduce that $G_c\subseteq\ker \psi$, and thus $\psi\in \mathcal{N}$. \qedhere.
\end{proof}
We now turn to analyze the limit points of $\Phi^t(\varphi)$ for $\varphi\in \mathcal{P}^0$. In fact, a pointwise convergence towards $\mathcal{N}$ for every $\varphi\in \mathcal{P}^0$ would be not difficult to obtain from the qualitative properties of $\mathbb{R}$-focal actions. To obtain a {uniform} convergence we will need a quantitative control on the size of the intervals in the CF-family of intervals associated with $\varphi$. Recall that given an $\mathbb{R}$-focal action $\varphi\in \mathcal{P}_+$, for given $x\in X$ and $\xi\in \suppphi(G_{(a, x)})$ we denote by $\Iphi(x, \xi)$ the connected component of $\suppphi(G_{(a, x)})$ containing $\xi$ (Definition \ref{d-I-phi}), and that the intervals of this form allow to define an invariant CF-family for $\varphi$ (see Proposition \ref{c-C-focal}).
In the case of $\mathbb{R}$-focal actions in $\mathcal{P}_-$, one can introduce the analogous notation for connected components of $\suppphi(G_{(x,b)})$.
The key point in the proof of Proposition \ref{p-lr-attractor} is the following uniform bound on the size of these intervals in harmonic coordinates.
\begin{lem}[Bound on the CF-family in harmonic coordinates]\label{l-lr-bound} Under Assumption \ref{a-lr}, there exists a constant $C=C(\mu)$ such that the following holds.
For every $x\in X$, there exists a constant $D=D(x, \mu)$ such that for every $\varphi\in \mathcal{P}_+$ and every $\xi\in \suppphi(G_{(a, x)})$ with $|\xi-\xi_\varphi| \ge D$, we have $|\Iphi(x, \xi)|\le C$.
The analogous result holds for actions $\varphi\in \mathcal{P}_-$.
\end{lem}
\begin{proof}
For given $g\in G$, set
\[\Delta_{g}:=\max\left \{ |\varphi(g)(\xi)-\xi|\colon \varphi\in \Der_\mu(G), \xi \in \mathbb{R}\right \}.\]
Note that this quantity is positive and finite for every $g\in G$, by compactness of the space $\Der_\mu(G)$.
We will establish the lemma for actions $\varphi\in \mathcal{P}_+$, the other case $\varphi \in \mathcal{P}_-$ being totally symmetric.
Fix $\varphi \in \mathcal{P}_+$. First of all we observe that the unique fixed point $\xi_\varphi$ of $\varphi(f)$ satisfies $\xi_\varphi\in \suppphi(G_{(a, x)})$ for every $x\in X$. Indeed if there exists $y\in X$ such that $\xi_\varphi\in \operatorname{\mathsf{Fix}}^\varphi(G_{(a, y)})$ we would have $\xi_\varphi\in \operatorname{\mathsf{Fix}}^\varphi(G_{(a, f^n(y))})$ for every $n\in \mathbb N$, so that $\xi_\varphi$ must be fixed by the normal subgroup $G_+=\bigcup_{n\in \mathbb N} G_{(a, f^n(y))}$. Since $\varphi$ is minimal, it follows that $G_+$ acts trivially, which is a contradiction with the fact that $\varphi\in \mathcal{P}$ is faithful. Thus, the interval $\Iphi(x, \xi_\varphi)$ is well-defined and non-empty for every $x\in X$.
Fix now $x\in X$. Since $G$ belongs to the class $\mathcal F${}, we can fix a finite subset $S\subset G_{(x, b)}$ such that $\langle S \rangle$ contains $G_{(y, b)}$ for some $y\ge x$. Set $D=\max\{ \Delta_{s}: s\in S\}$ (note that it depends on $\mu$ and on $x$).
\begin{claim}
We have $\sup\left\{|\Iphi(x, \xi_\varphi)|\colon \varphi \in \mathcal{P}_+ \right\}\le D$.
\end{claim}
\begin{proof}[Proof of claim]
Assume by contradiction that for some $\varphi\in \mathcal{P}_+$, we have $|\Iphi(x, \xi_\varphi)|>D$. Since every $s\in S$ fixes $x$, the interval $\varphi(s)(\Iphi(x, \xi_\varphi))=\Iphi(x, \varphi(s)(\xi_\varphi))$ is a connected component of $\suppphi(G_{(a, x)})$ and thus must be either equal to $\Iphi(x, \xi_\varphi)$ or disjoint from it. But the assumption that $|\Iphi(x, \xi_\varphi)|>\Delta_{s}$ rules out the second possibility. Thus we have that $\varphi(s)$ must preserve $\Iphi(x, \xi_\varphi)$ for every $s\in S$, and hence $\Iphi(x, \xi_\varphi)$ is $\varphi(G_{(y, b)})$-invariant. In particular,
the subgroup $N_{(y,b)}=[G_c,G_c]_{(y,b)}$ has fixed points, which contradicts Proposition \ref{p-lm-trichotomy}.
\end{proof}
Given $\varphi\in \mathcal{P}_+$, write $ \Iphi(x, \xi_\varphi)=(\alpha_0, \beta_0)$ and for $n\in \mathbb N$ set $\alpha_n=\varphi(f^n)(\alpha_0)$ and $\beta_n=\varphi(f^n)(\beta_0)$. On the one hand, note that for every $n\in \mathbb N$, the points $\alpha_n$ and $\beta_n$ belong to $\operatorname{\mathsf{Fix}}^\varphi(G_{(a, f^n(x))})$ and hence to $\operatorname{\mathsf{Fix}}^\varphi(G_{(a, x)})$. Thus for every $\xi \in \suppphi(G_{(a,x)})\setminus (\alpha,\beta)$, there exists $n\in \mathbb N$ such that the corresponding connected component $\Iphi(x,\xi)$ is contained in one of the two intervals $(\alpha_{n+1},\alpha_n)$ and $(\beta_n,\beta_{n+1})$. In particular, after the claim, this happens whenever $|\xi-\xi_\varphi|\ge D$.
On the other hand, setting $C= \Delta_{f}$ (note it only depends on $\mu$ only), we must have that both lengths $|\alpha_{n+1}-\alpha_n|$ and $|\beta_{n+1}-\beta_n|$ are upper-bounded by $C$.
This gives the desired conclusion.\qedhere
\end{proof}
\begin{proof}[Proof of Proposition \ref{p-lr-attractor}]
Assume by contradiction that we can find sequences $(\varphi_n)_{n\in\mathbb N}\subset \mathcal{P}^0$ and $(t_n)_{n\in\mathbb N}\subset \mathbb{R}$, with $|t_n|\to \infty$, such that $\psi_n:=\Phi^{t_n}(\varphi_n)$ converges to a limit $\psi\notin \mathcal{N}$ as $n\to \infty$. Up to extracting a subsequence we can assume that $(\varphi_n)\subset \mathcal{P}^0_+$ (the case $(\varphi_n)\subset\mathcal{P}^0_-$ being analogous). Note that since $\xi_{\varphi_n}=0$ we have $\xi_{\psi_n}=-t_n$. Fix $x\in X$ and the corresponding constants $C=C(\mu)$ and $D=D(x,\mu)$ given by Lemma \ref{l-lr-bound}. For $n$ large enough we have $|t_n|\ge D$, which implies that $|\IphiN(x,\xi_{\psi_n})|=|\IpsiN(x,0)|\le C$, so that there exists $\eta_n\in [-C, C]$ which is fixed by $\psi_n(G_{(a, x)})$.
Since every accumulation point of $(\eta_n)$ is fixed by $\psi(G_{(a, x)})$ we obtain that $\operatorname{\mathsf{Fix}}^{\psi}(G_{(a, x)})\cap [-C, C]\neq \varnothing$. Since $C$ does not depend on $x$, a compactness argument shows that $\operatorname{\mathsf{Fix}}^{\psi}(G_+)=\bigcap_{x\in X} \operatorname{\mathsf{Fix}}^{\psi}(G_{(a, x)}) \neq \varnothing$. As $\psi$ is minimal and $G_+$ is normal, we deduce that $\psi$ is not faithful, and this contradicts the assumption that $\psi\notin \mathcal{N}$.
\end{proof}
\subsubsection{Application to local rigidity}
\label{sec local rigidity}
Theorem \ref{t-space} has the following immediate consequence.
\begin{thm} \label{t-local-rigidity}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be a subgroup in the class $\mathcal{F}_0${}. Then the set of actions in $\Homirr(G, \homeo_0(\mathbb{R}))$ which are semi-conjgate to the standard action of $G$ on $X$ is open. In particular the standard action of $G$ on $X$ is locally rigid.
\end{thm}
\begin{rem}
An arbitrary locally moving group $G\subseteq \homeo_0(X)$ is not always locally rigid: the construction of actions in \S \ref{s-escaping-sequence} shows that this is never the case for any countable group $G\subseteq \homeo_0(X)$ consisting of elements of compact support. However the assumption that $G$ belongs to $\mathcal{F}_0${}\;is probably not optimal. We are able to prove Theorem \ref{t-local-rigidity} rigidity under some variants of this assumptions, all of which involve the finite generation of $G$ and some additional finite generation condition related to the subgroups $G_I$ for proper intervals $I\subset X$. It would be interesting to know whether these assumptions can be completely removed, namely whether Theorem \ref{t-local-rigidity} holds for every finitely generated locally moving group.
\end{rem}
\begin{rem}
In the setting of Theorem \ref{t-local-rigidity}, the natural action of $G$ on $X$ is not always the \emph{unique} locally rigid action of $G$: we will see in Section \ref{s-few-actions} that there are examples of groups in $\mathcal{F}_0${}\;that admit a finite number of faithful $\mathbb{R}$-focal actions up to conjugacy (i.e.\ for which the subset $\mathcal{P}$ contains finitely many $\Phi$-orbits), which are therefore locally rigid as well.
\end{rem}
We point out the following consequence.
\begin{cor} Let $H\subseteq\homeo_0(\mathbb{R})$ be a countable group. Then, there exists a finitely generated subgroup $G\subset \homeo_0(\mathbb{R})$ containing $H$ such that the action of $G$ on $\mathbb{R}$ is locally rigid.
\end{cor}
\begin{proof} By Proposition \ref{prop.superF} there exists a finitely generated group $G\subseteq\homeo_0(\mathbb{R})$ that contains $H $ and belongs to class $\mathcal{F}_0${}, and by Theorem \ref{t-local-rigidity}, its action is locally rigid.
\end{proof}
\subsubsection{Smoothness of the semi-conjugacy relation} \label{s-class-borel}
Recall from the discussion in \S \ref{ssc.complexity} that the Deroin space can be used to study when the semi-conjugacy relation on $\Homirr(G, \homeo_0(\mathbb{R}))$ is smooth, that is when there is a Borel complete invariant that classifies actions of $G$ up to semi-conjugacy.
When $G$ is in $\mathcal{F}_0${}, an obvious obstruction for this to hold is that this may fail already for action of the largest quotient $G/[G_c, G_c]$. However our analysis of $\Der_\mu(G)$ has the following straightforward consequence.
\begin{cor}\label{c-class-borel}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be a subgroup in the class $\mathcal{F}_0${}. Let $\mathcal{M}$ be the subset of $\Homirr(G, \homeo_0(\mathbb{R}))$ consisting of actions that are minimal and faithful. Then the following hold.
\begin{enumerate}[label=(\roman*)]
\item The topological conjugacy relation on $\mathcal{M}$ is smooth.
\item The semi-conjugacy relation on the space $\Homirr(G, \homeo_0(\mathbb{R}))$ is smooth if and only if the same holds true for the space $\Homirr(G/[G_c, G_c], \homeo_0(\mathbb{R}))$.
\end{enumerate} \end{cor}
\begin{proof}
The first claim follows from Lemma \ref{l-lr-section} and from Theorem \ref{t.retraction_Deroin}. The second is a direct consequence of the previous analysis of $\Der_\mu(G)$ and of Corollary \ref{cor.cross-section}. \qedhere
\end{proof}
\section{Deroin spaces and preorders}
\label{s.Deroin}
Recall that we denote by $\Homirr(G,\homeo_0(\mathbb{R}))$ the space of representations of $G$ without fixed points. The following notion is classical in dynamical systems.
\begin{dfn}
An action $\varphi:G\to\homeo_0(\mathbb{R})$ without fixed points is \emph{locally rigid} if there exists a neighborhood $\mathcal{U}$ of $\varphi$ in $\Homirr(G,\homeo_0(\mathbb{R}))$ consisting of representations all positively semi-conjugate to $\varphi$.
\end{dfn}
In other terms, every sufficiently small perturbation of the action $\varphi$ gives a positively semi-conjugate action.
The main goal of this section is to characterize local rigidity in terms of the topology of the orbits along the translation flow in the Deroin space (see \S \ref{ssubsec.Deroin} for a definition of this space). This will be a consequence of the following result.
\begin{thm}\label{t.retraction_Deroin} Let $G$ be a finitely generated group, and consider a symmetric probability measure $\mu$ whose support is finite and generates $G$. There exists a continuous retraction $$r:\Homirr(G,\homeo_0(\mathbb{R}))\to \Der_\mu(G)$$ which preserves positive semi-conjugacy classes.
\end{thm}
\begin{dfn}
The map $r:\Homirr(G,\homeo_0(\mathbb{R}))\to \Der_\mu(G)$ from Theorem \ref{t.retraction_Deroin} is called the \emph{harmonic retraction} of the representation space $\Homirr(G,\homeo_0(\mathbb{R}))$.
\end{dfn}
\begin{cor}[Local rigidity criterion]\label{prop.rigidityderoin}\label{c-rigidity-Deroin}
Let $G$ be a finitely generated group, and consider a symmetric probability measure $\mu$ whose support is finite and generates $G$. For $\varphi\in \Homirr(G, \homeo_0(\mathbb{R}))$, let $\psi\in \Der_\mu(G)$ be a normalized $\mu$-harmonic action which is positively semi-conjugate to $\varphi$. If the orbit along the translation flow of $\psi$ is open in $\Der_\mu(G)$, then $\varphi$ is locally rigid. The converse hold provided $\varphi$ is minimal.
\end{cor}
\begin{proof}
Let $\mathcal{I}=\{\Phi^t(\psi):t\in\mathbb{R}\}$ be the orbit of $\psi$ in $\Der_\mu(G)$ and let $r\colon \Homirr(G, \homeo_0(\mathbb{R})) \to \Der_\mu(G)$ be the harmonic retraction (Theorem \ref{t.retraction_Deroin}). Then $r(\varphi)$ is an element of $\Der_\mu(G)$ which is positively semi-conjugate to $\psi$ and thus belongs to $\mathcal{I}$, i.e.\ $\varphi\in r^{-1}(\mathcal{I})$. If $\mathcal{I}$ is open, then $r^{-1}(\mathcal{I})$ is an open neighborhood of $\varphi$ consisting only of actions positively semi-conjugate to $\psi$, so the claim follows.
Conversely, assume that $\varphi$ is minimal. Since local rigidity is preserved under conjugacy we can assume that $\varphi=\psi$. Thus if $\mathcal{I}$ is not open, using the translation flow we find that $\varphi$ can be approximated by $\mu$-harmonic actions in different $\Phi$-orbits. Since actions in $\Der_\mu(G)$ belonging to different $\Phi$-orbits are not positively semi-conjugate (Corollary \ref{rem.conjclass}), this implies that $\varphi$ is not locally rigid. \qedhere
\end{proof}
\begin{rem}\label{rem.stable_non-isolated} The requirement that $\varphi$ be minimal in the second part of Corollary \ref{prop.rigidityderoin} is essential. To see this consider the action $\varphi:\mathbb{F}_2\times\mathbb{Z}\to\homeo_0(\mathbb{R})$ obtained as the lift of the ping-pong action $\varphi_0:\mathbb{F}_2\to\mathsf{PSL}(2,\mathbb{R})\subset \homeo_{0}(\mathbb{S}^1)$, given by the action of the fundamental group of a hyperbolic \emph{one-holed} torus on the boundary at infinity of the hyperbolic plane; to make this more concrete, one can consider the subgroup of $\mathsf{PSL}(2,\mathbb{R})$ generated by the two homographies $\begin{bmatrix}5& 3\\3 & 2 \end{bmatrix}$ and $\begin{bmatrix}1& 1\\mathtt{1} & 2 \end{bmatrix}$.
The action $\varphi_0$ is then non-minimal and locally rigid so the same holds for $\varphi$. However, the canonical model of $\varphi_0$ corresponds to the action of the fundamental group of a hyperbolic \emph{one-punctured} torus on the boundary at infinity of the hyperbolic plane, and so this canonical model has a parabolic element (the commutator of the generators). In particular, this minimal action and its corresponding lift to the line are not locally rigid. Thus the representative of $\varphi$ in $\Der_\mu(\mathbb{F}_2\times\mathbb Z)$ has non-isolated orbit along the translation flow.
\end{rem}
In order to prove Theorem \ref{t.retraction_Deroin}, we obtain an order-theoretic description of the Deroin space.
Note that not every group action is the dynamical realization of some order, so we are forced to consider the much more flexible notion of preorder. However, different preorders may correspond to the same conjugacy class of a minimal action, and we have to translate this into an equivalence relation on preorders. The appropriate equivalence relation on preorders is introduced in \S \ref{sc.equivalence_preo}. In \S \ref{sc.preoders_dynamics}, we make explicit the relations between properties of preorders and their dynamical realizations. In \S \ref{sc.topology_preo} we investigate the topology of the corresponding quotient space. Finally, in \S \ref{sc.deroin_preo} we identify such quotient space with the Deroin space and provide the proof of Theorem \ref{t.retraction_Deroin}.
As a by-product of this, we will get that the topology of $\Der_\mu(G)$ does not depend on the choice of the probability measure $\mu$.
\subsection{Equivalence of preorders}\label{sc.equivalence_preo} We start with the following definition which gives an order-theoretic analogue of continuous positive semi-conjugacies between representations in $\homeo_0(\mathbb{R})$. Part of our discussion is close to the exposition of Decaup and Rond \cite{decaup2019preordered}.
\begin{dfn} \label{d-preorder-dominates}Let $G$ be a group and let $\leq,\preceq\in\LPO(G)$ be two preorders. We say that $\leq$ \emph{dominates} $\preceq$ if the identity map $\mathsf{id}:(G,\preceq)\to (G,\leq)$ is order-preserving.
\end{dfn}
\begin{rem}The direction in the definition of domination could appear counterintuitive, but in fact it is justified when thinking in terms of preorders (it is even clearer in terms of dynamical realization). Indeed, take an element $g\in G$, and suppose that we want to know whether $g\succneq 1$. Then we first check whether $g\gneq 1$ (or $g\lneq 1$), and only in the case $g\in [1]_{\leq}$, we take the preorder $\preceq$ into consideration.
\end{rem}
Note that when $\leq$ dominates $\preceq$, the subgroup $H=[1]_{\le}$ is $\preceq$-convex, thus the quotient preorder $\preceq_{H}$ is well-defined.
The following lemma characterizes domination in terms of positive cones.
\begin{lem}\label{lem.dominequiv} Let $G$ be a group and let $\leq,\preceq\in\LPO(G)$ be two preoders.
Then, the following are equivalent. \begin{enumerate}
\item $\leq$ dominates $\preceq$.\label{i.dominequiv}
\item $P_{\leq}\subseteq P_{\preceq}$.\label{ii.dominequiv}
\item The subgroup $H=[1]_{\leq}$ is $\preceq$-convex, and $P_\le=P_\preceq\setminus H$ (so that $\le$ coincides with the quotient preorder $\preceq_H$).\label{iii.dominequiv}
\end{enumerate}
\end{lem}
\begin{proof}
We prove that \eqref{i.dominequiv} implies \eqref{iii.dominequiv}. As $\mathsf{id}:(G,\preceq)\to (G,\le)$ is order-preserving, the subgroup $H=[1]_\le$, which coincides with its preimage, is $\preceq$-convex.
Moreover, we have the following inclusions:
\[
P_{\preceq}\subseteq P_\le \sqcup [1]_\leq,\quad [1]_{\preceq}\subseteq [1]_{\leq},\quad N_{\preceq}\subseteq [1]_\leq\sqcup N_{\leq}.
\]
As $G=P_\preceq\sqcup[1]_\preceq\sqcup N_{\preceq}$, we deduce $P_\preceq = P_\leq \setminus [1]_\leq$.
Clearly \eqref{iii.dominequiv} implies \eqref{ii.dominequiv}.
Finally, by left-invariance, we have $g\preceq h$ if and only if $h^{-1}g\in G\setminus P_\preceq$, and clearly the same holds for the preorder $\le$. This gives that \eqref{i.dominequiv} and \eqref{ii.dominequiv} are equivalent.
\qedhere
\end{proof}
Let $G$ be a group. For a given a preorder $\leq\in\LPO(G)$,
the collection of proper $\le$-convex subgroups is totally ordered by inclusion. The subgroup $[1]_\le$ is always the least such subgroup, however a maximal proper $\le$-convex subgroup may not exist.
This issue is the order-theoretic analogue of the problem of existence of a minimal invariant set (Remark \ref{r.maynotexist}). To simplify the discussion, we will systematically assume finite generation in the rest of the section, in which case there exists a maximal proper $\le$-convex subset, that we will denote by $H_\leq$.
This leads to the notion of minimal model of a preorder which is an order-theoretic analogue of canonical models for actions on the line.
\begin{dfn} Let $G$ be a finitely generated group. The \emph{minimal model} of a preorder $\leq\in\LPO(G)$ is the quotient preorder $\le_H$, where $H=H_{\le}$ is the maximal proper $\le$-convex subgroup. The minimal model will be denoted by $\leq^\ast$. When the preorder $\leq$ coincides with $\leq^\ast$, we say that $\leq\in\LPO(G)$ is a minimal model.
\end{dfn}
\begin{rem}\label{r.domination_minimal}\label{prop.minimalconvex}
Minimal preorders are exactly those whose structure of convex subgroups is the simplest possible.
Indeed, after Lemma \ref{lem.dominequiv}, the minimal model dominates the original preorder. Therefore, $H_\le$ is the unique proper $\le^*$-subgroup of $(G,\le^*)$, and thus $H_\le=[1]_{\leq^*}$.
Reversely, if $[1]_\le=H_\le$ is the unique proper $\le$-convex subgroup, then $\le$ is a minimal model.
\end{rem}
It should be not surprising that the minimal model is unique, in the following sense.
\begin{lem}\label{prop.caracterdom}
Let $G$ be a finitely generated group and consider preorders $\leq,\preceq\in\LPO(G)$ such that $\leq$ is a minimal model and dominates $\preceq$. Then, $\leq$ is the minimal model of $\preceq$.
\end{lem}
\begin{proof}
As $\le$ dominates $\preceq$, the subgroup $H=[1]_\leq$ is $\preceq$-convex, so $H\subseteq H_\preceq$. This gives $P_\leq\supseteq P_{\preceq^*}$, so by Lemma \ref{lem.dominequiv}, the map
\[
\mathsf{id}:(G,\le)\to (G,\preceq^*)
\]
is order-preserving. Using Remark \ref{prop.minimalconvex}, as $H_\preceq=[1]_{\preceq^*}$, we deduce that $H_\preceq$ is $\le$-convex, and thus that $H_\preceq=H$.
\end{proof}
We next introduce the following equivalence relation on preorders.
\begin{dfn} Let $G$ be a finitely generated group. We say two preorders are \emph{equivalent} if they have the same minimal model. We denote by $[\leq]$ the equivalence class of $\leq\in\LPO(G)$ and we write $[\LPO](G)=\{[\leq]:\leq\in\LPO(G)\}$ for the corresponding quotient.\end{dfn}
It is immediate to verify that this is indeed an equivalence relation.
We have the following result.
\begin{prop}\label{prop.starequiv} Let $G$ be a finitely generated group and consider preorders $\leq_1,\leq_2\in\LPO(G)$. Then, there exists a preorder $\le$ that dominates $\leq_1$ and $\leq_2$, if and only if the equivalence classes $[\leq_1]$ and $[\leq_2]$ are the same.
\end{prop}
\begin{proof}
We have that the minimal order $\le^*$ dominates $\le$ and thus both $\le_1$ and $\le_2$. From Lemma \ref{prop.caracterdom}, we conclude that $\le^*$ is the minimal model of both $\le_1$ and $\le_2$. The converse statement follows directly from Remark \ref{r.domination_minimal}.\qedhere
\end{proof}
\subsection{Relations between preorders and actions on the real line}\label{sc.preoders_dynamics}
We can now make explicit the relation between minimal models and their dynamical counterpart. For this, note that if $G$ is a countable group and $\le\in \LPO(G)$ is a preorder, we can consider a \emph{dynamical realization} $\varphi_{\leq}$ for the action of $G$ on the totally ordered set $(G/H,<_{G/H})$, which exists by Lemma \ref{lem.dynreal}.
Conversely, given an action $\varphi:G\to\homeo_{0}(\mathbb{R})$ without fixed points, we denote by $\leq_\varphi\in \LPO(G)$ the preorder induced by the $\varphi(G)$-orbit of $0$.
\begin{prop}\label{lem.minimalisminimal} Let $G$ be a finitely generated group and let $\varphi:G\to\homeo_0(\mathbb{R})$ be a canonical model. Then the preorder $\leq_\varphi$ is a minimal model.
\end{prop}
\begin{proof} Suppose by way of contradiction that there exists a proper $\leq_\varphi$-convex subgroup $H\neq [1]_{\leq_\varphi}$. Then, for each $gH\in G/H$, denote by $I_{gH}\subset \mathbb{R}$ the interior of the convex hull of $\varphi(gH)(0)$. Since $H$ is $\leq_\varphi$-convex, we get that $\mathbb{R}\setminus \bigcup_{g\in G}I_{gH}$ is a proper closed $\varphi(G)$-invariant subset. On the other hand, since $H$ strictly contains $[1]_{\leq_\varphi}$ the $\varphi(G)$-stabilizer of each $I_g$ acts non-trivially on $I_g$, and this implies that the action $\varphi$ is not cyclic, a contradiction.
\end{proof}
\begin{prop}\label{prop.minimalmodel} Let $G$ be a finitely generated group and consider a preorder $\leq\in\LPO(G)$. Then the dynamical realization $\varphi_\leq$ is a canonical model if and only if $\leq$ is a minimal model.
Moreover, the dynamical realization $\varphi_{\leq^\ast}$ is a canonical model for $\varphi_{\leq}$.
\end{prop}
The rest of the subsection is devoted to the proof of Proposition \ref{prop.minimalmodel}. We will need a preliminary result.
\begin{lem}\label{lem.domsemicon}
Let $G$ be a group.
Consider preorders $\leq_1,\leq_2\in\LPO(G)$ so that $\leq_2$ dominates $\leq_1$. Then the dynamical realizations $\varphi_{\leq_1}$ and $\varphi_{\leq_2}$ are positively semi-conjugate.
\end{lem}
\begin{proof} For $i\in \{1,2\}$ write $H_i:=[1]_{\leq_i}$ and $\iota_i:G/H_i\to \mathbb{R}$ for the corresponding good embeddings. As $\mathsf{id}:(G,\leq_1)\to(G,\leq_2)$ is order-preserving, we have $H_1\subseteq H_2$ and the quotient map
\[i_0:(G/H_1,<_1)\to(G/H_2,<_2)\]
is order-preserving and equivariant
(here $<_i$ denotes the total ordered induced by $\leq_i$ for $i\in\{1,2\}$). Write $X=\iota_1(G/H_1)$ and define $j_0:X\to\mathbb{R}$ as $$j_0(\iota_{1}(gH_1))=\iota_{2}(i_0(gH_1))=\iota_{2}(gH_2).$$ It follows directly from the definitions that $X$ if $\varphi_{\leq_1}(G)$-invariant and that $j_0$ is order-preserving and equivariant. We conclude using Lemma \ref{lem.semiconjugacy}.
\end{proof}
Let us immediately point out the following conclusion.
\begin{cor}\label{cor.equivsemicon}For finitely generated groups, equivalent preorders have positively semi-conjugate dynamical realizations.
\end{cor}
\begin{proof} After Lemma \ref{lem.domsemicon}, the dynamical realizations of equivalent preorders are positively semi-conjugate to the dynamical realization of their minimal model. As positive semi-conjugacy is an equivalence relation, the conclusion follows.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop.minimalmodel}] Suppose first that $\varphi_\leq$ is not a canonical model. Then $\varphi_\leq$ is neither minimal nor cyclic and so $\varphi_\leq$ has an exceptional minimal invariant set $\Lambda\subset \mathbb{R}$
or $\varphi_\leq$ has a closed discrete orbit but its image is not cyclic.
We write $X=\iota(G/[1]_\leq)$ for the image of the good embedding $\iota:G/[1]_\leq\to\mathbb{R}$ associated with $\varphi_\leq$, and remark that $X$ consists of a single $\varphi_{\leq}(G)$-orbit.
\begin{case}\label{c.minimalmodel}$\varphi_\leq$ has an exceptional minimal invariant set.
\end{case}
We first show the following.
\begin{claim}
$X$ is contained in $\mathbb{R}\setminus\Lambda$.
\end{claim}
\begin{proof}[Proof of claim]
We assume by contradiction that $X\cap \Lambda\neq\varnothing$. As $X$ consists of a single orbit, by invariance of $\Lambda$ we get that $X\subset \Lambda$ and thus $\Lambda=\overline{X}$ by minimality of $\Lambda$.
Consider a connected component $I=(\xi,\eta)$ of $\mathbb{R}\setminus\Lambda$ and note that, by Definition \ref{dfn.goodbehaved} of good embedding, we have $\{\xi,\eta\}\subseteq X$ and thus the points $\xi$ and $\eta$ are in the same orbit. This shows that $\Lambda$ is discrete, which is a contradiction.
\end{proof}
Consider now the connected component $U$ of $\mathbb{R}\setminus\Lambda$ that contains $\xi_0=\iota([1]_\leq)$ and write $H=\stab^{\varphi_{\leq}}_{G}(U)$. Note that $H=\iota^{-1}(U)$ is a $\leq$-convex subgroup. We will show that $H$ strictly contains $[1]_\leq$, which by Remark \ref{prop.minimalconvex}, implies that $\leq$ is not a minimal model. Equivalently, we need to show that $U\cap X$ strictly contains $\xi_0$. Assume that $\xi_0$ is an isolated point of $X$, otherwise there is nothing to prove.
Let $\lambda\in \Lambda$ be the rightmost point of $U$. Note that $\overline X$ is $\varphi_{\leq}(G)$-invariant, so we must have $\overline X\supseteq \Lambda$. Thus $\lambda\in \overline{X}$. If $X\cap (\xi_0,\lambda)=\varnothing$, then $(\xi_0,\lambda)$ is a connected component of $\mathbb{R}\setminus \overline{X}$. As $\iota$ is a good embedding, this gives $\lambda\in X$, contradicting the claim.
\begin{case} $\varphi_\leq$ has a closed discrete orbit but its image is not cyclic.
\end{case}
\setcounter{case}{0}
Take a point $\xi\in\mathbb{R}$ with closed discrete orbit and write $H=\stab^{\varphi_{\leq}}_G(\xi)$ and $F=\operatorname{\mathsf{Fix}}(H)$. Note that $H$ is the kernel of the morphism $G\to \mathbb Z$, induced from the orbit of $\xi$.
Since $\varphi_\leq$ is assumed to be non-cyclic we have that $F$ is a proper closed subset, and it is $\varphi_{\leq}(G)$-invariant, for $H$ is normal.
If $X\cap F\neq\varnothing$, then the fact that $X$ is a single $\varphi_{\leq}(G)$-orbit gives $X\subseteq F=\operatorname{\mathsf{Fix}}(H)$, and as $\varphi_{\leq}$ is a dynamical realization of $\le$, this implies $H=\varphi_{\leq}([1]_{\leq})$, so the image $\varphi_{\leq}(G)$ is cyclic. A contradiction.
Thus $X\subseteq\mathbb{R}\setminus F$. More precisely, we have the following.
\begin{claim}
$F=\operatorname{\mathsf{Fix}}(H)$ is contained in $\overline X\setminus X$.
\end{claim}
\begin{proof}[Proof of claim]
Assume for contradiction that there is a point of $F$ in the complement of $\overline X$, and let $I$ be corresponding connected component of $I$. As the action $\varphi_{\leq}$ is a dynamical realization (Definition \ref{d-dynamical-realisation}) and $F=\operatorname{\mathsf{Fix}}(H)$, this gives that the closure $\overline I$ is fixed by $H$, so that $\overline I\subseteq F$. However, $\iota$ is a good embedding, so $\partial I\in X$, which contradicts the fact that $X\cap F=\varnothing$.
\end{proof}
We can now argue analogously as in Case \ref{c.minimalmodel} to find a proper $\leq$-convex subgroup strictly containing $[1]_\leq$.
\smallskip
For the converse, assume that $\leq$ is not a minimal model and take a proper $\leq$-convex subgroup $H\neq[1]_\leq$. Denote by $U$ the interior of the convex hull of $\iota(H)\subset \mathbb{R}$ and notice that the orbit $\varphi_\leq(G)(U)$ is a proper open $\varphi_\leq(G)$-invariant subset. Also notice that the stabilizer $\stab_G^{\varphi_{\leq}}(U)$ acts non-trivially on $U$ which implies that $\varphi_\leq$ is not a cyclic action. The last two facts together imply that $\varphi_\leq$ is not a canonical model.
It remains to prove that $\varphi_{\leq^\ast}$ is a canonical model for $\varphi_\leq$. After Remark \ref{r.domination_minimal} and Lemma \ref{lem.domsemicon}, the actions $\varphi_{\leq^\ast}$ and $\varphi_\leq$ are positively semi-conjugate. On the other hand, by the first part of this proposition, the action $\varphi_{\leq^\ast}$ is a canonical model.
\end{proof}
\subsection{Topology of the space of equivalence classes of preorders}
\label{sc.topology_preo}
Recall that we consider $\LPO(G)$ as a subspace of $\{\leq,\gneq\}^{G\times G}$ endowed with the product topology, and that this makes $\LPO(G)$ a metrizable and totally disconnected topological space, which is compact when $G$ is finitely generated.
\begin{lem}\label{lem.closedrelation}Let $G$ be a finitely generated group. The subset \[\mathcal{E}=\{(\leq_1,\leq_2)\in\LPO(G)\times\LPO(G):[\leq_1]=[\leq_2]\}\]
is closed in $\LPO(G)\times\LPO(G)$. Therefore, $[\LPO](G)$ considered with the quotient topology is a Hausdorff topological space.
\end{lem}
\begin{proof} Consider a convergent sequence $(\preceq_n,\preceq_n')\to(\preceq_{\infty},\preceq_\infty')$ so that $(\preceq_n,\preceq_n')\in\mathcal{E}$ for every $n\in\mathbb N$. We want to show that $[\preceq_\infty]=[\preceq'_{\infty}]$ and this amounts to show, after Proposition \ref{prop.starequiv}, that there exists a preorder that dominates both $\preceq_\infty$ and $\preceq_\infty'$.
For every $n\in \mathbb N$, using Proposition \ref{prop.starequiv} we find a preorder $\leq_n$ that dominates both $\preceq_n$ and $\preceq_n'$. As $\LPO(G)$ is compact, upon passing to a subsequence, we can suppose that $\leq_n$ has a limit $\leq_{\infty}$. We shall prove that $\leq_{\infty}$ dominates both $\preceq_{\infty}$ and $\preceq_{\infty}'$, and by Lemma \ref{lem.dominequiv} this amounts to show that $P_{\leq_\infty}\subseteq (P_{\preceq_\infty}\cap P_{\preceq_\infty'})$. For this, note that by definition of product topology, if $g\in P_{\leq_\infty}$ then $g\in P_{\leq_n}$ for $n$ large enough, which by Lemma \ref{lem.dominequiv}, implies that $g\in P_{\preceq_n}\cap P_{\preceq_n'}$ for $n$ large enough. This implies that $g\in P_{\preceq_\infty}\cap P_{\preceq_\infty'}$ as desired.
\end{proof}
\begin{lem}\label{lem.convcriteria} Let $G$ be a finitely generated group. Consider preorders $(\leq_n)_{n\in\mathbb N}$ and $\leq$ in $\LPO(G)$. Assume that for every finite subset $F\subseteq P_{\leq}$ there exists $k_0\in\mathbb N$ such that $F\subseteq P_{\leq_k}$ for every $k\geq k_0$. Then $[\leq_k]\to[\leq]$ in $[\LPO](G)$.
In particular, if $\varphi_n,\varphi:G\to\homeo_0(\mathbb{R})$ are representations such that $\varphi_n\to\varphi$ in the compact open topology we get $[\leq_{\varphi_n}]\to[\leq_\varphi]$ in $[\LPO](G)$.
\end{lem}
\begin{proof} By hypothesis, we have that the limit of every convergent subsequence $\leq_{{k_n}}\to \leq_\infty$ satisfies $P_{\leq}\subseteq P_{\leq_\infty}$. This implies that $\leq$ dominates $\leq_\infty$ and therefore, by Proposition \ref{prop.starequiv}, we get $[\le]=[\le_\infty]$. Since every convergent subsequence of $(\leq_{n})_{n\in\mathbb N}$ converges to a preorder equivalent to $\leq$, we conclude that $[<_n]\to[<]$ in $[\LPO](G)$ as desired.
\end{proof}
\subsection{Retraction to the Deroin space}\label{sc.deroin_preo}
As in \S\ref{sec.harmder}, we fix a finitely generated group $G$, a symmetric probability measure $\mu$ on $G$ whose support is finite, and generates $G$.
We will show that $[\LPO](G)$ with the quotient topology is homeomorphic to $\Der_\mu(G)$. In particular, this will show that the topology of the Deroin space does not depend on the choice of the probability measure $\mu$.
\begin{thm}\label{thm.homeoderoin} With notation as above, the Deroin space $\Der_\mu(G)$ is homeomorphic to $[\LPO](G)$. In particular, the homeomorphism type of $\Der_\mu(G)$ does not depend on $\mu$.
\end{thm}
\begin{proof} Consider the map
\[\dfcn{I}{\Der_\mu(G)}{[\LPO](G)}{\varphi}{[\leq_\varphi]}.\]
We will show that $I$ is a homeomorphism. By Proposition \ref{prop.minimalmodel} we know that if $\preceq\in\LPO(G)$ is a minimal model, then $\varphi_\preceq$ is a canonical model. On the other hand, by Proposition \ref{prop.minimalisharmonic}, we can choose $\varphi_\preceq$ to be $\mu$-harmonic.
We introduce the map
\begin{equation}\label{eq.J}J:[\LPO](G)\to \Der_\mu(G),\end{equation}
where $J$ assigns to $[\preceq]$ the $\mu$-harmonic dynamical realization of $\preceq^\ast$, whose associated good embedding satisfies $\iota_{\preceq^\ast}([1]_{\preceq^\ast})=0$.
We will show that $J$ is the inverse of $I$.
The equality $I\circ J=\operatorname{\mathsf{id}}$ is given by the fact that
$\leq_{\varphi_\leq}$ coincides with $\leq$ for every preorder $\leq\in\LPO(G)$, which is a direct consequence of the definitions.
To verify that $J\circ I=\operatorname{\mathsf{id}}$, fix $\varphi\in \Der_\mu(G)$. Since $\varphi$ is a canonical model (Proposition \ref{prop.minimalisharmonic}), Proposition \ref{lem.minimalisminimal} implies that $\leq_\varphi$ is a minimal model. Therefore, $J([\leq_\varphi])$ is a dynamical realization of $\leq_\varphi$.
Set $X=\varphi(G)(0)$ and notice that there exists an order-preserving map $j_0:X\to\mathbb{R}$ which is equivariant for the actions $\varphi$ and $J([\leq_\varphi])$. Since both actions are canonical models, it follows from Lemma \ref{lem.semiconjugacy} and Corollary \ref{cor.basica} that $j_0$ can be extended to a positive conjugacy. Moreover, since $j_0(0)=0$, this conjugacy fixes $0$ and therefore, by Corollary \ref{rem.fixes0}, we conclude that $\varphi=J([\leq_\varphi])$ as desired.
The continuity of $I$ follows directly from Lemma \ref{lem.convcriteria}. Finally, since $\Der_\mu(G)$ is compact (Theorem \ref{t.def_Deroin}) and $[\LPO](G)$ is a Hausdorff topological space (Lemma \ref{lem.closedrelation}), we conclude that $I$ is a homeomorphism.
\end{proof}
We can now prove the main result of this section.
\begin{proof}[Proof of Theorem \ref{t.retraction_Deroin}]
We consider the map
\[
\dfcn{r}{\Homirr(G,\homeo_0(\mathbb{R}))}{\Der_\mu(G)}{\varphi}{J([\leq_\varphi])},
\]
where $J$ is the map introduced in \eqref{eq.J} and $\leq_\varphi$ is the preorder induced by $0$. It follows directly from the definitions that $r$ is the identity in restriction to $\Der_\mu(G)$. Also, by Lemma \ref{lem.convcriteria} and Theorem \ref{thm.homeoderoin} we conclude that $r$ is continuous. Finally, to check that $r$ preserves positive semi-conjugacy classes first note that $\varphi$ is positively semi-conjugate to the dynamical realization of $\leq_\varphi$ and that, by Corollary \ref{cor.equivsemicon}, the dynamical realizations of $\leq_\varphi$ and $\leq_\varphi^\ast$ are also positively semi-conjugate.
\end{proof}
\subsection{Application to the Borel reducibility of the semi-conjugacy relation}\label{ssc.complexity}
For a given group $G$ it is natural to try to determine how difficult it is to distinguish actions in $\Homirr(G, \homeo_0(\mathbb{R}))$ up to semi-conjugacy, and whether it is possible to classify them.
To conclude this section, we highlight that the Deroin space and of Theorem \ref{t.retraction_Deroin} shed light on this question, and suggest a line of research for group actions on the line.
An appropriate framework to formalize the question is the theory of Borel reducibility of equivalence relations. Most notions of isomorphisms can be naturally interpreted as equivalence relations on some standard Borel space, so that this theory offers tools to study various classification problems and compare the difficulty of one with respect to another.
Let us recall some basic notions from this setting following the exposition of Kechris \cite{KechrisCBE} (to which we refer for more details). Recall that a standard Borel space $\mathcal{Z}$ is a measurable space isomorphic to a complete separable metric space endowed with its Borel $\sigma$-algebra. A subset of such a space is \emph{analytic} if it is the image of a Borel set under a Borel map from a standard Borel space. An analytic equivalence relation on $\mathcal{Z}$ (henceforth just \emph{equivalence relation}) is an equivalence relation $\mathcal{R}$ which is an analytic subset $\mathcal{R}\subset \mathcal{Z}\times \mathcal{Z}$. This standing assumption is convenient to develop the theory, and general enough to model most natural isomorphism problems. It includes in particular the class of Borel equivalence relations (those which are Borel subsets of $\mathcal{Z}\times \mathcal{Z}$), which is much better behaved but too restricted for some purposes (see below). We write $x\mathcal{R}y$ if $(x, y)\in \mathcal{R}$
For $i\in\{1, 2\}$ let $\mathcal{R}_i$ be an equivalence relation on a standard Borel space $\mathcal{Z}_i$. We say that $\mathcal{R}_1$ is \emph{reducible} to $\mathcal{R}_2$, and write $\mathcal{R}_1\le_B \mathcal{R}_2$, if there exists a Borel map $q\colon \mathcal{Z}_1\to \mathcal{Z}_2$ such that $x\mathcal{R}_1y$ occurs if and only if $q(x)\mathcal{R}_2 q(y)$. This means that distinguishing classes of $\mathcal{R}_1$ is ``easier'' than for $\mathcal{R}_1$. If $\mathcal{R}_1\le_B\mathcal{R}_2$ and $\mathcal{R}_2\le_B \mathcal{R}_1$, we say that $\mathcal{R}_1$ and $\mathcal{R}_2$ are \emph{bireducible}.
The simplest type of equivalence relations from the perspective of reducibility are the \emph{smooth} ones. An equivalence relation $\mathcal{R}$ on $\mathcal{Z}$ is said {smooth} if there exist a standard Borel space $\mathcal{W}$ and a Borel map $q\colon \mathcal{Z}\to \mathcal{W}$ which is a complete invariant for the equivalence classes of $\mathcal{R}$, in the sense that $x\mathcal{R} y$ if and only if $q(x)=q(y)$.
The next class of equivalence relations are the \emph{hyperfinite} ones. An equivalence relation $\mathcal{R}$ is called \emph{hyperfinite} if it is a countable union $\mathcal{R}=\bigcup\mathcal{R}_n$ of equivalence relations whose classes are finite, and {essentially hyperfinite} if it is bireducible to a hyperfinite one. An example of hyperfinite equivalence relation which is not smooth, denoted $\mathcal{E}_0$, is the equivalence relation on the set of one-sided binary sequence $\{0,1\}^\mathbb N$, where $(x_n)$ and $(y_n)$ are equivalent if $x_n=y_n$ for large enough $n$. In fact, up to bireducibility the relation $\mathcal{E}_0$ is the unique essentially hyperfinite equivalent relation which is not smooth, and moreover every Borel equivalence relation $\mathcal{R}$ is either smooth or satisfies $\mathcal{E}_0\le_B \mathcal{R}$ \cite{HKL}, so that $\mathcal{E}_0$ can be thought of as the {simplest} non-smooth Borel equivalence relation.
The essentially hyperfinite equivalence relations form a strict subset of the essentially countable ones (those that are bireducible to a relation whose classes are countable). These are precisely those induced by orbits of actions of countable groups, and the poset of bireducibility types of such relations is quite complicated (see \cite{AdamsKechris}). Essentially countable relations are themselves a strict subset of Borel equivalence relations, after which we find general (analytic) equivalence relations.
The bireducibility type of the conjugacy relation of various classes of dynamical systems and group actions has been extensively studied. For many classes of topological or measurable dynamical systems, the conjugacy relations is known or conjectured to be quite complicated from the perspective of bireducibility \cite{MR2800720, GJS, TS, LR-Borel}. As an example, the conjugacy relation of elements of $\homeo_0(\mathbb{R})$ is bireducible to the isomorphism relation of all countable (but not necessarily locally finite) graphs, which is analytic but {not} Borel (see \cite[Theorem 4.9]{Hjorth}), and the conjugacy of homeomorphisms of the plane is strictly more complicated by a result of Hjorth \cite[Theorem 4.17]{Hjorth}. As a consequence, one may expect that the semi-conjugacy relation on the space of irreducible action of a given group $G$ should also be complicated and might not even be Borel. In contrast, the existence of Deroin space and Theorem \ref{t.retraction_Deroin} imply the following.
\begin{cor} \label{c-borel-hyperfinite}
Let $G$ be a finitely generated group. Then the semi-conjugacy relation on the space $\Homirr(G, \homeo_0(\mathbb{R}))$ is essentially hyperfinite (in particular, it is Borel).
\end{cor}
\begin{proof}
Theorem \ref{t.retraction_Deroin} shows that this relation is bireducible to the orbit equivalence relation on the Deroin space $\Der_\mu(G)$ induced by the translation flow $\Phi$.
On the other hand, the orbit equivalence relation of any Borel flow on a standard Borel space is essentially hyperfinite \cite[Theorem 8.32]{KechrisCBE}. \qedhere
\end{proof}
After Corollary \ref{c-borel-hyperfinite} and the previous discussion we may further distinguish two cases: either the semi-conjugacy relation on $\Homirr(G, \homeo_0(\mathbb{R}))$ is smooth, or it is not, in which case it is bireducible to $\mathcal{E}_0$. As mentioned above, the first case corresponds to groups for which actions up to semi-conjugacy can be completely classified by a Borel invariant, so that it is natural to ask which groups have this property. This can be rephrased using the Deroin space as follows.
\begin{cor}\label{cor.cross-section}
Let $G$ be a finitely generated group endowed with a symmetric finitely supported probability measure $\mu$. Then the semi-conjugacy relation on $\Homirr(G, \homeo_0(\mathbb{R}))$ is smooth if and only if the translation flow $\Phi$ on the Deroin space $\Der_\mu(G)$ admits a Borel cross section (that is a Borel subset which intersects every $\Phi$-orbit in exactly one point).
\end{cor}
\begin{proof}
Given a flow $\Phi$ on a Borel space $\mathcal{Z}$, the orbit equivalence relation induced by $\Phi$ is smooth if and only if $\Phi$ admits a Borel cross section \cite[Proposition 3.12]{KechrisCBE}. \qedhere
\end{proof}
We propose to keep Corollary \ref{cor.cross-section} in mind as a goal when studying Deroin space of groups. We will present some applications in \S \ref{s-class-borel}.
\begin{rem}\label{rem non smooth}
There exist finitely generated groups $G$ for which the semi-conjugacy relation on $\Homirr(G, \homeo_0(\mathbb{R}))$ is not smooth (and in particular, the space of semi-conjugacy classes is not a standard measurable space). For example it is not difficult to show this for the free group $\mathbb{F}_2$ as follows. Fix a ping-pong pair of homeomorphisms $g, h$ of $\mathbb{R}/\mathbb Z$, with $\operatorname{\mathsf{Fix}}(g)=\{0, 1/2\}$ and $\operatorname{\mathsf{Fix}}(h)=\{1/4, 3/4\}$, where both $g, h$ have one attracting and one repelling fixed point and such that $\langle g, h\rangle$ acts minimally on $\mathbb{R}/\mathbb Z$. Let $\tilde{g}, \tilde{h}\in \homeo_0(\mathbb{R})$ be two lifts, with $\operatorname{\mathsf{Fix}}(\tilde{g})=\frac{1}{2}\mathbb Z$ and $\operatorname{\mathsf{Fix}}(\tilde{h})=\frac{1}{2}\mathbb Z+\frac{1}{4}$. Given a sequence $\omega=(\omega_n)\in \{+1, -1\}^\mathbb Z$, define an element $\tilde{g}_\omega\in \homeo_0(\mathbb{R})$ by $\tilde{g}_\omega(x)=\tilde{g}^{\omega_n}(x)$ if $x\in [\frac{1}{2}n, \frac{1}{2}n+1]$, and if $\mathbb{F}_2$ is the free group with free generators $a, b$, define a representation $\varphi_\omega\colon \mathbb{F}_2\to \homeo_0(\mathbb{R})$ by $\varphi_\omega(a)=\tilde{g}_\omega$ and $\varphi_\omega(b)=\tilde{h}$. It is not difficult to check that the map $\omega\mapsto \varphi_\omega$ is Borel (actually continuous).
since $\varphi_\omega(\mathbb F_2)=\langle \tilde{g}_\omega, \tilde{h}\rangle$ acts on $\mathbb{R}$ with the same orbits as $\langle \tilde{g}, \tilde{h}\rangle$, every action $\varphi_\omega$ is minimal, and this implies for $\omega, \omega'\in \{\pm 1\}^\mathbb Z$ the actions $\varphi_\omega$ and $\varphi_{\omega'}$ are positively semi-conjugate if and only if they are positively conjugate, and this happens if and only if $\omega$ and $\omega'$ belong to the same orbit of the $\mathbb Z$-shift. Since the orbit equivalence relation of the shift is not smooth, the conclusion follows. \end{rem}
\begin{rem}
In contrast, for every countable group $G$ the semi-conjugacy relation on the space $\Homirr(G, \homeo_0(\mathbb{S}^1))$ of irreducible actions on the circle is always smooth. One way to prove this is to repeat the above argument by fixing a generating probability measure $\mu$ on $G$ and recall that every action is semi-conjugate to an action for which the Lebesgue measure is stationary (unlike for the case of the real line, this is a straightforward consequence of compactness). This can be used to construct an analogue of the Deroin space, where the translation flow should be replaced by an action of the group $\mathbb{S}^1$, and since this is a compact group the action must always admit a Borel cross section. Another route would be to prove that the bounded Euler class is a Borel complete invariant under semi-conjugacy.
This difference can be seen as a formalization of the observation, mentioned in the introduction, that studying actions on the circle up to semi-conjugacy is easier than for the real line thanks to compactness.
\end{rem}
\section{Differentiable actions of locally moving groups}\label{s-differentiable}
In this section we are interested in actions on a closed interval, or on the real line, which are by diffeomorphisms of class $C^1$.
First, let us observe that a rather direct consequence of Theorem \ref{t-lm-trichotomy} is a rigidity result for actions on the real line by diffeomorphisms of class $C^2$, namely every action $\varphi : G \to \Diff^2_0(\mathbb{R})$ of a locally moving group (without fixed points) is semi-conjugate to the standard action or to an action of $G/[G_c, G_c]$. Indeed, recall that Kopell's lemma states that whenever $f$ and $g$ are non-trivial commuting $C^2$ diffeomorphisms of a compact interval, and if $f$ has no fixed point in its interior, then neither does $g$. If $\varphi$ falls into the exotic case of Theorem \ref{t-lm-trichotomy}, it is easy to check (as in the proof of Corollary \ref{c-lm-property-exotic}) that the image of $G$ must contain an abundance of commuting pairs of elements which preserve a compact interval and do not satisfy the conclusion of Kopell's lemma. However we do not elaborate on this, because we will show in this section that this rigidity actually holds for $C^1$ actions (for which Kopell's lemma fails, see e.g.\ Bonatti and Farinelli \cite{BonattiFarinelli}).
We start by recalling some classical results in this setting.
\subsection{Conradian actions and $C^1$ actions} Let $G\subseteq \homeo_0(\mathbb{R})$ be a subgroup.
A \emph{pair of successive fixed points} for $G$ is a pair $a, b\in\mathbb{R}\cup\{\pm\infty\}$ with $a < b$ such that there is an element
$g\in G$ for which $(a, b)$ is a connected component of $\supp (g)$. A \emph{linked pair of fixed points} for $G$ consists of pairs $a, b$ and $c , d$ in $\mathbb{R}\cup\{\pm\infty\}$ such that:
\begin{enumerate}
\item there are elements $f, g\in G$ such that $a, b$ is a pair of successive fixed points
of $f$ and $c , d$ is a pair of successive fixed points of $g$;
\item either $\{a,b\}\cap (c,d)$ or $(a,b)\cap \{c,d\}$ is a point.
\end{enumerate}
As pointed out by Navas \cite{Navas2010}, the previous notion is the dynamical counterpart of Conradian orderings on groups. Following the terminology of \cite{NavasRivas-Conrad}, we will say that an action of a group $G$ on an interval is \emph{Conradian} if it has no pair of linked fixed points and there is no (global) fixed point. We have the following fundamental fact (see for instance \cite[Theorem 2.14]{NavasRivas-Conrad}).
\begin{thm}\label{t-conrad}
Any Conradian action $\varphi:G\to \homeo_0(\mathbb{R})$ of a finitely generated group $G$ on the real line is semi-conjugate to an action by translations. In particular, there exists a non-trivial morphism $\tau:G\to \mathbb{R}$ (the \emph{Conrad homomorphism}), unique up to positive rescaling, such that if $g\in G$ is such that $\tau(g)>0$ and $f\in \ker \tau$, then for every $n\in \mathbb Z$ and $\xi\in \mathbb{R}$ one has
$f^n.\xi\le g.\xi$.
\end{thm}
Moreover, assume that $\varphi\colon G\to \homeo_0(\mathbb{R})$ is Conradian, let $H\subseteq G$ be a finitely generated subgroup, and let $I \subset \mathbb{R}$ be a connected component of $\suppphi(H)$. Then the action of $H$ induced on $I$ by restriction is still Conradian, therefore it is also semi-conjugate to an action by translations.
The following result is the version of Sacksteder's theorem for $C^1$ pseudogroups, as established by Deroin, Kleptsyn, and Navas in \cite{DKN-acta} (see also Bonatti and Farinelli \cite{BonattiFarinelli} for a simplified proof).
\begin{thm}
\label{p.sacksteder}
Let $G\subseteq \Diff_0^1([0,1])$ be a subgroup acting with a linked pair of fixed points. Then there exists a point $x\in (0,1)$, and an element $h\in G$ for which $x$ is a hyperbolic fixed point: $h(x)=x$ and $h'(x)<1$.
\end{thm}
\begin{rem}\label{r.sacksteder}
In fact, the proof of Theorem \ref{p.sacksteder} gives a more precise statement, which we point out as it will be useful for the sequel.
Write $I=[0,1]$. It is not difficult to see that the existence of a linked pair of fixed points for $G$, as in Theorem~\ref{p.sacksteder}, gives a subinterval $J\subset I$ and two elements $f,g\in G$ such that the images $f(J)$ and $g(J)$ are both contained in $J$, and are disjoint: $f(J)\cap g(J)=\varnothing$. (This situation is the analogue of a Smale's horseshoe for one-dimensional actions.)
It follows that every element $h\in \langle f,g\rangle_+$ in the (free) semigroup generated by $f$ and $g$ satisfies $h(J)\subset J$, and moreover the images $h(J)$, where $h$ runs through the $2^n$ elements of length $n$ in the semigroup $\langle f,g\rangle_+$ (with respect to the generating system $\{f,g\}$), are pairwise disjoint. Clearly the inclusion $h(J)\subset J$ gives that every $h$ admits a fixed point in $h(J)$. Using a probabilistic argument, and uniform continuity of $f'$ and $g'$ on $J$, one proves that as $n$ goes to infinity, most of the elements $h$ of length $n$ are uniform contractions on $J$. This implies that most elements $h\in \langle f,g\rangle_+$ of length $n$, when $n$ is large enough, have a unique fixed point in $J$, which is hyperbolic.
We deduce that if $\Lambda\subset J$ is an invariant Cantor set for $f$ and $g$ (and thus for $\langle f,g\rangle_+$), then the hyperbolic fixed point for a typical long element $h\in \langle f,g\rangle_+$ will never belong to the closure of a gap $J_0$ of $\Lambda$: otherwise, $h$ would fix the whole gap $J_0$, and therefore there would be a point $y\in J_0$ for which $h'(y)=1$, contradicting the fact that $h$ is a uniform contraction.
\end{rem}
We point out a straightforward consequence of Theorem \ref{p.sacksteder} (largely investigated in \cite{BonattiFarinelli}).
\begin{cor}\label{t-Bo-Fa}
Let $G\subseteq \Diff^1_0([0,1])$ be a subgroup acting with a linked pair of fixed points, then there is no non-trivial element $f\in \Diff^1_0([0,1])$ without fixed points in $(0, 1)$, centralizing $G$.
\end{cor}
\subsection{Conradian $C^1$ actions of Thompson's group $F$}
Before discussing $C^1$ actions of general locally moving groups, we first prove a preliminary result in the case of Thompson's group $F$, namely we rule out the existence of Conradian $C^1$ faithful actions of $F$. In fact, this will be used when studying general locally moving groups.
The first step is to analyze actions which are sufficiently close to the trivial action, in a spirit similar to the works of Bonatti \cite{BonattiProches} and McCarthy \cite{McCarthy} (see also the related works \cite{BMNR,BKKT}). For the statement, we recall that the $C^1$ topology on $\Diff_0^1([0,1])$ is defined by the $C^1$ distance
\[d_{C^1}(f,g)=\sup_{\xi\in [0,1]} |f(\xi)-g(\xi)|+\sup_{\xi\in [0,1]}|f'(\xi)-g'(\xi)|.\]
When $G$ is a finitely generated group endowed with a finite symmetric generating set $S$, we consider the induced topology on $\Hom\left (G,\Diff_0^1([0,1])\right )$, saying that two representations $\varphi$ and $\psi$ are $\delta$-close if $d_{C^1}(\varphi(g),\psi(g) )\le \delta$ for every $g\in S$.
Moreover, given $\varphi \in\Hom\left (G,\Diff_0^1([0,1])\right )$, an element $g\in G$, and a point $\xi\in [0,1]$, we write
$\Delta_\xi^\varphi(g)=g.\xi-\xi$ for the displacement of the point $\xi$. Clearly $\Delta_\xi^\varphi(g)=0$ for every $\xi\in [0,1]$ if and only if $g\in \ker \varphi$.
\begin{lem}\label{l-Conrad-C1-local} There exists a neighborhood $\mathcal V$ of the trivial representation in $\Hom\left (F,\Diff_0^1([0,1])\right )$ such that if $\varphi\in\mathcal V$ has no linked pair of fixed points, then $[F,F]\subseteq \ker \varphi$.
\end{lem}
\begin{proof}
Let $\varphi \in\Hom\left (F,\Diff_0^1([0,1])\right )$ be an action with no linked pair of fixed points.
Consider two dyadic open subintervals $I\Subset J\Subset X=(0,1)$ (i.e.\ with dyadic rational endpoints).
Let $h\in F$ be such that $h(J)=I$. We choose an element $f\in F_I\cong F$ without fixed points in $I$, and set $g=h^{-1}fh\in F_J$ for the conjugate element (which acts without fixed points on $J$). Note that both $f$ and $g$ belong to the subgroup $H:=\langle g, [F_J, F_J]\rangle$. The group $H$ is finitely generated (it is generated by $g$ and by the group $F_L$ for any dyadic subinterval $L\Subset J$ such that $g(L)\cap L\neq \varnothing)$, and since $[F_J, F_J]$ is simple and normal in $H$, the abelianization of $H$ is infinite cyclic, generated by the image of $g$. By Theorem \ref{t-conrad} (applied to every action obtained by taking restriction of $\varphi$ to a connected component of $\suppphi(H)$), this implies that
\begin{equation}\label{eq:conrad-F}
\left |\Delta_\xi^\varphi(f^n)\right | \leq \max\left \{ \left |\Delta_\xi^\varphi(g) \right |,\left |\Delta_\xi^\varphi(g^{-1})\right |\right \}\quad
\text{for every $\xi\in [0,1]$ and $n\in \mathbb Z$}.\end{equation}
\begin{claim}
Fix $\gamma>0$. If $\varphi(f)$ and $\varphi(g)$ are sufficiently close to the identity in the $C^1$ topology, for all $\xi\in [0,1]$ we have
\[(2-\gamma) \left |\Delta_\xi^\varphi(f) \right |\le (1+\gamma) \left |\Delta_\xi^\varphi(g) \right |.\]
\end{claim}
\begin{proof}[Proof of claim]
Fix $\gamma>0$. Using the mean value theorem (in more general form, this is Bonatti's approximate linearization estimate \cite{BonattiProches}; see also \cite{BKKT}), we get that if $\varphi(f)$ and $\varphi(g)$ are sufficiently $C^1$ close to the identity, for all $\xi\in [0,1]$ we have
\[
\left |\Delta_\xi^\varphi(f^2)-2\Delta_\xi^\varphi(f)\right |\le \gamma \left |\Delta_\xi^\varphi(f)\right |
\]
and
\[
\left| |\Delta_\xi^\varphi(g^{-1}) |-| \Delta_\xi^\varphi(g) | \right|=\left|\Delta_\xi^\varphi(g^{-1}) + \Delta_\xi^\varphi(g) \right|\le \gamma \left |\Delta_\xi^\varphi(g)\right |.
\]
From \eqref{eq:conrad-F} and the two estimates above, we obtain the inequalities \[(2-\gamma)\left |\Delta_\xi^\varphi(f)\right |\le \left |\Delta_\xi^\varphi(f^2)\right |\le (1+\gamma)\left |\Delta_\xi^\varphi(g)\right |.\qedhere\]
\end{proof}
\begin{claim}
Fix $\gamma>0$. If $\varphi(g)$ and $\varphi(h)$ are sufficiently close to the identity in the $C^1$ topology, for all $\xi\in [0,1]$ we have
\[
(1-\gamma)\left |\Delta_\xi^\varphi(g)\right |\le
\left |\Delta^\varphi_{h.\xi}(f) \right |.
\]
\end{claim}
\begin{proof}[Proof of claim]
We have
\begin{align*}
\Delta^\varphi_{h.\xi}(f)&=fh.\xi-h.\xi=hg.\xi-h.\xi\\
&=\varphi(h)'(\xi)\cdot \Delta^\varphi_\xi(g)+o\left (\Delta^\varphi_\xi(g)\right ).
\end{align*}
Assume that $o\left ( |\Delta^\varphi_\xi(g) |\right )\le \frac{\gamma}{2} \left |\Delta^\varphi_\xi(g)\right |$ and $\left |\varphi(h)'(\xi)-1\right |\le \frac{\gamma}2$. Then by the triangle inequality we get
\[
\left |\Delta^\varphi_{h.\xi}(f)- \Delta_\xi^\varphi(g)\right |\le \left |\varphi(h)'(\xi)-1\right |\cdot \left |\Delta^\varphi_\xi(g)\right |+o\left ( |\Delta^\varphi_\xi(g) |\right )\le \gamma \left |\Delta_\xi^\varphi(g)\right |.\qedhere
\]
\end{proof}
Choose now $\gamma>0$ small enough, so that $\frac{(2-\gamma)(1-\gamma)}{1+\gamma}=\lambda>1$, and consider the appropriate neighborhood $\mathcal V$ of the trivial representation in $\Hom\left (F,\Diff_0^1([0,1])\right )$, so that the conditions in the claims are satisfied for every Conradian action $\varphi\in \mathcal V$. Fix $\varphi\in \mathcal V$ Conradian. Combining the claims, we conclude that for every $\xi\in [0,1]$, one has
$\lambda \left |\Delta_\xi^\varphi(f) \right |\leq \left |\Delta^\varphi_{h.\xi}(f)\right |$.
As this holds for any $\xi\in [0,1]$, we get that for any $n\in \mathbb N$ and $\xi\in [0,1]$ one has
$\lambda^n \left |\Delta_\xi^\varphi(f) \right |\leq \left |\Delta^\varphi_{h^n.\xi}(f) \right |$. As $ \left |\Delta^\varphi_{h^n.\xi}(f) \right |$ is bounded by $1$ (the length of $[0,1]$),
we deduce that the element $f$ belongs to the kernel of $\varphi$, as desired. In particular $\ker\varphi$ is a non-trivial normal subgroup of $F$, and thus it contains $[F, F]$. \end{proof}
The previous statement, which is of local nature (perturbations of the trivial actions), is used to obtain a global result. For this, we recall a trick attributed to Muller \cite{Muller} and Tsuboi \cite{Tsuboi} (see also \cite{BMNR}) after which, given any $C^1$ action $\varphi:G\to \Diff_0^1([0,1])$ of a group $G$ on $[0,1]$ and a finite subset $S\subset G$, for any $\varepsilon>0$ there exists $\delta>0$ and a $C^1$ action $\psi:G\to \Diff_0^1([0,1])$, which is $C^0$ conjugate to the original one, such that
\[|\psi(g)(\xi)-\xi|+|\psi(g)'(\xi)-1|\le \varepsilon\quad \text{for every }\xi\in [0,\delta]\text{ and }g\in S.\]
\begin{lem}\label{l-Conrad-C1-global} Every Conradian $C^1$ action of $F$ on the closed interval $[0,1]$ has abelian image.
\end{lem}
\begin{proof}
Let $\varphi\in \Hom\left (F,\Diff_0^1([0,1])\right )$ be a Conradian action.
We want to prove that $\ker \varphi$ contains $[F,F]$, and for this it is enough to show that $\ker\varphi$ is non-trivial.
Let us fix a dyadic subinterval $I\Subset X=(0,1)$.
After Theorem \ref{t-conrad}, $\varphi$ is semi-conjugate to an action by translations, and this is given by the Conrad homomorphism $\tau:F\to \mathbb{R}$. We can find a minimal non-empty closed $\varphi(F)$-invariant subset $\Lambda\subset (0,1)$ such that $\ker \tau$ pointwise fixes $\Lambda$. We write $\mathscr J$ for the collection of connected components of $(0,1)\setminus \Lambda$.
Note that $F_I\subseteq \ker \tau$ and therefore, given an interval $J\in \mathscr{J}$, we have an induced action $\varphi_J\in \Hom\left (F_I,\Diff_0^1(\overline J)\right )$ obtained by restriction of $\varphi$. Note that as $\varphi$ is Conradian, then $\varphi_J$ has no linked pair of fixed points. We also remark that if $J$ and $J'$ are in the same $\varphi(F)$-orbit, then $\varphi_J$ and $\varphi_{J'}$ are conjugate.
Let $\mathcal V$ be the neighborhood of the trivial representation provided by Lemma \ref{l-Conrad-C1-local}, and denote by $\mathcal V_J\subset \Hom\left (F_I,\Diff_0^1(\overline J)\right )$ the corresponding neighborhood obtained after considering an identification $\Hom\left (F_I,\Diff_0^1(\overline J)\right )\cong \Hom\left (F,\Diff_0^1([0,1])\right )$.
Using the trick of Muller and Tsuboi to the action $\varphi$, we can assume that there exists $\delta>0$ such that if $J\subset (0,\delta)$, then $\varphi_J\in \mathcal V_J$. In particular for such intervals $J\in \mathscr{J}$, we have $[F_I,F_I]\subseteq\ker \varphi_J$. As the $\varphi(F)$-orbit of every $J\in \mathscr J$ has an element contained in $(0,\delta)$, and two intervals in the same $\varphi(F)$-orbit lead to conjugate actions, we deduce that $[F_I,F_I]\subseteq\ker \varphi_J$ for every $J\in \mathscr{J}$. As $\varphi(F_I)$ pointwise fixes $\Lambda$, which is the complement of $\bigcup_{J\in \mathscr{J}}J$ in $(0,1)$, we deduce that $[F_I,F_I]\subseteq\ker \varphi$, as desired.
\end{proof}
With similar proof, we can extend the previous result to $C^1$ actions on the real line.
\begin{prop}\label{p-Conrad-C1-global} Every Conradian $C^1$ action of $F$ on the real line has abelian image.
\end{prop}
\begin{proof}
We proceed as in the proof of Lemma \ref{l-Conrad-C1-global}, and for this reason we skip some detail. We start with a Conradian action $\varphi\in \Hom\left (F,\Diff_0^1(\mathbb{R})\right )$, and consider the Conradian homomorphism $\tau:F\to \mathbb{R}$, and a minimal invariant subset $\Lambda\subset \mathbb{R}$, pointwise fixed by $\ker \tau$. We denote by $\mathscr J$ the collection of connected components of $\mathbb{R}\setminus \Lambda$. For a given dyadic subinterval $I\Subset X$, this gives rise to actions $\varphi_J\in \Hom\left (F_I,\Diff_0^1(J)\right )$ without linked pairs of fixed points ($J\in \mathscr{J}$). After Lemma \ref{l-Conrad-C1-global} (applied to the restriction of $\varphi_J$ to the closure of every connected component of $\supp^{\varphi_J}(F_I)$), we deduce that $[F_I,F_I]\subseteq \ker \varphi_J$ and therefore $[F_I,F_I]\subseteq \ker \varphi$.
\end{proof}
\subsection{$C^1$ actions of general locally moving groups}
Using that any locally moving group contains a copy of $F$ (Proposition \ref{p-chain}), we get the following consequence of Proposition \ref{p-Conrad-C1-global}.
\begin{prop}\label{p-lm-Conrad-C1} Locally moving groups admit no faithful Conradian $C^1$ actions on the real line.
\end{prop}
We actually have the following alternative, which is a more precise formulation of Theorem \ref{t-intro-C1} from the introduction.
\begin{thm}\label{t-lm-C1}
For $X= (a, b)$, let $G\subseteq \homeo_0(X)$ be locally moving. Then every action $\varphi\colon G\to \Diff^1_0(\mathbb{R})$ without fixed points, satisfies one of the following:
\begin{enumerate}[label=\roman*)]
\item either $\varphi$ is semi-conjugate to the standard action of $G$ on $X$;
\item or $\varphi$ is semi-conjugate to an action that factors through the quotient $G/[G_c, G_c]$.
\end{enumerate}
Moreover the second case occurs if and only if $\varphi([G_c, G_c])$ has fixed points, in which case the action of $[G_c, G_c]$ on each connected component of its support is semi-conjugate to its standard action on $X$.
\end{thm}
\begin{proof}[Proof of Theorem \ref{t-lm-C1}]
Assume by contradiction that $\varphi\colon G\to \Diff^1_0(\mathbb{R})$ is an action which is not semi-conjugate to the standard action on $X$, nor to any action of the quotient $G/[G_c,G_c]$. In particular $\varphi$ must be faithful.
Note that $G_c$ is locally moving, so that by Proposition \ref{p-chain} we can find a subgroup $\Gamma\subseteq G_c$ isomorphic to $F$. In particular, $\Gamma$ is finitely generated, and we can apply Corollary \ref{c-lm-property-exotic}. Let $I\subset \supp^\varphi(\Gamma)$ be a connected component, and take an element $f\in [G_c,G_c]$ centralizing $\Gamma$, such that $f.I\cap I=\varnothing$ (this always exists, as pointed out in the proof of Corollary \ref{c-lm-property-exotic}.\ref{i-lm-fixed}). Let $J$ be the connected component of $\supp^\varphi(f)$ containing $I$, which is bounded after Corollary \ref{c-lm-property-exotic}.\ref{i-lm-support}. Applying Corollary \ref{t-Bo-Fa} to the induced action of $\Gamma$ on $\overline J$ obtained by restriction of $\varphi$, we deduce from Lemma \ref{l-Conrad-C1-global} that the restriction of $\varphi(\Gamma)$ to $I$ is abelian. As $I\subset \supp^\varphi(\Gamma)$ was arbitrary, we get that $\varphi(\Gamma)$ is abelian, and therefore $\varphi:G\to \Diff^1_0(\mathbb{R})$ is not faithful, which is an absurd.
The last statement is a consequence of the fact that the group $[G_c, G_c]$ is still locally moving (by Lemma \ref{l-rigid}), thus we can apply the first part of the theorem to its action on the connected components of the support, and since it is simple (Proposition \ref{p-micro-normal}) only the first case can occur.
\end{proof}
The last result of this section corresponds to Corollary \ref{c-intro-C1}, with a more precise statement.
\begin{dfn}
For $X=(a,b)$, let $G\subseteq \homeo_{0}(X)$ be locally moving. We say that $G$ has \emph{independent groups of germs at the endpoints} if for every pair of elements $\gamma_1\in \Germ(G, a)$ and $\gamma_2\in \Germ(G, b)$, there exists $g\in G$ such that $\mathcal{G}_a(g)=\gamma_1$ and $\mathcal{G}_b(g)=\gamma_2$. \end{dfn}
Note that this is equivalent to the condition that the natural injective homomorphism $G/G_c\hookrightarrow \Germ(G, a)\times \Germ(G, b)$ be onto.
\begin{cor}\label{c-lm-C1-interval}
For $X=(a,b)$, let $G\subseteq \homeo_{0}(X)$ be locally moving with independent groups of germs at the endpoints. Then the following hold.
\begin{enumerate}[label=\roman*)]
\item \label{i-C1-interval} Every faithful action $\varphi:G\to \Diff_0^1([0,1])$ without fixed points is semi-conjugate to the standard action on $X$.
\item \label{i-C1-line} Every faithful action $\varphi:G\to \Diff_0^1(\mathbb{R})$ without fixed points is either semi-conjugate to the standard action on $X$, or to a cyclic action.
In the latter case, if $\tau :G\to \mathbb Z$ is the homomorphism giving the semi-conjugate action, then the action of $\ker \tau$ on each connected component of its support is semi-conjugate to its standard action on $X$.
\end{enumerate}
\end{cor}
To prove Corollary \ref{c-lm-C1-interval} we need the following lemma.
\begin{lem}\label{l-independent-germs}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be locally moving with independent groups of germs at the endpoints. Then for every pair of points $c<d$ in $X$, the subgroup $\langle G_{(a, c)},G_{(d, b)}\rangle $ projects onto the largest quotient $G/[G_c, G_c]$.
\end{lem}
\begin{proof}
Given $\gamma\in \Germ(G, a)$ the assumption implies that there exists $g\in G$ with $\mathcal{G}_a(g)=\gamma$ and $\mathcal{G}_b(g)=\operatorname{\mathsf{id}}$. Thus, we have $g\in G_{(a, x)}$ for some $x\in X$. If we choose $h\in G_c$ such that $h(x)<c$ the element $g'=hgh^{-1}$ belongs to $G_{(a, c)}$ and satisfies $\mathcal{G}_a(g')=\gamma$. We conclude that the group $G_{(a, c)}$ projects onto $\Germ(G, a)$ and similarly $G_{(d, b)}$ projects onto $\Germ(G, b)$. Therefore the subgroup $\langle G_{(a, c)}, G_{(d, b)}\rangle$ projects onto $\Germ(G, a)\times \Germ(G, b)\simeq G/G_c$. It is therefore enough to show that its image in $G/[G_c, G_c]$ contains $G_c/[G_c, G_c]$. This is the case because every $g\in G_c$ is conjugate inside $G_c$ to an element of $\langle G_{(a, c)}, G_{(d, b)}\rangle$, which has the same image in the abelianisation $G_c/[G_c, G_c]$. \qedhere
\end{proof}
\begin{proof}[Proof of Corollary \ref{c-lm-C1-interval}]
In the proof we set $N=[G_c, G_c]$. Let $Y=[0, 1]$ or $Y=\mathbb{R}$ according to the case in the statement. By Theorem \ref{t-lm-C1} the $\varphi$-action of $G$ on $Y$ is either semi-conjugate to the standard action on $X$ or to an action that factors through $G/N$. Assume that the second condition holds: after Theorem \ref{t-lm-C1}, we know more precisely that if $I$ is a connected component of $\suppphi(N)$, then the induced action of $N$ on $I$ is semi-conjugate to the standard action on $X$. In particular, it admits linked pairs of fixed points and by Theorem \ref{p.sacksteder} we can find $h\in N$ with a hyperbolic fixed point $\xi\in I$. We make a slightly more elaborate argument than the one that is needed for Corollary \ref{t-Bo-Fa}. Let $(c, d)\Subset X$ be such that $h\in N_{(c, d)}$. Then the subgroup $H=\langle G_{(a,c)},G_{(d,b)}\rangle$ centralizes $h$ and thus every point in the orbit $H.\xi$ is fixed by $h$, with derivative always equal to $\varphi(h)'(\xi)\neq 1$. As the derivative $\varphi(h)'$ is continuous, it must be that the orbit $H.\xi$ is discrete in $Y$. In the case $Y=[0,1]$, the only possibility is that $\xi$ is fixed by $\varphi(H)$, and thus after Lemma \ref{l-independent-germs}, the quotient action of $G/N$ would have a fixed point as well. Similarly, in the case $Y=\mathbb{R}$, $\varphi(H)$ cannot fix $\xi$, so that the orbit $H.\xi$ is infinite and discrete.
Using Lemma \ref{l-independent-germs} again, we deduce that the quotient action of $G/N$ has an infinite discrete orbit, which means that it is semi-conjugate to a cyclic action.
For the last statement, apply the case $Y=[0,1]$ to the action of $\ker \tau$.
\qedhere
\end{proof}
\subsection{An application to non-smoothability}\label{ssc:Stein}
In Theorem \ref{t-lm-C1} it may happen that the standard action of $G$ is not semi-conjugate to an action of a given regularity, so that the first possibility is not realizable for actions in that regularity. Here we discuss two applications of this to certain groups of piecewise linear homeomorphisms, which improve results on non-smoothability of such groups of Bonatti, Lodha and the fourth author \cite{BLT}.
\subsubsection{Thompson--Brown--Stein groups} Here we discuss an application to differentiable actions of Thompson--Brown--Stein groups $F_{n_1,\ldots,n_k}$ introduced in Definition \ref{d-Thompson-Stein}, which are natural generalizations of Thompson's group $F$. Such groups are clearly locally moving. It was shown in \cite{BLT} that when $k\geq 2$ the standard action of $F_{n_1,\ldots,n_k}$ cannot be conjugate to any $C^2$ action. Here we show the following.
\begin{thm}\label{t-Thompson-Stein}
Let $r>1$. For any $k\ge 2$ and choice of $n_1,\ldots,n_k$ as in Definition \ref{d-Thompson-Stein}, the Thompson--Brown--Stein group $F_{n_1,\ldots,n_k}$ admits no faithful $C^{r}$ action on the real line.
\end{thm}
We first need a lemma for the usual Higman--Thompson groups $F_n$. Corollary \ref{c-lm-C1-interval} applies to them, so every faithful action of $F_n$ of class $C^1$ is semi-conjugate to its standard action on the interval $[0,1]$. However, as pointed out by Ghys and Sergiescu \cite{GhysSergiescu}, the group $F$ (and every $F_n$) actually admits $C^1$ (even $C^\infty$) actions which are not conjugate to the standard action (see Remark \ref{r-F-C1}). In these examples, the action is obtained by blowing up the orbit of dyadic rationals.
The next lemma, which is a consequence of Sacksteder's theorem (Theorem \ref{p.sacksteder}) and the structure of the group, gives a restriction on possible semi-conjugate but not conjugate $C^1$ actions.
\begin{lem}\label{l-semic-F-C1}
For $n\ge 2$, let $\varphi:F_n\to \Diff_0^1(\mathbb{R})$ be a faithful action which is semi-conjugate to the standard action $\varphi_0:F_n\to \PL([0,1])$, but not conjugate. Let $h:\mathbb{R}\to (0,1)$ be the corresponding continuous monotone map such that $h\varphi=\varphi_0 h$. Then there exists a rational point $p\in [0,1]$ which is not $n$-adic (i.e.\ $p\in (\mathbb Q\setminus \mathbb Z[1/n])\cap [0,1]$), such that the preimage $\xi=h^{-1}(p)$ is a singleton, and an element $g\in F$ for which $\xi$ is a hyperbolic fixed point.
\end{lem}
\begin{proof}
The proof is a tricky refinement of arguments in \cite[\S 5.1]{BLT}.
%
In the following, we write $G=F_n$. Given an action $\varphi$ as in the statement, denote by $\Lambda\subset \mathbb{R}$ the corresponding minimal invariant Cantor set. Fix an open $n$-adic subinterval $I\Subset (0,1)$, then $\varphi(G_I)$ preserves the interval $h^{-1}(I)$ and $\Lambda_I:=\Lambda\cap h^{-1}(I)$ is the minimal invariant subset for the restriction $\varphi_I:G_I\to \Diff_0^1\left (h^{-1}(\overline I)\right )$ induced by $\varphi$.
After Proposition \ref{p.sacksteder} and subsequent Remark \ref{r.sacksteder} applied to $\varphi_I$, there exists an element $g\in G_I$ and a point $\xi\in \Lambda_I$ such that $g.\xi=\xi$ and $\varphi(g)'(\xi)<1$. Moreover, such a point $\xi$ cannot belong to the closure of a gap of $\Lambda_I$ (see Remark \ref{r.sacksteder}), or in other terms the semi-conjugacy $h$ must be injective at $\xi$.
%
It is well known that if a point $p\in I$ is an isolated fixed point for some element of $G_I$ in the standard action, then $p$ is rational (the point $p$ must satisfy a rational equation $n^kp+\frac{a}{n^b}=p$; see Lemma \ref{l:flt} below). From this we deduce that the point $p=h(\xi)$ is rational.
%
Moreover, the point $p$ cannot be $n$-adic: take any element $f\in G_I$ such that $\varphi_0(f)$ coincides with $\varphi_0(g)$ in restriction to $[p,1]$ and is the identity in restriction to $[0,p]$. Then the right derivative of $\varphi(f)$ at $\xi$ must be equal to $\varphi(g)'(\xi)<1$ and the left derivative of $\varphi(f)$ at $\xi$ must be equal to $1$, contradicting the fact that $\varphi$ is a $C^1$ action. This concludes the proof.
\end{proof}
As a consequence of Lemma \ref{l-semic-F-C1}, we get a strong improvement of \cite[Theorem 3.4]{BLT}, on regularity of actions of Thompson--Brown--Stein groups.
The idea is to replace the use of the Szekeres vector field (which requires $C^2$ regularity), with Sternberg's linearization theorem, which works in $C^{r}$ regularity ($r>1$), but requires hyperbolicity (granted from Lemma \ref{l-semic-F-C1}). In this form, these results can be found in \cite[Appendice 4]{Yoccoz} or \cite[Theorems 3.6.2 and 4.1.11]{Navas-book} (a detailed proof when $r<2$ appears in \cite[\S 6.2.1]{Triestino}). A similar approach, although less technical, appears in \cite{MannWolff} to exhibit examples of groups at ``critical regularity''.
\begin{thm}[Sternberg]\label{t-sternberg0}
Fix $r>1$ and let $f$ be a $C^r$ diffeomorphism of the half-open interval $[0,1)$ with no fixed point in $(0,1)$ and such that $f'(0)\neq 1$. Then there exists a diffeomorphism $h:[0,1)\to [0,+\infty)$ of class $C^r$ such that
\begin{enumerate}
\item $h'(0)=1$,
\item the conjugate map $hfh^{-1}$ is the scalar multiplication by $f'(0)$.
\end{enumerate}
\end{thm}
\begin{thm}[Szekeres]\label{t-sternberg}
Fix $r>1$ and let $f$ be a $C^r$ diffeomorphism of the half-open interval $[0,1)$ with no fixed point in $(0,1)$ and such that $f'(0)\neq 1$. Then there exists a unique $C^{r-1}$ vector field $\mathcal X$ on $[0,1)$ with no singularities on $(0,1)$ such that
\begin{enumerate}
\item $f$ is the time-1 map of the flow $\left \{\phi_{\mathcal X}^s\right \}$ generated by $\mathcal X$,
\item the flow $\left \{\phi_{\mathcal X}^s\right \}$ coincides with the $C^1$ centralizer of $f$ in $\Diff_0^1([0,1))$.
\end{enumerate}
\end{thm}
The following statement is the analogue of \cite[Proposition 7.2]{BLT}.
\begin{prop}\label{p-local-flow}
Take $a\in (0,1)$ and $r>1$. Assume that two homeomorphisms $f,g\in \homeo_{0}([0,1])$ satisfy the following properties.
\begin{enumerate}
\item The restrictions of $f$ and $g$ to $[0,a]$ are $C^2$ contractions, namely the restrictions are $C^2$ diffeomorphisms onto their images such that
\[
f(x)<x\quad\text{and}\quad g(x)<x\quad\text{for every }x\in (0,a].
\]
\item $f$ and $g$ commute in restriction to $[0,a]$, that is,
\[
fg(x)=gf(x)\quad\text{for every }x\in [0,a].
\]
\item The $C^2$ germs of $f$ and $g$ at $0$ generate an abelian free group of rank 2.
\end{enumerate}
Then, for every homeomorphism $\psi \in \homeo_{0}([0,1])$ such that:
\begin{enumerate}
\item $\psi f\psi^{-1}$ and $\psi g \psi^{-1}$ are $C^r$ in restriction to $[0,\psi(a)]$,
\item $(\psi f\psi^{-1})'(0)<1$,
\end{enumerate}
one has that the restriction of $\psi$ to $(0,a]$ is $C^r$.
\end{prop}
\begin{proof}[Sketch of proof]
The proof is basically the same as in \cite{BLT}, and we only give a sketch. As $f$ and $g$ are $C^2$ contractions near $0$ and commute, they can be simultaneously linearized by considering the Szekeres vector field $\mathcal X$ for $f$. As their germs generate a rank 2 abelian group, we can find a dense subset of times $A\subset \mathbb{R}$, such that for every $\alpha\in A$, there exists an element $h_\alpha\in \langle f,g\rangle$ such that the restriction of $h_\alpha$ to $[0,a]$ coincides with the time-$\alpha$ map of the flow $\phi_{\mathcal X}^\alpha$. Given a map $\psi$ as in the statement, we can also simultaneously linearize $\psi f\psi^{-1}$ and $\psi g \psi^{-1}$, using Sternberg's linearization theorem (Theorem \ref{t-sternberg0}). If $\mathcal Y$ is the corresponding vector field from Theorem \ref{t-sternberg}, we deduce that the restriction of $\psi$ to $(0,a]$ is $C^1$ and sends one vector field to the other: $\left (\psi_{[0,p]}\right )_*\mathcal X=\mathcal Y$. Writing this relation more explicitly, we get
\[
\psi'(x)=\frac{\mathcal Y(\psi(x))}{\mathcal X(x)}\quad\text{for every }x\in (0,a],
\]
whence we deduce that $\psi$ is $C^r$ in restriction to $(0,a]$.
\end{proof}
The next technical result is an adaptation of classical arguments in one-dimensional dynamics, which can be traced back to Hector and Ilyashenko (see specifically \cite[Proposition 3.5]{Navas-QS} and \cite[Lemma 3]{Ilyashenko}).
\begin{prop}\label{prop.stabilizer_dense}
For $r\in (1,2]$, let $f,g\in \Diff_0^r(\mathbb{R})$ be two diffeomorphisms fixing $0$ with the following properties:
\begin{enumerate}[label=(\arabic*)]
\item $f'(0)=\lambda<1$ and $g'(0)=\mu\ge 1$,
\item\label{i.ilyashenko} for every $(l,m)\in \mathbb N^2\setminus \{(0,0)\}$, there exists $\varepsilon>0$ such that $g^mf^l(x)\neq x$ for every $x\in (0,\varepsilon)$.
\end{enumerate}
Then there exists $\delta>0$ such that the $\langle f,g\rangle$-orbit of every point $x\in (0,\delta)$ is dense in $(0,\delta)$.
\end{prop}
\begin{proof}
By Sternberg's linearization theorem (Theorem \ref{t-sternberg0}), we can take a $C^r$ coordinate $h:U\to \mathbb{R}$ on a neighborhood $U$ of $0$ so that the map $f$ becomes the scalar multiplication by $\lambda$ on $\mathbb{R}$ (more precisely, we take as $U$ the maximal open interval containing no other fixed points for $f$). Write $V=h(U\cap g^{-1}(U))$, $\overline f=h f h^{-1}$ and $\overline g = h g h^{-1}\restriction_{V}$. Note that $\overline g'(0)=g'(0)=\mu$.
We first rule out the case where $\log \lambda$ and $\log \mu$ are rationally dependent. So take $(l,m)\in \mathbb N^2\setminus \{(0,0)\}$ such that $\lambda^l\mu^m=1$ and consider the composition $\gamma:=g^m\circ f^l$, which satisfies $\gamma'(0)=1$, and write $\overline \gamma= \overline g^m\circ \overline f^l$ for the corresponding map defined on an appropriate open subinterval $V'\subset V$ containing $0$.
Then, for every $x\in V'$ and $n\in \mathbb N$, we have $\overline f^{-n}\overline \gamma \overline f^{n}(x)=\frac{\overline \gamma(\lambda^nx)}{\lambda^n}$, from which we deduce that $\overline f^{-n}\overline \gamma \overline f^{n}\to \mathsf{id}$ uniformly on compact subsets of $V'$, as $n\to \infty$. Going back to the original coordinates, we get that there exists $\delta>0$ such that $f^{-n}\gamma f^{n}\to \mathsf{id}$ uniformly on $[0,\delta]$ as $n\to \infty$.
Take now $x\in (0,\delta)$ and let $K$ be the closure of the $\langle f,g\rangle$-orbit of $x$, which clearly contains the point $0$. Assume by contradiction that $K\cap [0,\delta]\neq [0,\delta]$, and let $I$ be a connected component of $[0,\delta]\setminus K$. By $\langle f,g\rangle$-invariance of $K$ and the established uniform convergence to the identity, there exists $n_0\in \mathbb N$ such that $f^{-n}\gamma f^{n}(I)=I$ for every $n\ge n_0$. This gives that the element $\gamma=g^m\circ f^l$ preserves all the intervals of the form $f^n(I)$, for $n\ge n_0$, and in particular it admits infinitely many fixed points accumulating on $0$, which contradicts the assumption \ref{i.ilyashenko}.
We assume next that $\log \lambda$ and $\log \mu$ are rationally independent, and in particular that $\mu>1$. Applying Sternberg's linearization theorem again, take a $C^r$ coordinate $k:W\to \mathbb{R}$ on a neighborhood $W$ of $0$ so that the map $\overline g$ becomes the scalar multiplication by $\mu$ on $\mathbb{R}$; more precisely, upon exchanging the roles of $f$ and $g$, we take as $W$ the maximal open interval containing no other fixed points for $\overline g$, and assume that $W\subset U$. Note that after Theorem \ref{t-sternberg0}, we can take $k$ such that $k'(0)=1$, so that $k(x)=x+O(x^r)$ as $x\to 0$.
Given $\nu >0$, there exist two increasing sequences $(l_n)_{n\in \mathbb N},(m_n)_{n\in \mathbb N}\subset \mathbb N$, such that $\lambda^{l_n}\mu^{m_n}\to \nu$ as $n\to \infty$. For $n\in \mathbb N$, the composition $g_n=\overline g^{m_n} \circ \overline f^{l_n}$ is defined on $W$, as $\overline f$ contracts $W$ and $\overline g$ preserves it.
Fix $x\in W$, so that
\begin{align*}
g_n(x)&=k^{-1}\left (\mu^{m_n}k(\lambda^{l_n}x)\right )
=
k^{-1}\left (\mu^{m_n} (\lambda^{l_n}x+O(\lambda^{rl_n}x^r))\right )\\
&=k^{-1}\left (\mu^{m_n} \lambda^{l_n}x+O(\lambda^{(r-1)l_n}x^r)\right )\quad\text{as }n\to \infty.\end{align*}
We deduce the convergence $g_n(x)\to k^{-1}(\nu x)$ as $n\to \infty$. As $\nu>0$ was arbitrary, this gives that the orbit of every $x\in W\cap (0,+\infty)$ is dense in $W\cap (0,+\infty)$, as desired.\qedhere
\end{proof}
Finally, we also need a basic fact.
\begin{lem}\label{l:flt}
Let $\ell \in \mathbb N$ be an integer, and $p\in \mathbb Q$ any rational. Then there exists a non-trivial $\ell$-adic affine map $g\in\Aff(\mathbb Z[1/\ell],\langle \ell\rangle_*)\subseteq \Aff(\mathbb{R})$ such that $g(p)=p$.
\end{lem}
\begin{proof}
Write $g(x)=\ell^kx+\frac{a}{\ell^b}$, with $a\in\mathbb Z$ and $b,k\in\mathbb N$, for a generic $\ell$-adic affine map.
Note that the condition $g(p)=\ell^kp+\frac{a}{\ell^b}=p$ gives $p=\frac{a}{\ell^b(\ell^k-1)}$, which can be any rational number (choosing appropriate $a\in\mathbb Z$ and $b,k\in\mathbb N$).
\end{proof}
We can now prove the main result of this section.
\begin{proof}[Proof of Theorem \ref{t-Thompson-Stein}]
We argue by way of contradiction. Write $G=F_{n_1,\ldots,n_k}$ and let $\varphi:G\to \Diff_0^r(\mathbb{R})$ be a faithful action ($r>1$) without fixed points. After Corollary \ref{c-lm-C1-interval}, $\varphi$ is either semi-conjugate to the standard action on $X$, or to a cyclic action. Assume first that the former occurs and write $h:\mathbb{R}\to X=(0,1)$ for the semi-conjugacy. Using Lemma \ref{l-semic-F-C1} applied to the action of $F_{n_1}\subseteq G$, we find a rational point $p\in X$ and an element $f\in F_{n_1}$, such that $\xi=h^{-1}(p)$ is a hyperbolic fixed point for $\varphi(f)$. Using Lemma \ref{l:flt}, we can find an element $g\in F_{n_2}$ for which $p\in X$ is an isolated fixed point. In particular, $f$ and $g$ commute in restriction to a right neighborhood $[p,q]$ of $p$, and their right germs at $p$ generate an abelian free group of rank 2 (they are scalar multiplications by powers of $n_1$ and $n_2$ respectively).
%
Thus, up to considering inverse powers, the assumptions of Proposition \ref{prop.stabilizer_dense} are satisfied by the maps $\varphi(f)$ and $\varphi(g)$, from which we deduce that
the action of $\langle \varphi(f),\varphi(g)\rangle$ is minimal in restriction to an interval of the form $(\xi,\xi+\delta)$, with $\delta>0$. Hence, the semi-conjugacy $h:\mathbb{R}\to X$ considered above is a conjugacy (that is, $h$ is a homeomorphism).
On the other hand, up to considering inverse powers, the elements $f$ and $g$ satisfy the assumptions of Proposition \ref{p-local-flow}.
We deduce that $h$ is $C^r$ in restriction to $[p,q]$.
We conclude as in \cite[Proof of Theorem 7.3]{BLT}. Take an element $\gamma \in G$ with a discontinuity point $r\in [p,q]$ for its derivative; then also the derivative $\varphi(\gamma)'$ has a discontinuity point at $h^{-1}(r)$. This gives the desired contradiction.
In the case of cyclic action, considering the corresponding homomorphism $\tau:G\to \mathbb Z$, we have after Corollary \ref{c-lm-C1-interval} that $\ker \tau$ acts on every connected component of its support semi-conjugate to the standard action. Thus one can reproduce the previous argument, adapted to $\ker \tau$.
This is a little tricky, as the abelianization $F_{n_i}/[F_{n_i},F_{n_i}]\cong \mathbb Z^{n_i}$ is larger than the quotient $F_{n_i}/(F_{n_i})_c\cong \mathbb Z^2$. Start with an element $f\in F_{n_1}\cap \ker \tau$ with a hyperbolic fixed point $\xi$, as in the previous case, and then choose $g_1\in (F_{n_2})_c$ fixing $p=h^{-1}(\xi)$ playing the role of $g$ in the previous case.
However, it could be that $g_1\notin \ker \tau$, so for this, take an element $g_2\in G$ such that $g_2\left (\supp(g_1)\right )\cap \supp(g_1)=\varnothing$. Then the commutator $g=[g_1,g_2]$ coincides with $g_1$ on $\supp(g_1)$, and belongs to $\ker \tau$.
\end{proof}
The method presented here cannot be improved further to go down to exclude $C^1$ smoothability. We believe however this can be achieved with a different approach. Let us point out that all known examples of $C^1$ smoothable groups of PL homeomorphisms embed in Thompson's $F$. It would be tempting to conjecture that this is also a necessary condition. However, we estimate that little is known about other groups of PL homeomorphisms, such as those defined by irrational slopes (see however the very recent work \cite{hyde2021subgroups}, where it is proved that several such groups do not embed in $F$). So, let us highlight the following concrete problem.
\begin{ques}
Fix an irrational $\tau\in \mathbb{R}\setminus \mathbb Q$, and write $\Lambda=\langle \tau\rangle_*$ and $A=\mathbb Z[\Lambda]$ (as an explicit case, one can take the golden ratio $\tau=\frac{\sqrt{5}-1}{2}$). Consider the irrational slope Thompson's group $F_\tau=G([0,1];A,\Lambda)$. Is the action of $F_\tau$ on the interval $C^1$ smoothable?
\end{ques}
\subsubsection{An application to Bieri--Strebel groups on the line}
Given a real number $\lambda>1$, we consider the Bieri--Strebel groups acting on the line $G(\lambda)=G(\mathbb{R};\mathbb Z[\lambda,\lambda^{-1}],\langle \lambda\rangle_*)$. It was remarked in \cite{BLT} that the standard action of $G(\lambda)$ cannot be conjugate to any $ C^1$ action. One of the main results of \cite{BLT} states that for certain choices of $\lambda$ the group $G(\lambda)$ does not admit any faithful $C^1$ action on the line. Here we generalize this result by removing all restrictions on $\lambda$.
\begin{cor}
For $\lambda>1$, there is no faithful $C^1$ action of the Bieri--Strebel group $G(\lambda)$ on the closed interval.
\end{cor}
\begin{proof}
Indeed, it is proved in \cite[Theorem 6.10]{BLT} that the standard action of $G(\lambda)$ on the line cannot be conjugate to any $C^1$ action on the closed interval, but a closer look at the proof (notably using the results from \cite[\S 4.2]{BMNR}) shows that even a semi-conjugacy is impossible, as the action of the affine subgroup of $G(\lambda)$ must be minimal. Hence the result follows from Theorem \ref{t-lm-C1}.
\end{proof}
In Section \ref{s-few-actions} we will classify $C^0$ actions of $G(\lambda)$ up to semi-conjugacy whenever $\lambda$ is algebraic.
\section{Finitely generated locally moving groups with few exotic actions}\label{s-few-actions}
\subsection{Bieri--Strebel groups over the real line}\label{ssc:BieriStrebel} In this section we study actions of Bieri--Strebel groups $G(X; A,\Lambda)$ in the case $X=\mathbb{R}$ (see Definition \ref{d.BieriStrebel}). These are close relatives of Thompson's group $F$, yet their minimal $\mathbb{R}$-exotic actions turn out to be much more rigid (and, in some cases, there are only finitely many such actions).
Here we denote by $\langle S \rangle_*$ the multiplicative group generated by a subset $S\subset \mathbb{R}_+^*$.
The results of \cite[\S\S A--B]{BieriStrebel} allow to characterize under which conditions the group $G(\mathbb{R}; A, \Lambda)$ is in the class $\mathcal F${}. (Note that since they all contain translation, this is the same as asking whether they belong to $\mathcal{F}_0${}.) As for many properties of the groups $G(X; A, \Lambda)$, this depends on the $\mathbb Z[\Lambda]$-submodule
\[I\Lambda\cdot A:=\langle (\lambda-1)a\colon\lambda\in \Lambda,a\in A\rangle.\]
Indeed, we have the following.
\begin{lem}\label{p.BieriStrebel_fg_germs}
The Bieri--Strebel group $G(\mathbb{R}; A, \Lambda)$ is in the class $\mathcal F${}\;if and only if the following conditions are satisfied:
\begin{enumerate}[label=(BS\arabic*)]
\item\label{i:LambdaBS} $\Lambda$ is finitely generated as a group,
\item\label{i:ABS} $A$ is finitely generated as $\mathbb Z[\Lambda]$-module,
\item\label{i:quotientBS} the quotient $A/I\Lambda\cdot A$ is finite.
\end{enumerate}
\end{lem}
\begin{proof}
Conditions \ref{i:LambdaBS}--\ref{i:ABS} characterize when the group $G(\mathbb{R}; A, \Lambda)$ is finitely generated \cite[Theorem B7.1]{BieriStrebel}. On the other hand, Bieri--Strebel groups of the form $G((-\infty, x]; A, \Lambda)$ and $G([x,+\infty); A, \Lambda)$ are finitely generated if and only if $x\in A$ and all three conditions \ref{i:LambdaBS}--\ref{i:quotientBS} are satisfied (this is the statement of \cite[Theorem B8.2]{BieriStrebel}); this immediately implies that $G(\mathbb{R}; A, \Lambda)$ is in the class $\mathcal F${}. For the converse, recall that two points $p,q\in A\cap (-\infty, x)$ are in the same orbit under the group $G((-\infty, x]; A, \Lambda)$ if and only if $p-q\in I\Lambda\cdot A$ \cite[Corollary A.5.1]{BieriStrebel}. If $G$ is in the class $\mathcal F${}\;there exists a finitely generated subgroup $H\subseteq G((-\infty, x]; A, \Lambda)$ which contains a subgroup of the form $G((-\infty, y]; A, \Lambda)$ with $y<x$. Since all points of $A\cap (-\infty, y)$ occur as breakpoints for elements in $G((-\infty, y]; A, \Lambda)$, they must all belong to the $H$-orbit of one of the finitely many breakpoints of a finite generating set of $H$ (see \cite[\S B6]{BieriStrebel}). Thus condition \ref{i:quotientBS} is also necessary. \qedhere
\end{proof}
As an important special case, the reader can have in mind the following example.
\begin{ex} \label{e-G-lambda}
For $\lambda>1$, we denote by $G(\lambda)$ the Bieri--Strebel group $G(\lambda):=G(\mathbb{R};A,\Lambda)$ corresponding to the cyclic group $\Lambda=\langle \lambda\rangle_*$ and to $A:=\mathbb Z[\lambda,\lambda^{-1}]$.
One has $I\Lambda\cdot A=(\lambda-1)\, \mathbb Z[\lambda,\lambda^{-1}]$. It is not difficult to see that the quotient $A/I\Lambda\cdot A$ is finite if and only if $\lambda$ is algebraic (see \cite[Illustration A4.3]{BieriStrebel}).
For instance, for $\lambda=p/q$ rational (with $p$ and $q$ coprime), one has $|A/I\Lambda\cdot A|=p-q$.
Therefore, by Lemma \ref{p.BieriStrebel_fg_germs}, the group $G(\lambda)$ is in the class $\mathcal F${}\;exactly for algebraic $\lambda$.
\end{ex}
Thus if $G(\mathbb{R}; A, \Lambda)$ satisfies conditions \ref{i:LambdaBS}--\ref{i:quotientBS}, its actions on the real line satisfy Theorem \ref{t-C-trichotomy}, so that all its exotic actions are semi-conjugate to a minimal $\mathbb{R}$-focal action horograded by the standard action on $\mathbb{R}$. Our goal is to give an explicit description of all such minimal $\mathbb{R}$-focal actions, thus yielding a complete classification of the faithful actions of such groups up to semi-conjugacy. For this, recall that in \S \ref{ss.Bieri-Strebelfocal} we gave a construction of actions of Bieri--Strebel groups $G(X; A, \Lambda)$ parameterized by preorders on the group $\Lambda$. In what follows, we will denote by $\varphi_{+, \leq_\Lambda}$ (respectively, $\varphi_{-, \leq_\Lambda}$) the dynamical realization of the right (respectively, left) jump preorder associated with $\leq_\Lambda\in\LPO(\Lambda)$ (see Definition \ref{dfn.cocyclepreorder}). The main result is that these are the only exotic actions of $G$.
\begin{thm} \label{t-BBS}
Let $G=G(\mathbb{R};A,\Lambda)$ be a Bieri--Strebel group satisfying conditions \ref{i:LambdaBS}--\ref{i:quotientBS}. Then every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ without fixed points, is semi-conjugate to an action in one of the following families.
\begin{enumerate}
\item \emph{(Non-faithful)} An action induced from the quotient $G/[G_c,G_c]$.
\item \emph{(Standard)} The standard piecewise linear action of $G$ on $\mathbb{R}$.
\item \emph{($\mathbb{R}$-focal)} An action of the form $\varphi_{\pm, \leq_\Lambda}$, obtained as the dynamical realization of a jump preorder (see Definition \ref{dfn.cocyclepreorder}). \end{enumerate}
\end{thm}
In particular, this provides a classification of all minimal faithful actions of $G$ up to conjugacy.
\begin{cor}
Under the assumptions of Theorem \ref{t-BBS},
every minimal faithful action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is topologically conjugate either to the standard action on $\mathbb{R}$ or to the dynamical realization of a jump preorder.
\end{cor}
The following special case gives Theorem \ref{t-intro-Glambda} from the introduction.
\begin{ex}
\label{ex:focal_G_lambda}
Let us consider the special case $G=G(\lambda)$ as in Example \ref{e-G-lambda}, with $\lambda>1$ algebraic. Since in this case the group $\Lambda$ is infinite cyclic, it admits only two non-trivial preorders, namely the usual order $<_\Lambda$ and its opposite. Thus, the jump preorder construction gives exactly two actions $\varphi_+:=\varphi_{+, <_\Lambda}$ and $\varphi_-:=\varphi_{-, <_\Lambda}$, which are the dynamical realizations of the right and left jump preorders associated with $<_\Lambda\in\LPO(\Lambda)$. Note indeed, that after Remark \ref{r.opposite}, the dynamical realization of the jump preorder corresponding to $<_\Lambda^{op}$ is conjugate to that for $<_\Lambda$.
Thus in this case the group $G$ admits finitely many (more precisely, two) $\mathbb{R}$-focal actions.
Note that such actions $\varphi_\pm$ are both locally rigid, which shows that groups in the class $\mathcal{F}_0${}\;may admit locally rigid actions other than the standard action.
\end{ex}
\begin{rem}\label{r.quotients_BieriStrebel}
In the setting of Theorem \ref{t-BBS}, let us comment on the actions coming from the largest quotient $G/[G_c,G_c]$. The structure of this quotient highly depends on $A$ and $\Lambda$. Note that $G/G_c$ embeds in the product $\Aff(A,\Lambda)\times \Aff(A,\Lambda)$ via the product $\mathcal G_{-\infty}\times\mathcal G_{+\infty}$ of the germ homomorphisms, and thus $G/[G_c,G_c]$ is solvable of derived length at most 3. More precisely (see \cite[Corollary A5.5]{BieriStrebel}), writing $\tau_\pm(f)\in A$ for the translation part of the germ of $f\in G$ at $\pm\infty$, respectively, the image $(\mathcal G_{-\infty}\times\mathcal G_{+\infty})(G)$ in $\Aff(A,\Lambda)\times \Aff(A,\Lambda)$ is the subgroup
\[
\left\{(f,g)\in \Aff(A,\Lambda)\colon \tau_+(g)-\tau_-(f)\in I\Lambda\cdot A \right\}.
\]
The investigation of the abelianization $G_c/[G_c,G_c]$ is much less understood; partial results are discussed in \cite[\S C12]{BieriStrebel}. If $G_c/[G_c,G_c]$ is trivial, then one must have $I\Lambda\cdot A=A$ \cite[Theorem C12.14]{BieriStrebel}, but the converse is not true in general.
In the case $G=G(\lambda)$ for $\lambda>1$ algebraic (Example \ref{e-G-lambda}) one can check that the image of $G/G_c$ under the map induced by $\mathcal G_{-\infty}\times\mathcal G_{+\infty}$ has finite index in $\Aff(A,\Lambda)\times \Aff(A,\Lambda)$ (see Example \ref{e-G-lambda}).
Moreover if $\lambda=p/q$ is rational (with $p$ and $q$ coprime), the abelianization $G_c/[G_c,G_c]$ is free of rank $p-q-1$ (see \cite[Corollary C12.12]{BieriStrebel}). For other algebraic values of $\lambda$, up to our knowledge, only the case $\lambda=\frac{\sqrt{5}+1}{2}$ has been studied in detail: after work of Burillo, Nucinkis, and Reeves \cite{burillo2020irrationalslope}, we know that $G_c/[G_c,G_c]\cong\mathbb Z_2$ (note that in this case $I\Lambda\cdot A=A$).
Note also that for any algebraic $\lambda>1$, the abelianization $G/[G,G]$ is free of rank $|A/I\Lambda\cdot A|+1$ (see \cite[Corollary C12.2]{BieriStrebel}).
The situation is particularly simple in the case $\lambda=2$, where we have
\[G/[G_c,G_c]\cong \Aff(\mathbb Z[1/2],\langle 2\rangle_*)\times \Aff(\mathbb Z[1/2],\langle 2\rangle_*)\cong \BS(1,2)\times \BS(1,2),\]
where $\BS(1,2)=\langle a,b\mid aba^{-1}=b^2\rangle$ is the solvable Baumslag--Solitar group (see Proposition \ref{p-BSxBS} below).
\end{rem}
Before getting to the proof of Theorem \ref{t-BBS}, we need further preliminary discussions. Recall that the group $G$ contains the affine subgroup $\Aff(A,\Lambda)\cong A\rtimes \Lambda$ of transformations of the form $x\mapsto \lambda x+a$, with $\lambda\in \Lambda$ and $a\in A$.
Given $a\in A$ and $\lambda\in \Lambda$, we will denote by $g(a,\lambda)$ the affine transformation $x\mapsto \lambda x+(1-\lambda)a$, which is the unique element of $\Aff(A,\Lambda)$ which fixes $a$ and has slope $\lambda$. We will also consider the elements
\begin{equation}\label{eq BBS}
g_+(a,\lambda):x\mapsto \left\{\begin{array}{lr}x & x\in (-\infty,a],\\[.5em]
g(a, \lambda)(x) & x\in [a,+\infty), \end{array} \right.
\end{equation}
and $g_-(a,\lambda):=g(a,\lambda)\,g_+(a,\lambda)^{-1}$. Note that $g_+(a,\lambda)\in G_{(a,+\infty)}$ and $g_-(a,\lambda)\in G_{(-\infty,a)}$. For $a\in A$ we also denote by $t_a$ the translation $x\mapsto x+a$.
For every $a\in A$ and $\lambda\in \Lambda$, if $h\in G(\mathbb{R};A,\Lambda)$ is an element with no breakpoint on $(a,+\infty)$ (respectively on $(-\infty, a)$), we have \[hg_+(a,\lambda)h^{-1}=g_+(h(a),\lambda)\quad\text{(respectively, }hg_-(a,\lambda)h^{-1}=g_-(h(a),\lambda)).\]
In particular we have the following relations for such elements (see \cite[\S B7]{BieriStrebel}):
\begin{equation}
\label{e-bx-equivariance1} h \,g_\pm(a,\lambda)\, h^{-1}=g_\pm(h(a),\lambda) \quad\text{for every } h \in \Aff(A,\Lambda),
\end{equation}
as well as
\begin{equation*}
g_+(a,\lambda)\,g_+(b,\mu)\,g_+(a,\lambda)^{-1}=g_+(g(a,\lambda)(b),\mu) \quad\text{for every } a>b,
\end{equation*}
\begin{equation}
\label{e-bx-equivariance2}
g_-(a,\lambda)\,g_-(b,\mu)\,g_-(a,\lambda)^{-1}=g_-(g(a,\lambda)(b),\mu) \quad\text{for every } a<b.
\end{equation}
We also remark that the subset
\[
\left\{t_a\right\}_{a\in A}\cup \left \{g(0,\lambda),g_+(0,\lambda)\right \}_{\lambda\in \Lambda}
\]
is generating for $G(\mathbb{R}; A, \Lambda)$ (see \cite[Theorem B7.1]{BieriStrebel}).
For what follows, the reader can keep in mind the following example.
\begin{ex}
For $\lambda>1$, the Bieri--Strebel group $G(\lambda)$ is generated by the finite subset $\{ g(0,\lambda),g_+(0,\lambda), t_1\}$.
\begin{rem}
The group $\left \langle g(0,\lambda),g_+(0,\lambda), t_1\right \rangle$ appears in \cite{BLT} (denoted $G_\lambda$), where it was shown that, for certain algebraic numbers $\lambda>1$ (called Galois hyperbolic \textit{ibid.}) it admits no faithful $C^1$ action on the closed interval. The fact that this group coincides with the Bieri--Strebel group $ G(\lambda)$ was unnoticed in \cite{BLT}.
\end{rem}
\end{ex}
We also need some preliminary results, stated in the following more flexible setting.
\begin{assumption}\label{ass.BieriStrebel}
Fix a non-trivial multiplicative subgroup $\Delta\subseteq \mathbb{R}_*$, and a $\Delta$-submodule $A\subseteq \mathbb{R}$, and let $H=G(\mathbb{R}; A, \Delta)$ be the corresponding Bieri--Strebel group. Moreover we let $G \subseteq\homeo_0(\mathbb{R})$ be a subgroup in the class $\mathcal F${}\;which contains $H$ as a subgroup. Finally we assume that $\mathcal{L}=\{L_x\}_{x\in \mathbb{R}}$ is a family of subgroups of $G$ with the following properties.
\begin{enumerate}[label=(C\arabic*)]
\item \label{i-L-nested} For each $x\in\mathbb{R}$ we have $\bigcup_{y<x} G_{(-\infty, y)} \subseteq L_x \subseteq G_{(-\infty, x)}$.
\item \label{i-L-equivariant} For every $x\in \mathbb{R}$ and every $g\in G$ we have $gL_xg^{-1}=L_{g(x)}$.
\item \label{i-L-contains-broken} For every $x\in A$ and every $\delta\in \Delta$ we have $g_-(x, \delta)\in L_x$.
\end{enumerate}
We write $L_+=\bigcup_{x\in\mathbb{R}} L_x$ which, by \ref{i-L-equivariant}, defines a normal subgroup of $G$.
Finally we assume that $\varphi\colon G\to \homeo_0(\mathbb{R})$ is a faithful minimal action of $G$ which is $\mathbb{R}$-focal and increasingly horograded by its standard action on $\mathbb{R}$. \end{assumption}
\begin{rem}
For the proof of Theorem \ref{t-BBS}, the reader can have in mind the case where $G=G(\mathbb{R}; A, \Lambda)$ is itself a Bieri--Strebel group, $H$ is a subgroup corresponding to some $\Delta\subseteq \Lambda$, and the $L_x$ are subgroups of $G_{(-\infty, x)}$ consisting of elements whose left-derivative at $x$ belongs to some intermediate group $\Delta\subseteq \Lambda_1\subseteq \Lambda$. However a different choice of $G$ will be used later in \S \ref{s-no-actions}.
\end{rem}
We will write $X=\mathbb{R}$ for the real line on which the standard action of $G$ is defined. We resume notation of \S \ref{subs.construction}. As introduced in Definition \ref{d-I-phi}, for every $x\in X$ and $\xi\in \mathbb{R}$ with $\xi\notin \operatorname{\mathsf{Fix}}^\varphi\left (G_{(-\infty,x)}\right )$, we write $\Iphi(x,\xi)$ for the connected component of $\suppphi\left (G_{(-\infty, x)}\right )$ containing $\xi$. Recall that intervals of this form define a CF-cover. Moreover we will consider the planar directed tree $(\mathbb T, \triangleleft, \prec)$ constructed in \S \ref{subs.construction}, whose dynamical realization is $\varphi$. Recall that vertices of $\mathbb T$ are intervals of the form $\Iphiout(x, \xi)=\Int \left (\bigcap_{y>x} \Iphi(y, \xi)\right )$ and that the map $\pi\colon \mathbb T\to \mathbb{R}$ given by $\pi\left (\Iphiout(x, \xi)\right )=x$ is an equivariant horograding. Recall also that we write $\Iphiinn(y, \xi)=\bigcup_{x<y}\Iphi(x, \xi)$.
\begin{lem} \label{l-bs-fix}
Under Assumption \ref{ass.BieriStrebel}, the group $\varphi\left (\Aff(A,\Delta)\right )$ has a unique fixed point $\eta\in \mathbb{R}$. This point satisfies $\eta\in \suppphi\left (G_{(-\infty, x)}\right )$ for every $x\in X$.
Moreover, the map
\[
\dfcn{q_+}{\mathbb{R}}{(\eta,+\infty)}{x}{\sup \Iphi(x,\eta)}\] is monotone increasing and $\Aff(A,\Delta)$-equivariant. In particular the standard affine action of $\Aff(A,\Delta)$ on $X=\mathbb{R}$ is positively semi-conjugate to its action $\varphi$ on $(\eta, +\infty)$.
Similarly the map \[
\dfcn{q_-}{\mathbb{R}}{(-\infty,\eta)}{x}{\inf \Iphi(x,\eta)}\] defines a negative semi-conjugacy.
\end{lem}
\begin{proof}
Every translation $t_a$, with $a\in A$, acts without fixed points, therefore by Proposition \ref{prop.dynamic-class-horo}, its $\varphi$-image is a homothety. As the subgroup of translations is abelian, its $\varphi$-image has a unique fixed point $\eta\in \mathbb{R}$. Moreover, as such subgroup is normal in $\Aff(A,\Delta)$, this is a fixed point for $\varphi\left (\Aff(A,\Delta)\right )$.
Assume first that there exists $x\in X$ such that $\eta\in\operatorname{\mathsf{Fix}}^\varphi\left (G_{(-\infty,x)}\right )$. Then we would have $\eta=g.\eta\in \operatorname{\mathsf{Fix}}^\varphi\left (G_{(-\infty,x)}\right )$ for every $g\in \Aff(A,\Delta)$, so that by minimality of the standard action of $\Aff(A,\Delta)$ on $X$ we get $\eta\in \operatorname{\mathsf{Fix}}^\varphi\left (G_+\right )$. Since $G_+$ is a normal subgroup of $G$, and $\varphi$ is minimal, this implies that $\varphi\left (G_+\right )$ acts trivially, contradicting faithfulness of $\varphi$ in Assumption \ref{ass.BieriStrebel}. Thus $\eta\in \suppphi\left (G_{(-\infty,x)}\right )$, and the interval $\Iphi(x,\eta)$ is well-defined for every $x\in X$.
The second statement is now a direct consequence of the properties of these two families of intervals: the family of $\{\Iphi(x,\eta)\}_{x\in X}$ is increasing with respect to $x\in X$ and moreover one has the equivariance relation $g.\Iphi(x,\eta)=\Iphi(g(x),g.\eta)$ for every $x\in X$ and $g\in G$ (see \S \ref{ssc.CF_family}). \qedhere
\end{proof}
In what follows we will always denote by $\eta$ the unique fixed point of $\varphi\left (\Aff(A,\Lambda)\right )$ provided by Lemma \ref{l-bs-fix}. For $x\in X$ and $\xi\in \suppphi(L_x)$ we will denote by $\Iphi(\mathcal{L}, x, \xi)$ the connected component of $\suppphi(L_x)$ containing $\xi$. Note that condition \ref{i-L-nested} implies that $\Iphi(\mathcal{L}, x, \xi)$ is increasing with respect to $x\in\mathbb{R}$, and moreover
\[\Iphiinn(x, \xi)\subset \Iphi(\mathcal{L},x, \xi)\subset \Iphi(x, \xi).\]
The key point is to establish the following strict inclusion when $x\in A$.
\begin{lem}\label{l-bbs-lex} Under Assumption \ref{ass.BieriStrebel}, assume that there exists $g\in H$ such that $g.\eta\neq \eta$. Then for every $x\in A$ we have a strict inclusion $\Iphiinn(x, \eta)\subsetneq \Iphi(\mathcal{L}, x, \eta)$.
\end{lem}
\begin{proof}
Assume by contradiction that $\Iphiinn(x, \eta)=\Iphi(\mathcal{L}, x, \eta)$ for some $x\in A$. Note that then this is automatically true for every $x\in A$, since the group $\Aff(A, \Delta)$ acts transitively on $A$ (it contains all translations by elements in $A$) and fixes $\eta$, so that for $h\in \Aff(A, \Delta)$ we have $h.\Iphiinn(x, \eta)=\Iphiinn(h(x), \eta)$ and
\begin{equation}\label{eq:L-equivariance}h.\Iphi(\mathcal{L}, x, \eta)=\Iphi(\mathcal{L}, h(x), \eta)\end{equation}
(after condition \ref{i-L-equivariant}).
Fix $a\in A$ and choose $\delta\in \Delta$ with $\delta>1$ such that the element $g_-(a, \delta).\eta\neq \eta$. Such a $\delta$ exists because the elements $g_-(a, \delta)$ together with $\Aff(A, \Delta)$ generate $H$, and we assume that $\varphi(H)$ does not fix $\eta$. Note also that once such an element $g_-(a, \delta)$ is found, it follows that $g_-(x, \delta).\eta\neq \eta$ for every $x\in A$, since these elements are all conjugate to each other by elements of $\Aff(A, \Lambda)$.
Note that by condition \ref{i-L-contains-broken} the image $\varphi(g_-(x, \delta))$ must preserve $\Iphi(\mathcal{L}, x, \eta)$ for every $x\in A$. We claim that it acts on it as an expanding homothety. To this end, we look at the action of $H$ on the planar directed tree $(\mathbb T, \triangleleft, \prec)$ constructed in \S \ref{subs.construction}. Consider the subset $\mathbb T_0$ of $\mathbb T$ given by
\[\mathbb T_0=\left \{\Iphiout(y, \xi)\colon y<x, \xi \in \Iphi(\mathcal{L}, x, \eta)\right \}.\] Then the equality $\Iphiinn(x, \eta)=\Iphi(\mathcal{L}, x, \eta)$ implies that $\mathbb T_0$ is a directed planar subtree of $\mathbb T$ invariant under the subgroup $L_x$ (note that $\mathbb T_0$ is equal to the direction below the vertex $v=\Iphiout(x, \eta)$ corresponding to $\Iphiinn(x, \eta)$). Moreover the restriction of the horograding $\pi\colon \mathbb T \to \mathbb{R}$ to $\mathbb T_0$ takes values in $(-\infty, x)$ and is $L_x$-equivariant. Finally since the intervals $\Iphiout(y, \eta)$ for $y<x$ are relatively compact inside $\Iphiinn(x, \eta)$ and every $I\in \mathbb T_0$ is contained in one such interval, the restriction of the planar order on $\mathbb T_0$ is proper. Thus we can apply Proposition \ref{prop.dynclasselements0} to the action of $L_x$ on $\mathbb T_0$, and since the element $g_-(x, \delta)$ satisfies $g_-(x, \delta)(y)>y$ for every $y\in(-\infty, x)$, it follows that it has no fixed points in $\mathbb T_0$ and thus acts on $(\partial \mathbb T_0, \prec)$ as an expanding homothety, and we conclude as in the proof of Proposition \ref{prop.dynamic-class-horo} that its $\varphi$-image is an expanding homothety on $\Iphi(\mathcal{L}, x, \eta)$, as desired.
Now for $x\in A$ let us denote by $\xi_x\in \Iphi(\mathcal{L}, x, \eta)$ the unique fixed point of $\varphi(g_-(x, \delta))\restriction_{\Iphi(\mathcal{L}, x, \eta)}$. Note that $\xi_x\neq \eta$, by the choice of $\delta$. Without loss of generality, we assume that $\xi_{b}>\eta$ for some $b\in A$. Then we have the following.
\begin{claim}
For every $x\in A$ we have $\xi_x>\eta$ and the map $ x\mapsto \xi_x$ is monotone increasing.
\end{claim}
\begin{proof}[Proof of claim]
The relations \eqref{e-bx-equivariance1} and \eqref{eq:L-equivariance} give that the map $x\mapsto \xi_x$ is $\Aff(A,\Delta)$-equivariant. The conclusion follows using that $\Aff(A, \Delta)$ acts transitively on $A$ and that, after Lemma \ref{l-bs-fix}, we know that the action of $\Aff(A,\Delta)$ on $(\eta, +\infty)$ is positively semi-conjugate to the standard affine action.
\end{proof}
Now fix $x\in X$. By the assumption that $\Iphiinn(x, \eta)=\Iphi(\mathcal{L}, x, \eta)$ and by \ref{i-L-nested} we can find $y\in A$ with $y<x$ such that $\Iphi(\mathcal{L}, y,\eta)$ contains $\xi_x$. After the claim, we have $\eta<\xi_y<\xi_x$. Since $\xi_x$ is a repelling fixed point for $\varphi(g_-(x,\delta))$, we have the inclusion $g_-(x,\delta).\Iphi(\mathcal{L}, y,\eta)\supset \Iphi(\mathcal{L}, y,\eta)$, and since the latter contains $\eta$ we have $g_-(x,\delta).\Iphi(\mathcal{L}, y,\eta)=\Iphi\left (\mathcal{L}, g_-(x,\delta)(y),\eta\right )$. Then \eqref{e-bx-equivariance2} implies that $g_-(x,\delta).\xi_y=\xi_{g_-(x,\delta)(y)}$. However, $g_-(x,\delta).\xi_y<\xi_y$ since $\xi_y$ lies to the left of $\xi_x$. On the other hand $\xi_{g_-(x,\lambda)(y)}>\xi_y$ since $g_-(x,\lambda)(y)>y$, and we know that the map $y\mapsto \xi_y$ is increasing after the claim. This is a contradiction, giving end to the proof of the lemma. \qedhere
\end{proof}
\begin{prop}\label{c-bs-L-semiconjugate}
Under the same assumptions as in Lemma \ref{l-bbs-lex}, for every $x\in A$ the action of $\varphi(L_{x})$ on $\Iphi(\mathcal{L}, x,\eta)$ is semi-conjugate to a non-faithful action induced from an action of the group of left germs $\Germ\left (L_{x}, x\right )$.
\end{prop}
\begin{proof}
Lemma \ref{l-bbs-lex} implies that the normal subgroup $\left (G_{(-\infty, x)}\right )_+=\bigcup_{y<x} G_{(-\infty, y)}$ of $L_x$ has fixed points in $\Iphi(\mathcal{L}, x, \eta)$ (namely the endpoints of $\Iphiinn(x, \eta)$). Thus the action of $\varphi(L_x)$ on $\Iphi(\mathcal{L}, x, \eta)$ is semi-conjugate to an action induced from the quotient $L_x/\left (G_{(-\infty, x)}\right )_+\cong \Germ\left (L_{x}, x\right )$. \qedhere
\end{proof}
The next result is the only place where a particular choice of the family $\mathcal{L}=\{L_x\}$ is needed.
\begin{cor}\label{cor.intmov} Let $G=G(\mathbb{R};A,\Lambda)$ be a Bieri--Strebel group satisfying conditions \ref{i:LambdaBS}--\ref{i:quotientBS}. Let $\varphi:G\to \homeo_0(\mathbb{R})$ be a minimal faithful $\mathbb{R}$-focal action, increasingly horograded by the standard action on $\mathbb{R}$. Assume there exists an element $g=g_{-}(x,\lambda)\in G$ such that $g.\eta\neq \eta$. Then $g.\Iphiinn(x, \eta)\cap\Iphiinn(x, \eta)=\emptyset$.
\end{cor}
\begin{proof} Take $g=g_{-}(x,\lambda)$ not fixing $\eta$ and consider the Bieri--Strebel group $H=G(\mathbb{R};A,\langle\lambda\rangle_\ast)$, which is a subgroup of $G$. We consider the family $\mathcal{L}=\{L_x:x\in X\}$ defined by
\[
L_x=\left \{ h\in G_{(-\infty,x)}\colon D^-h(x)\in \langle \lambda\rangle_\ast\right \}.
\]
It is straightforward to verify that Assumption \ref{ass.BieriStrebel} is fulfilled by such choices.
Therefore, by Proposition \ref{c-bs-L-semiconjugate}, the action of $L_x$ on $\Iphi(\mathcal{L}, x,\eta)$ is semi-conjugate to an action that factors through the germ homomorphism $\mathcal{G}_x:L_x\to\Germ\left (L_{x}, x\right )$. Since in this case $\Germ\left (L_{x}, x\right )$ is generated by $\mathcal{G}_x(g)$, we conclude that $\operatorname{\mathsf{Fix}}^\varphi(g)\cap \Iphi(\mathcal{L}, x,\eta)=\emptyset$.
On the other hand, by Lemma \ref{l-bbs-lex}, we get that $\Iphi(\mathcal{L}, x, \eta)$ strictly contains $\Iphiinn(x, \eta)$. Then, since $\{\Iphiinn(x, \xi):x,\xi\in\mathbb{R}\}$ is a CF-cover preserved by the action, we must have $g.\Iphiinn(x, \eta)\cap\Iphiinn(x, \eta)=\emptyset$ as desired.
\end{proof}
The next two lemmas analyse properties of the jump preorders. The first one gives decompositions for elements in $G_+= \bigcup_{x<\infty} G_{(-\infty, x)} $ which are well suited for our purposes. The second one allows to identify dynamical realizations of jump preorders.
\begin{lem}\label{lem.decomp} Let $G=G(\mathbb{R};A,\Lambda)$ be a Bieri--Strebel group and let $\leq_\Lambda$ be a preorder on $\Lambda$. Let $\preceq$ be the corresponding right jump preorder and take $g\in G_+$ with $\mathsf{id}\precneq g$. Then, there exist elements $h,k,g_{-}(y,\lambda)\in G_+$ such that \begin{enumerate}
\item\label{i.decomp1} $g=kg_{-}(y,\lambda)h$,
\item\label{i.decomp2} $h\in [1]_{\preceq}$,
\item\label{i.decomp3} $1\lneq_\Lambda \lambda$, and
\item\label{i.decomp4} $k\in G_{(-\infty,z)}$ for some $z<y$.
\end{enumerate}
The analogous result holds for the left jump preorder.
\end{lem}
\begin{proof} The condition $g\in G_+$ gives the equality $j^+(g,y)=D^-g(y)$ for every $y\in X$. Write $\Lambda_0=[1]_{\leq_\Lambda}$. Since we are assuming $\mathsf{id}\precneq g$, we can consider $x=x_{g, \Lambda_0}$ as in \eqref{ekuation0}. Thus the restriction of $g$ to $(x, +\infty)$ coincides with some element $h\in G(\mathbb{R};A,\Lambda_0)_+$. Consider the product $f=gh^{-1}$. The rightmost point of $\supp(f)$ is $y=g(x)=h(x)$, and by the chain rule we have
\[D^{-}f(y)=D^{-}g(x)\,D^-h^{-1}(y)=D^{-}g(x)\,D^-h(x)^{-1}.\]
Since $D^-h(x)\in \Lambda_0$ and $D^-g(x)\in \Lambda\setminus\Lambda_0$, we get $1\lneq_\Lambda D^{-}f(y)$. Write $\lambda=D^{-}f(y)$. As before, we have that the rightmost point of the support of $fg_{-}(y,\lambda)^{-1}$ coincides with $g_{-}(y,\lambda)(z)$ where $z$ is the second largest breakpoint of $f$ (the one before $y$). Then write $k=fg_{-}(y,\lambda)^{-1}$. It is direct to check that the decomposition $g=kg_{-}(y,\lambda)h$ satisfies conditions (\ref{i.decomp1}--\ref{i.decomp4}) in the statement.
\end{proof}
\begin{lem}\label{lem.equalpreorder} Consider a Bieri--Strebel group $G=G(\mathbb{R};A,\Lambda)$ and a preorder $\preceq\in\LPO(G)$ containing $\Aff(A,\Lambda)$ in its residue subgroup. Assume further that $\preceq'\in\LPO(G)$ is a right (respectively, left) jump preorder coinciding with $\preceq$ over $G_+$ (respectively, $G_-$). Then $\preceq$ and $\preceq'$ are the same preorder.
\end{lem}
\begin{proof} Assume that $\preceq'$ is the right jump preorder associated with the preorder $\leq_\Lambda\in\LPO(\Lambda)$, the case where $\preceq'$ is a left jump preorder is analogous. Denote by $\Lambda_0$ the residue of $\leq_\Lambda$ and recall from \S \ref{ss.Bieri-Strebelfocal}, that in this case the residue of $\preceq'$ is the subgroup
\[H:=\{g\in G:j^+(g,x)\in\Lambda_0\ \forall x\in\mathbb{R}\}\]
(see \eqref{eq.P}). Since elements of $\Aff(A,\Lambda)$ have constant derivative it holds that $j^+(g,x)=1$ for every $g\in\Aff(A,\Lambda)$ and $x\in\mathbb{R}$. In particular we have $\Aff(A,\Lambda)\subseteq H$.
Note that $G$ decomposes as $G=G_+\rtimes\Aff(A,\Lambda)$. Then, for every $g\in G$ we can write $g=g_+a_g$ with $g_+\in G_+$ and $a_g\in\Aff(A,\Lambda)$. Denote by $P$ and $P'$ the positive cones of $\preceq$ and $\preceq'$ respectively. Since $\Aff(A,\Lambda)$ is contained in the residue of both $\preceq$ and $\preceq'$, it holds that $g\in P$ if and only if $g_+\in G_+\cap P$ and also that $g\in P'$ if and only if $g_+\in G_+\cap P'$. Finally, since by hypothesis it holds $G_+\cap P=G_+\cap P'$, the lemma follows.
\end{proof}
\begin{proof}[Proof of Theorem \ref{t-BBS}]
The assumptions on $G=G(\mathbb{R};A,\Lambda)$ ensure that $G$ is in the class $\mathcal F${}\;(see Lemma \ref{p.BieriStrebel_fg_germs}). After Theorem \ref{t-C-trichotomy}, we only need to show that a minimal faithful $\mathbb{R}$-focal action $\varphi\colon G\to \homeo_0(\mathbb{R})$, increasingly (respectively, decreasingly) horograded by its standard action on $\mathbb{R}$, is conjugate to an action of the form $\varphi_{+, \leq_\Lambda}$ (respectively, $\varphi_{-, \leq_\Lambda}$) for some preorder $\leq_\Lambda$. We will only discuss the case of increasing horograding by the standard action, the decreasing case being totally analogous.
We write $\eta\in \mathbb{R}$ for the unique fixed point of $\varphi\left (\Aff(A, \Lambda)\right )$ given by Lemma \ref{l-bs-fix} (applied to the case $\Delta=\Lambda$).
In what follows, let $\preceq$ be the preorder on $G$ induced by $\eta$, namely by declaring $g\precneq h$ if and only if $g.\eta<h.\eta$. We will show that this preorder coincides with a right jump preorder associated with some $\leq_\Lambda\in\LPO(\Lambda)$.
Let us first identify such preorder. For $x\in A$, the set of elements $T_x:=\left \{g_-(x, \lambda)\colon \lambda \in \Lambda\right \}$ is a subgroup of $G$ isomorphic to $\Lambda$, which is a section inside $G_{(-\infty, x)}$ of the group of germs $\Germ\left (G_{(-\infty, x)}, x\right )$. We put on $\Lambda$ the preorder $\leq_\Lambda$ given by restricting $\preceq$ to this subgroup, namely by setting $\lambda\lneq_\Lambda \mu$ if $g_-(x, \lambda).\eta<g_-(x, \mu).\eta$. Note that this preorder does not depend on the choice of $x\in A$, as for $x, y\in A$ the groups $T_{x}$ and $T_{y}$ are conjugate by an element of $\Aff(A, \Lambda)$ (see \eqref{e-bx-equivariance1}), which fixes $\eta$. Denote $\preceq'$ the right jump preorder in $G$ associated with $\leq_\Lambda$. We proceed to show that $\preceq'$ and $\preceq$ coincide on $G_+$ which, by Lemma \ref{lem.equalpreorder} will conclude the proof.
Denote by $\Lambda_0$ the residue of the preorder $\leq_\Lambda$ and recall that in this case the residue of $\preceq'$ equals $H=\{g\in G:j^+(g,x)\in\Lambda_0\ \forall x\in \mathbb{R}\}$ (see \eqref{eq.P}).
As observed in the proof of Lemma \ref{lem.decomp}, for $g\in G_+$ we have $j^+(g,x)=D^-g(x)$ for every $x\in X$, so that we have the equality
$H\cap G_+=G(\mathbb{R}; A, \Lambda_0)_+$. Note that $H\cap G_+$ fixes $\eta$, since it is generated by $\left \{g_-(x, \lambda)\colon x\in A, \lambda\in \Lambda_0\right \}$ (this can be easily checked from \cite[\S 8.1]{BieriStrebel}). Thus, we have \begin{equation}\label{ecuasion} G_{+}\cap [1]_{\preceq'}\subseteq G_+\cap[1]_{\preceq}. \end{equation}
Assume now that $g\in G_+$ satisfies $\mathsf{id}\precneq' g$. We proceed to show that in this case $\eta<g.\eta$ which implies that $\mathsf{id}\precneq g$. For this, consider the decomposition $g=kg_-(y,\lambda)h$ given by Lemma \ref{lem.decomp}. Then, by Lemma \ref{lem.decomp}.\eqref{i.decomp2} and \eqref{ecuasion}, we get that $h.\eta=\eta$ and therefore $g.\eta=kg_-(y,\lambda).\eta$. On the other hand, Lemma \ref{lem.decomp}.\eqref{i.decomp3} together with the definition of $\leq_\Lambda$ imply that $g_-(y,\lambda).\eta>\eta$. Then, by Corollary \ref{cor.intmov}, we get that $g_-(y,\lambda).\Iphiinn(y, \eta)\cap\Iphiinn(y, \eta)=\emptyset$. Finally, by Lemma \ref{lem.decomp}.\eqref{i.decomp4} we have that $k.\Iphiinn(y, \eta)=\Iphiinn(y, \eta)$ which, in light of what we have already done, shows that $g.\eta=kg_-(y,\lambda).\eta>\eta$ as desired. Analogously one shows that if $g\precneq' \mathsf{id}$ then $g\precneq \mathsf{id}$. This shows that the preorders $\preceq'$ and $\preceq$ coincide over $G_+$ and concludes the proof.
\end{proof}
\subsection{A finitely generated group in the class $\mathcal{F}_0${}\;with no $\mathbb{R}$-focal actions} \label{s-no-actions}
Here we construct an example of a finitely generated locally moving group in the class $\mathcal{F}_0${}, which admits no faithful $\mathbb{R}$-focal actions. In particular every minimal faithful action on $\mathbb{R}$ is conjugate to the standard action.
The starting point of the construction is the Bieri--Strebel group $G(2)=G(\mathbb{R};\mathbb Z[1/2],\langle 2\rangle_* )$ of all finitary dyadic PL homeomorphisms of $\mathbb{R}$, which we already proved to admit only two $\mathbb{R}$-focal actions (Theorem \ref{t-BBS} and Example \ref{ex:focal_G_lambda}). Moreover, as discussed in Remark \ref{r.quotients_BieriStrebel}, we have that its largest proper quotient is isomorphic to the direct product of two solvable Baumslag--Solitar groups $\BS(1,2)$. Therefore also the non-faithful case in Theorem \ref{t-BBS} is very restrictive. Indeed, we have the following result (compare with \cite[Theorem 6.12]{BLT}).
\begin{prop} \label{p-BSxBS}
Let $\Gamma_1$ and $\Gamma_2$ be two groups isomorphic to the Baumslag--Solitar group $\BS(1,2)$, and consider their direct product $\Gamma:=\Gamma_1\times \Gamma_2$. Then every action $\varphi\colon \Gamma\to \homeo_0(\mathbb{R})$ without fixed points is semi-conjugate either to an action by translation of the abelianization $\Gamma^{ab}=\mathbb Z^2$, or to an action obtained by composing the projection to one of the factors $\Gamma_i$, $i\in\{1,2\}$, with the action of $\Gamma_i$ on $\mathbb{R}$ by affine dyadic maps.
\end{prop}
\begin{proof}
The following proof is based on the classification of actions of the Bausmlag--Solitar group $\BS(1,2)$ on the real line, without fixed points: up to semi-conjugacy, it is either an action of the abelianization $\mathbb Z$, or the standard affine action (see e.g.\ \cite{RivasBS,BMNR}).
Consider an action $\varphi\colon \Gamma\to \homeo_0(\mathbb{R})$ without fixed points.
After Proposition \ref{p-centralizer-fix}, without loss of generality, we can assume that the $\varphi$-image $[\Gamma_1,\Gamma_1]$ has fixed points. Thus the action $\varphi$ is semi-conjugate to an action of the quotient $\Gamma_1^{ab}\times \Gamma_2$. Assume that the factor $\Gamma_1^{ab}\cong \mathbb Z$ has no fixed point
and that the action $\varphi$ is not semi-conjugate to any action of the abelianization $\Gamma^{ab}=\Gamma_1^{ab}\times \Gamma_2^{ab}$. This means that the action of $\Gamma_2$ is semi-conjugate to the standard affine action. This is however not possible, because elements in $\Gamma_2$ corresponding to homotheties under the semi-conjugacy have a compact set of fixed points, which must be fixed by $\Gamma_1$, an absurd.
\end{proof}
As a consequence of Corollary \ref{c-exotic-pp}, if we want to avoid $\mathbb{R}$-focal actions, we must leave the setting of {\em finitary} PL transformations. We will consider groups whose elements are PL with a countably many breakpoints that accumulate at some finite subset of ``higher order'' singularities (with some control on these).
Given an open interval $X\subset \mathbb{R}$ we say that a homeomorphism $f\in \homeo_0(X)$ is \emph{locally $\PL$} if there is a finite subset $\Sigma\subset X$ such that $f$ is (finitary) $\PL$ in $X\setminus \Sigma$. For such an $f$, we denote by $\BP^2(f)\subset X$ the minimal subset such that $f$ is $\PL$ in $X\setminus \BP^2(f)$. The set $\BP^2(f)$ is the set of breakpoints of \emph{second order} of $f$.
Points $x\in X\setminus \BP^2(f)$ where $f$ has discontinuous derivative are called breakpoints of \emph{first order}, and we denote them by $\BP^1(f)$. Also, we write $\BP(f)=\BP^1(f)\cup \BP^2(f)$ for the set of breakpoints of $f$. Clearly, when $\BP^2(f)=\varnothing$ we have that $f$ is PL. We will silently use a couple of times the observation that for $f$ and $g$ locally $\PL$, we have that $\BP^2(fg)\subset \BP^2(g)\cup g^{-1}\BP^2(f)$.
\begin{dfn}
Let $X\subset \mathbb{R}$ be an open interval. We write $G(X)=G(X;\mathbb Z[1/2],\langle 2\rangle_*)$ for the Bieri--Strebel group (see Definition \ref{d.BieriStrebel}). We also denote by $G_\omega(X)$ the group of all locally $\PL$ homeomorphisms of $X$ with the following properties:
\begin{itemize}
\item $f$ is locally {\em dyadic} $\PL$, that is at each $x\in X\setminus \BP(f)$ the map $f$ is locally an affine map of the form $x\mapsto 2^nx+b$ for $n\in \mathbb Z$ and $b\in \mathbb Z[1/2]$;
\item breakpoints of $f$ are contained in a compact subset of $X$: $\BP(f)\Subset X$;
\item breakpoints of $f$ and their images are dyadic rationals: $\BP(f)\cup f(\BP(f))\subset \mathbb Z[1/2]$.
\end{itemize}
\end{dfn}
The group $G_\omega(X)$ is uncountable, so too big for our purposes. We will instead consider some subgroups defined in terms of the local behaviour at the breakpoints of second order.
Here we keep the notation from the previous subsection, such as $g(a,\lambda)$, $g_\pm(a,\lambda)$ (see \ref{eq BBS}), and $t_a:x\mapsto x+a$ which denote elements in $\PL(\mathbb{R})$. For $r\in \mathbb{R}$, we also write $h_r=g(r,\tfrac12)$, which corresponds to the homothety of ratio $1/2$ centered at $r$, and similarly we write $h_{r\pm}=g_{\pm}(r,\tfrac12)$ for shorthand notation.
\begin{dfn}
Let $g\colon I\to J$ be a homeomorphism between two open intervals. We say that $g$ has a \emph{$2^n$-scaling germ} at $r\in I$, if there exists a neighborhood $U$ of $r$ such that $g h_r^n\restriction_U=h_{g(r)}^n g\restriction_U$.
\end{dfn}
\begin{rem}
Note that when $g(r)=r$ this simply means that the germ of $g$ at $r$ commutes with the germ of $h_r^n$. More generally, if $g(r)\neq r$, and if $h$ is any PL map such that $hg(r)=r$, then $g$ has a $2^n$-scaling germ at $r$ if and only if the germs of $hg$ and $h_r$ at $r$ commute: this does not depend on the choice of $h$, since every PL map has $2^n$-scaling germ (and more generally $k$-scaling germ for any $k>0$, with the obvious extension of the definition) at every point, including breakpoints.
\end{rem}
\begin{dfn}
Given an open interval $X\subset \mathbb{R}$ and $n\ge 1$, we let $G_\omega^{(n)}(X)$ be the subgroup of $G_\omega(X)$ consisting of elements that have $2^n$-scaling germs at every breakpoint of second order (and thus at all points $x\in X$).
\end{dfn}
For every dyadic point $x\in X$, let $\mathcal{D}^{(n)}_{x}$ be the group of germs at $x$ of elements in $G_\omega^{(n)}(X)$ which fix $x$, that is, the group of germs of homeomorphisms that are locally dyadic PL away from $\{x\}$ and that commute with $h_x^n$. We denote by $\mathcal{D}^{(n)}_{x-}$ and $\mathcal{D}^{(n)}_{x+}$ the corresponding groups of left and right germs, respectively, so that $\mathcal{D}^{(n)}_{x}\cong \mathcal{D}^{(n)}_{x-}\times \mathcal{D}^{(n)}_{x+}$. The groups $\mathcal{D}^{(n)}_{x-}$ and $\mathcal{D}^{(n)}_{x+}$ are isomorphic to a well-known group, namely the lift $\widetilde{T}\subseteq \homeo_0(\mathbb{R})$ of Thompson's group $T$ acting on the circle. Explicitly, $\widetilde{T}$ is the group of all dyadic PL homeomorphisms of $\mathbb{R}$ which commute with the unit translation $t_1\colon x\mapsto x+1$. The point is that for every $n\ge 1$ and $x\ge 1$ dyadic, the map $h_{x}^n\restriction_{(-\infty,x)}$ can be conjugated to the translation $t_1$ by a dyadic PL homeomorphism $f\colon (-\infty, x)\to \mathbb{R}$. This establishes an isomorphism of $\mathcal{D}^{(n)}_{r-}$ with the group of germs of $\widetilde{T}$ at $+\infty$, which is isomorphic to $\widetilde{T}$ itself. Similarly one argues for $\mathcal{D}^{(n)}_{r+}$. This fact will be constantly used in what follows.
A first consequence is that the groups $\mathcal{D}^{(n)}_{x-}$ and $\mathcal{D}^{(n)}_{x+}$ are finitely generated, since $\widetilde{T}$ is so.
This leads to the following:
\begin{prop}\label{p-doubling-fg}
For every dyadic open interval $X=(a, b)\subset \mathbb{R}$ and every $n\ge 1$, the group $G:=G_\omega^{(n)}(X)$ satisfies the following.
\begin{enumerate}
\item $G$ is finitely generated, and belongs to the class $\mathcal F${}.
\item the subgroup $G_c$ of compactly supported elements is perfect (and thus simple). In particular the largest proper quotient of $G$ is $G/G_c=\Germ(G, a)\times \Germ(G, b)$.\end{enumerate}
\end{prop}
Note that, as breakpoints of every element in $G_\omega^{(n)}(X)$ are contained in a compact subset of $X$, the group $\Germ(G, a)$ is infinite cyclic if $a>-\infty$, and isomorphic to $\BS(1,2)$ if $a=-\infty$, and similarly for $\Germ(G, b)$.
\begin{proof}[Proof of Proposition \ref{p-doubling-fg}]
Fix a dyadic point $x \in (a, b)$. Since the group of germs $\mathcal{D}^{(n)}_x=\mathcal{D}^{(n)}_{r-}\times \mathcal{D}^{(n)}_{r+}$ is finitely generated, we can find a finite subset $S\subset G_\omega^{(n)}(X)$ which fix $x$ and whose germs generate $\mathcal{D}^{(n)}_x$, and that have no breakpoints of second order apart from $x$.
\begin{claim}
We have $G_\omega^{(n)}(X)=\langle G(X), S\rangle$.
\end{claim}
\begin{proof}[Proof of claim]
Let $g\in G_\omega^{(n)}$, and let us show that $g\in \langle G(X), S\rangle$ by induction on the number $k=|\BP^2(g)|$ of breakpoints of $g$ of second order. If $k=0$, then $g\in G(X)$. Assume that $k\ge 1$, and let $y\in \BP^2(g)$ be a breakpoint of second order of $g$. Since $G(X)$ acts transitively on dyadic rationals, we can choose $h_1\in G(X)$ such that $h_1(g(x))=x$. As $\BP^2(h_1)=\varnothing$, we have that the element $g'=h_1g$ satisfies $|\BP^2(g')|=k$, and moreover $x$ belongs to $\BP^2(g')$ and is fixed by $g'$. Choose $h_2\in \langle S\rangle$ whose germ at $x$ is equal to the germ of $g$. By the choice of $S$, we have $\BP^2(g)=\{x\}$, so that for the element $g''=h_2^{-1}g'$ we have
$\BP^2(g'')=\BP^2(g')\setminus \{x\}$, and thus $\BP^2(g'')=k-1<k$. By induction, we have $g''\in \langle G(X), S\rangle$, and it follows that $g=h_1^{-1}h_2g''\in \langle G(X), S\rangle$.
\end{proof}
Since the Bieri--Strebel group $G(X)=G(X;\mathbb Z[1/2],\langle 2\rangle_*)$ is finitely generated for $X$ dyadic, from the claim we get that $G$ is finitely generated as well.
The fact that $G^{(n)}_\omega(X)$ belongs to the class $\mathcal F${}\, follows from finite generation of the subgroups $G^{(n)}_\omega(Y)$ for $Y\subset X$ dyadic.
Finally the same argument for the claim shows that the group $G_\omega^{(n)}(X)_c$ of compactly supported elements is generated by $\left \langle G(X)_c, S\right \rangle$. Since $\mathcal{D}^{(n)}_x$ is a perfect group we can choose the set $S$ consisting of commutators. And since $G(X)_c$ is perfect as well, the group $G_\omega^{(n)}(X)_c$ is perfect. The last statement follows from Proposition \ref{p-micro-normal}.
\qedhere
\end{proof}
Here is the main result of this subsection, whose proof will need some preliminary lemmas.
\begin{thm} \label{t-doubling-actions}
For $n\ge 2$, every action $\varphi\colon G_\omega^{(n)}(\mathbb{R}) \to \homeo_0(\mathbb{R})$ without fixed points is semi-conjugate to one of the following.
\begin{enumerate}
\item A non faithful action induced by an action of the groups of germs
\[\Germ\left ( G_\omega^{(n)}(\mathbb{R}), -\infty\right ) \times \Germ\left (G_\omega^{(n)}(\mathbb{R}), +\infty\right )\cong \BS(1, 2)\times \BS(1, 2)\]
(these are classified in Proposition \ref{p-BSxBS}).
\item The standard action.
\end{enumerate}
In particular every faithful minimal action of $G_\omega^{(n)}(\mathbb{R})$ on the real line is conjugate to its standard action.
\end{thm}
Until the end of the subsection, for fixed $n\ge1$ we write $G=G_\omega^{(n)}(\mathbb{R})$ and $H=G(\mathbb{R})=G(2)$, so that $H\subseteq G$.
\begin{lem}[Upgrading fixed points] \label{l-fix-upgrade}
With the notations as above, let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be an action on the real line. Then every fixed point of $\varphi(H)$ must be fixed by $\varphi(G)$. In other words, $\operatorname{\mathsf{Fix}}^\varphi(H)=\operatorname{\mathsf{Fix}}^\varphi(G)$.
\end{lem}
\begin{proof}
Consider the subgroups
\[K_l=\left \{g\in G_{(-\infty,0)}\colon \BP^2(g)\subset\{0\}\right \}\quad\text{and}\quad
K_r=\left \{g\in G_{(0,+\infty)}\colon \BP^2(g)\subset\{0\}\right \},
\]
and set $K=\langle K_l, K_r\rangle \cong K_l\times K_r$. Note that group $K$ realizes the group of germs $\mathcal{D}^{(n)}_0$ and after the assumption on breakpoints of second order, the claim in the proof of Proposition \ref{p-doubling-fg} gives that $G=\langle H, K\rangle=\langle H,K_l, K_r\rangle$.
Consider the subgroup $H_l\subseteq K_l$ consisting of all elements whose germ at $0$ is given by a power of $h_{0-}^n$. In particular, every $g\in H_l$ satisfies $\BP^2(g)=\varnothing$, hence $H_l$ is a subgroup of $H$.
Since the germ of $h_{0-}^n$ is central in $\mathcal{D}^{(n)}_{0-}$, we have that $H_l$ is normal in $K_l$, with quotient $K_l/H_l\cong \mathcal{D}^{(n)}_{0-}/\langle h_{0-}^n\rangle$ which is isomorphic to Thompson's group $T$ acting on the circle. The same considerations hold for the subgroup $H_r\subseteq K_r$ defined analogously.
Assume now that $\varphi\colon G\to \homeo_0(\mathbb{R})$ is an action such $\operatorname{\mathsf{Fix}}^\varphi(H)\neq \varnothing$, so that $\operatorname{\mathsf{Fix}}^\varphi(H_l)$ is non-empty and contains $\operatorname{\mathsf{Fix}}^\varphi(H)$. Then $\varphi(K_l)$ preserves $\operatorname{\mathsf{Fix}}^\varphi(H_l)$, and the $\varphi$-action of $K_l$ on $\operatorname{\mathsf{Fix}}^\varphi(H_l)$ factors through the quotient $K_l/H_l\cong T$. Since $T$ is a simple group, and it contains elements of finite order, every order-preserving action on a totally ordered set is trivial. Thus the action of $K_l$ on $\operatorname{\mathsf{Fix}}^\varphi(H_l)$ is actually trivial, and in particular it fixes $\operatorname{\mathsf{Fix}}^\varphi(H)$. Similarly so does $K_r$. Since $G=\langle H, K_l, K_r\rangle$ this implies that every point in $\operatorname{\mathsf{Fix}}^\varphi(H)$ is fixed by $\varphi(G)$. \qedhere
\end{proof}
The next lemma makes use of the assumption that $n\ge 2$ in Theorem \ref{t-doubling-actions}, and leverages the fact that the group $\widetilde{T}$ admits only one action on the real line up to semi-conjugacy. In the statement, with abuse of notation, we identify $h_{x-}$ with its germ in $\mathcal{D}^{(n)}_{x-}$
\begin{lem} \label{l-Ttilde-semiorder}
For every $n\ge 2$, the group $\mathcal{D}^{(n)}_{x-}$ admits no non-trivial left-invariant preorder which is invariant under conjugation by the element $h_{x-}$.
\end{lem}
\begin{proof}
The natural isomorphism $\mathcal{D}^{(n)}_{x-}\cong \widetilde{T}$ maps $h_{x-}$ to an element $h\in \widetilde{T}$ which is an $n$th root of the translation $t_1$, i.e.\ $h^n=t_1$. So it is enough to show that $\widetilde{T}$ admits no non-trivial left-invariant preorder $\prec$ which is invariant under conjugation by such an $h$. Assume by contradiction that $\prec$ is such a preorder. By \cite[Theorem 8.7]{MatteBonTriestino} the dynamical realization of $\prec$ is semi-conjugate to the standard action of $\widetilde{T}$ on the real line, so that the maximal $\prec$-convex subgroup $K$ must be equal to the stabilizer $\widetilde{T}_y$ of some point $y\in \mathbb{R}$ for the standard action. On the other hand $K$ must be normalized by $h$, so that we must have $\widetilde{T}_y=\widetilde{T}_{h(y)}$. However since $h^n=t_1$ we have $h(y)\neq y$ and $|h(y)-y|<1$, so that $y$ and $h(y)$ have different projections to the circle $\mathbb{R}/\mathbb Z$. But any two distinct points in the circle have different stabilizers in Thompson's group $T$, and thus $y$ and $h(y)$ have different stabilizers in $\widetilde{T}$, which is a contradiction. \qedhere
\end{proof}
\begin{proof}[Proof of Theorem \ref{t-doubling-actions}]
Let $\varphi:G\to \homeo_0(\mathbb{R})$ be an action without fixed points.
Since $G$ is in the class $\mathcal F${}\;(Proposition \ref{p-doubling-fg}), we can apply Theorem \ref{t-C-trichotomy}. We then assume that $\varphi$ is faithful minimal and, by symmetry, it is enough to exclude that $\varphi$ is $\mathbb{R}$-focal, increasingly horograded by the standard action of $h$ on $\mathbb{R}$. Note that by Lemma \ref{l-fix-upgrade}, we know that $\varphi(H)$ has no fixed point.
In order to fulfill Assumption \ref{ass.BieriStrebel}, we will consider the family of subgroups $\mathcal{L}=\left \{L_x\right \}_{x\in \mathbb{R}}$, where $L_x=G_{(-\infty,x)}$, and with this choice we will simply have $\Iphi(\mathcal{L},x,\xi)=\Iphi(x,\xi)$.
We apply Lemma \ref{l-bs-fix}: let $\eta$ be the unique fixed point of $\varphi\left (\Aff(A,\Lambda)\right )$. Fix a dyadic rational $x\in \mathbb{R}$ and consider the preorder $\prec_{\eta}$ on $G_{(-\infty, x)}$ associated with the action of $G_{(-\infty, x)}$ on $\Iphi(x,\eta)$. By Proposition \ref{c-bs-L-semiconjugate}, this preorder descends to a non-trivial preorder $\bar{\prec}_{\eta}$ on $\Germ\left (G_{(-\infty, x)}, x\right )=\mathcal{D}_{x-}$.
Consider now the element $h_x\in \Aff(A,\Lambda)$. Since $h_x$ fixes $x$, it normalizes $G_{(-\infty, x)}$; moreover fixes $\eta$ and thus $\varphi(h_x)$ preserves $\Iphi(x,\eta)$. We also see that the preorder $\prec_{\eta}$ is invariant under the automorphism induced by $h_x$ on $G_{(-\infty, x)}$. But this automorphism coincides with the inner automorphism defined by conjugation by $h_{x-}$, so that the preorder $\prec_{\eta}$, and thus $\bar{\prec}_{\eta}$, must be invariant under conjugation by $h_{x-}$. This is in contradiction with Lemma \ref{l-Ttilde-semiorder}. \qedhere
\end{proof}
\section{$\mathbb{R}$-focal actions}\label{sec.focalgeneral}
In this section and the next we leave temporary aside the study of locally moving groups to introduce and study the notion of $\mathbb{R}$-focal action. In this section we give the definition and some first properties of $\mathbb{R}$-focal actions and, as initial motivation, we show that they arise naturally in some situations, for instance for actions of solvable groups. The meaning of this notion will be further clarified in Section \ref{sec_focal_trees}, where we will reinterpret $\mathbb{R}$-focal actions in terms of actions on planar directed trees, and study their dynamical properties more in details.
\subsection{Cross-free covers and $\mathbb{R}$-focal actions}\label{ss.focal}
\begin{dfn} \label{d-CF-cover} Let $\Omega$ be a set and let $I,J\subset \Omega$ be two subsets. We say that $I$ and $J$ \emph{do not cross} if either $I\subset J$ or $J\subset I$, or $I\cap J=\varnothing$. A collection of subsets $\mathcal{S}$ is \emph{cross-free} if $I$ and $J$ do not cross for every $I, J\in \mathcal{S}$.
When $(\Omega,\prec)$ is a totally ordered space, we say that a cross-free collection $\mathcal S$ is a \emph{cross-free cover} (CF-cover for short) if the following conditions are satisfied:
\begin{enumerate}[label=(C\arabic*)]
\item \label{i.CF}every element of $\mathcal S$ is $\prec$-convex, open and bounded (with respect to the order topology),
\item \label{ii.CF} there exists a subcollection of $\mathcal S$ which is totally ordered by inclusion and covers $\Omega$.
\end{enumerate}
\end{dfn}
\begin{rem}\label{r-CF-cover}
When $(\Omega,\prec)$ is the real line $\mathbb{R}$ with its standard order, condition \ref{i.CF} for a CF-cover $\mathcal S$ means that every element of $\mathcal S$ is a bounded open interval, whilst condition \ref{ii.CF} amounts to requiring that $\mathcal{S}$ be a cover of $\mathbb{R}$, justifying the terminology. Indeed if this holds, then if $\mathcal{C}_1$ and $\mathcal{C}_2$ are maximal totally ordered subcollections of $\mathcal{S}$ (with respect to inclusion), then the cross-free property implies that the unions $\bigcup_{I\in \mathcal{C}_1} I$ and $\bigcup_{J\in \mathcal{C}_2} J$ are either equal or disjoint, and hence $\mathbb{R}$ can be written as a disjoint union of open subsets of this form. By connectedness of $\mathbb{R}$, there can be only one such maximal subcollection (and there is at least one by Zorn's lemma).
\end{rem}
\begin{dfn} \label{d-R-focal}
An action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is said to be $\mathbb{R}$-\emph{focal} if it has no fixed points and there exists a bounded open interval $I\subset \mathbb{R}$ whose $\varphi(G)$-images form a CF-cover of $\mathbb{R}$.
\end{dfn}
\begin{rem}
Note that equivalently an action $\varphi: G\to \homeo_0(\mathbb{R})$ is $\mathbb{R}$-focal if and only if it admits an invariant CF-cover $\mathcal{S}$ and an interval $J\in \mathcal{S}$ whose $\varphi(G)$-orbit is \emph{cofinal} in $\mathcal{S}$ with respect to inclusion, meaning that for every $J'\in \mathcal{S}$ there exists $g\in G$ such that $J'\subset g.J$. Indeed the latter condition clearly implies that the orbit of $J$ is a CF-cover. Conversely if $I$ is a bounded open interval whose orbit is a CF-cover, then condition \ref{ii.CF} implies that the orbit of $I$ is cofinal.
\end{rem}
We now study some basic properties of $\mathbb{R}$-focal actions.
\begin{lem} \label{rem.fixedepoints}
Let $\varphi:G\to \homeo_{0}(\mathbb{R})$ be an action admitting an invariant CF-cover. Then for every element $g\in G$, the image $\varphi(g)$ has fixed points.
\end{lem}
A more precise analysis of the dynamics of individual elements in an $\mathbb{R}$-focal action will be given in \S \ref{s-dynclasselements}.
\begin{proof}[Proof of Lemma \ref{rem.fixedepoints}]
Let $\mathcal{S}$ be an invariant CF-cover and fix $g\in G$. For every $\xi \in\mathbb{R}$, from condition \ref{ii.CF} we can find an interval $J\in \mathcal S$ which contains $\xi$ and $g.\xi$, so that $g.J\cap J\neq \varnothing$. Then the cross-free property implies that either $g$ or $g^{-1}$ must map $J$ into itself. By the intermediate value theorem, $\varphi(g)$ has fixed points inside $J$.
\end{proof}
\begin{prop} \label{p-focal-semiconj} Let $G$ be a group, and let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be an $\mathbb{R}$-focal action. Then the following holds.
\begin{enumerate}[label=(\roman*)]
\item \label{i-focal-semiconj}
If $\psi\colon G\to \homeo_0(\mathbb{R})$ is an action semi-conjugate to $\varphi$ then $\psi$ is $\mathbb{R}$-focal.
\item \label{i-focal-minimal} $\varphi(G)$ has a unique minimal invariant set $\Lambda\subseteq \mathbb{R}$, which is not discrete.
\end{enumerate}
In particular, every $\mathbb{R}$-focal action is semi-conjugate to a minimal $\mathbb{R}$-focal action.
\end{prop}
\begin{proof}
Let $I\subset \mathbb{R}$ be a bounded open interval such that $\mathcal{S}=\{\varphi(g)(I)\colon g\in G\}$ is a CF-cover.
To prove \ref{i-focal-semiconj}, let $h\colon \mathbb{R}\to \mathbb{R}$ be a semi-conjugacy from $\varphi$ to $\psi$ (in the sense that \eqref{eq:semiconj} holds). After condition \ref{ii.CF}, $\mathcal{S}$ contains an increasing sequence of intervals that exhaust $\mathbb{R}$, thus, as $h$ is proper, we can find $g\in G$ such that $h(\varphi(g)(I))=\psi(g)(h(I))$ is not a singleton, so that actually $J=h(I)$ is a bounded open interval. The equivariance of $h$ implies that the $\psi(G)$-orbit of $J$ is a CF-cover.
To prove \ref{i-focal-minimal}, observe first that $\varphi(G)$ has no closed discrete orbit. If this was the case, then the number of points of such an orbit inside intervals in $\mathcal{S}$ must be finite and constant, which is clearly in contradiction with condition \ref{ii.CF}. Thus $\varphi(G)$ has at most one non-empty closed minimal subset, and it is enough to show that such a set exists. By condition \ref{ii.CF}, every non-empty closed $G$-invariant subset of $\mathbb{R}$ intersects an element of $\mathcal{S}$, so by $\varphi(G)$-invariance it intersects $I$. By compactness of $\overline{I}$, a standard application of Zorn's lemma gives a non-empty minimal invariant set.
Finally note that since $\varphi$ admits a non-discrete minimal invariant set, it is semi-conjugate to a minimal action (Corollary \ref{cor.basica}), which must be $\mathbb{R}$-focal by \ref{i-focal-semiconj}.
\end{proof}
After the previous proposition there is little loss of generality if we restrict to the study of $\mathbb{R}$-focal \emph{minimal} actions, and we will systematically do so. Note that in this case the notion of $\mathbb{R}$-focal action can be reformulated by simply requiring the existence of a CF-cover. More precisely, we have the following result.
\begin{prop}\label{prop.minimalimpliesfocal}
Let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be a minimal action. Then $\varphi$ is $\mathbb{R}$-focal if and only if it preserves a non-empty cross-free family of intervals. Moreover if this is the case, then $\varphi$ is {proximal}.
\end{prop}
\begin{proof}
The forward implication is obvious from the definition of $\mathbb{R}$-focal action, by taking the family consisting of $\varphi(G)$-images of $I$. Conversely assume $\varphi$ is minimal and that $\mathcal{S}$ is an invariant cross-free family of intervals. Then, for every $I\in \mathcal{S}$, the family $\mathcal{S}_0=\{\varphi(g)(I)\colon g\in G\}$ is also cross-free. The union of elements of $\mathcal{S}_0$ is an invariant open subset which is non-trivial, so that by minimality of the action, it has to coincide with the whole real line. In particular we have that $\mathcal{S}_0$ is a CF-cover, so $\varphi$ is an $\mathbb{R}$-focal action.
Let us now show that $\varphi$ must be proximal. By Theorem \ref{t-centralizer} if this is not the case then there exists an element $\tau\in \homeo_0(\mathbb{R})$ without fixed points which centralizes $\varphi(G)$. Upon conjugating the action, we can assume that $\tau(x)=x+1$. Now, given a bounded interval $I\subset \mathbb{R}$ denote by $n_I$ the largest $n\ge 0$ such that $\tau^n(I)\cap I\neq \varnothing$. Since $\tau$ commutes with $\varphi(G)$, we must have that $n_{\varphi(g)(I)}= n_I$ holds for every $g$. We deduce that the orbit of every bounded interval $I$ must consist of intervals whose length is bounded by $n_I+1$, and thus $\varphi$ cannot be $\mathbb{R}$-focal.
\end{proof}
We conclude with a simple lemma, which says that minimal $\mathbb{R}$-focal actions are determined, uniquely up to conjugacy, by the combinatorics of the CF-cover. This will be used in the next section.
\begin{lem}\label{l-focal-equivalence}
Let $G$ be a group and for $i\in\{1,2\}$, let $\varphi_i\colon G\to \homeo_0(\mathbb{R})$ be a minimal $\mathbb{R}$-focal action with invariant CF-cover $\mathcal{S}_i$. Assume there exists a $G$-equivariant map $f:\mathcal{S}_1\to\mathcal{S}_2$ such that for every $I,J\in\mathcal{S}_1$ with $\sup I\le \sup J$, one has $\sup f(I)\le \sup f(J)$.
Then the actions $\varphi_1$ and $\varphi_2$ are positively conjugate.
\end{lem}
\begin{rem}
After Proposition \ref{prop.minimalimpliesfocal}, the actions in the statement of Lemma \ref{l-focal-equivalence} are proximal, so by Remark \ref{r-centralizer} the positive conjugacy $h:\mathbb S^1\to \mathbb S^1$ between $\varphi_1$ and $\varphi_2$ is unique. One can actually prove that the conjugacy induces the map $f$, in the sense that $h(I)=f(I)$ for every $I\in \mathcal S_1$.
\end{rem}
\begin{proof}[Proof of Lemma \ref{l-focal-equivalence}] Given a bounded interval $I$, we write $I_+=\sup I$ for simplicity.
Consider the subset $\mathcal{S}_1^+=\{I_+:I\in\mathcal{S}_1\}\subseteq \mathbb{R}$ of rightmost points of elements of $\mathcal S_1$, and introduce
the function $j:\mathcal S_1^+\to \mathbb{R}$ given by \[j(\xi)=\sup\{f(I)_+:I\in \mathcal S_1,\,I_+=\xi\}.\] Note that $j$ is a monotone non-decreasing equivariant map: monotonicity follows from the assumption that $I_+\le J_+$ implies $f(I)_+\le f(J)_+$, and equivariance follows from equivariance of $f$. By Lemma \ref{lem.semiconjugacy}, the map $j$ extends to a positive semi-conjugacy $h$ between $\varphi_1$ and $\varphi_2$, which is actually a conjugacy because $\varphi_1$ and $\varphi_2$ are minimal.
\end{proof}
Probably the first example of $\mathbb{R}$-focal action appearing in the literature is an action of the lamplighter group $\mathbb Z\wr \mathbb Z$ studied by J. F. Plante \cite{Plante}. The following example generalizes this construction to arbitrary wreath products of countable groups. We give will use it repeteadly along this section and the next to illustrate the notions discussed.
\begin{ex}[Plante actions of wreath products]
\label{subsec.Plantefocal}
Recall that for general groups $G$ and $H$, the \emph{wreath product} $H\wr G$ is defined by the semidirect product $\left (\bigoplus_GH\right )\rtimes G$, where $G$ acts on the direct sum by shift of indices. More explicitly, considering the direct sum $\bigoplus_GH$ as the set of functions $\s:G\to H$ which are trivial at all but finitely elements of $G$, the action of $h\in G$ is given by $\sigma(g)(\s)(x)=\s(g^{-1}x)$.
Given left-invariant orders $<_G\in \LO(G)$ and $<_H\in \LO(H)$, we can consider an order $\prec$ of lexicographic type on $\bigoplus_GH$, as follows.
We denote by $\mathsf e$ the trivial element of $\bigoplus_GH$, that is the function satisfying $\mathsf e(x)=1_H$ for every $x\in G$, and we define
\[
P=\left \{\s\in \bigoplus_GH\colon \s\neq \mathsf e,\, \s(x_{\s})>_H1_H\right \},
\]
where for $\s\neq \mathsf e$ we set $x_{\s}=\max_{<_G}\{x\in G\colon \s(x)\neq 1_H\}$.
It is not difficult to check that $P$ defines a positive cone, and thus a left-invariant order $\prec$ on the direct sum $\bigoplus_GH$, which is also invariant under the shift action $\sigma$ of $H$. This gives an order-preserving action
$\Psi:H\wr G\to\Aut\left (\bigoplus_GH,\prec\right )$, that we call the \emph{Plante product} of $<_G$ and $<_H$. When $G$ and $H$ are countable so is $\bigoplus_GH$, and thus we may consider the dynamical realization of $\Psi$, which we call the \emph{Plante action} associated with $<_G$ and $<_H$. In this situation we let $\iota\colon (\bigoplus_GH, \prec)\to \mathbb{R}$ be the associated good embedding and let $\varphi\colon H\wr G\to \homeo_0(\mathbb{R})$ be the dynamical realization. When $G=H=\mathbb Z$ and $<_G$ and $<_H$ are the standard left orders of $\mathbb Z$, this construction yields the action of $\mathbb Z\wr\mathbb Z$ considered by Plante (see \cite[\S 3.3.2]{GOD}), an illustration of which appears in Figure \ref{fig-Plante-action}.
We claim that the Plante action $\varphi$ is minimal and $\mathbb{R}$-focal.
To prove minimality, note that the stabilizer of the trivial element $\mathsf e\in \bigoplus_GH$ coincides with $G$, and its orbit is the whole subgroup $\bigoplus_GH$. Thus, after Proposition \ref{p.minimalitycriteria}, it is enough to check that $\Psi(G)$ is of homothetic type. For this, consider four elements $\s_1,\s_2,\mathsf t_1,\mathsf t_2\in \bigoplus_GH$ such that $\mathsf t_1\prec \mathsf s_1\prec \mathsf e\prec \s_2\prec \mathsf t_2$. Set $y_*=\max_{<_G}\{x_{\mathsf t_1},x_{\mathsf t_2}\}$ and $x_*=\min_{<_G}\{x_{\s_1},x_{\s_2}\}$, and consider an element $g\in G$ such that $g^{-1}x_*>_Gy_*$. Then it is immediate to check that
\[
\sigma(g)(\s_1)\prec \mathsf t_1\prec \mathsf e\prec \mathsf t_2\prec \sigma(g)(\s_2),
\]
which gives the desired conclusion.
To check that $\varphi$ is $\mathbb{R}$-focal it is enough to check that $(\bigoplus_GH,\prec)$ admits a CF-cover $\mathcal S$ which is invariant under the $\Psi$-action of $H\wr G$.
Indeed, considering the collection of interiors of closures of images of elements of $\mathcal S$ by the good embedding $\iota:\bigoplus_GH\to \mathbb{R}$ associated with $\varphi$, we obtain a CF-family which is invariant for $\varphi(H\wr G)$ (an argument analogous to that in Proposition \ref{p-focal-semiconj} shows that this is non-trivial), so that we can conclude by Proposition \ref{prop.minimalimpliesfocal}.
For this, for $\s\in \bigoplus_GH$ and $g\in G$, we set \[C_{\s,g}=\left \{\mathsf t\in\bigoplus_GH:\mathsf t(x)=\s(g)\text{ for every }x >_G g\right \}.\]
Clearly every $C_{\s,g}$ is a convex and bounded subset of $(\bigoplus_GH, \prec)$ and \[\mathcal{S}_0=\left \{C_{\s,g}:\s\in\bigoplus_GH,g\in G\right \}\]
defines a cover of $\bigoplus_GH$, which is $\Psi(H\wr G)$-invariant. It only remains to check that $\mathcal S_0$ is a cross-free family. For this, take two elements $C_{\s,g}$ and $C_{\s',g'}$ in $\mathcal S_0$ with $g\le_G g'$, and assume there is some element $\mathsf t$ in their intersection. It follows that $\s$, $\mathsf t$, and $\s'$ all agree on $\{x\in G \colon x>_Gg' \}$ and so $C_{\s,g}\subseteq C_{\s',g'}$. This shows that $\mathcal S_0$ is a CF-cover and thus the Plante action is $\mathbb{R}$-focal.
\begin{figure}[ht]
\includegraphics[scale=1]{lamplighter1-1.pdf}
\caption{Plante action of $\mathbb Z\wr\mathbb Z$ on the line. One factor is generated by $g$ which acts as a homothety. The generator of the other factor is $h_0$, and we have $h_n=g^nh_0g^{-n}$ for every $n\in \mathbb Z$, where the $h_n$ commute and are a basis of the lamp group $\oplus_\mathbb Z \mathbb Z$. }\label{fig-Plante-action}
\end{figure}
\end{ex}
\subsection{A condition for $\mathbb{R}$-focality}\label{ss.focal_condition} The following criterion implies the $\mathbb{R}$-focality of a vast class of actions.
\begin{prop}\label{prop.maximal}Consider a minimal faithful action $\varphi:G\to\homeo_{0}(\mathbb{R})$ and let $N\triangleleft G$ be a normal subgroup which is not a cyclic subgroup of the center of $G$. Assume that $\varphi(N)$ does not act minimally on $\mathbb{R}$. Then $\varphi$ is an $\mathbb{R}$-focal action.
\end{prop}
First we recall the following lemma. Its proof is well-known but we include it for completeness.
\begin{lem}\label{l-normal-minimal}
Let $G$ be a group, and let $N$ be a non-trivial normal subgroup which is not a cyclic subgroup of the center of $G$. Assume that $\varphi\colon G\to \homeo_0(\mathbb{R})$ is a minimal faithful action of $G$. Then either the image of $N$ acts minimally, or it admits no minimal invariant set.
\end{lem}
\begin{proof}
Assume that there exists a non-empty minimal $\varphi(N)$-invariant set $\Lambda\subset \mathbb{R}$, and let us show that $\Lambda=\mathbb{R}$. Note that $\Lambda$ cannot be a fixed point otherwise we get a contradiction from Corollary \ref{cor.normalsemicon}. Then either $\Lambda$ is the unique minimal $\varphi(N)$-invariant set, or a closed orbit. In the first case, we have that $\Lambda$ is preserved by the whole group $G$ and thus $\Lambda=\mathbb{R}$ by minimality. In the second case we have that the action of $N$ must be semi-conjugate to a cyclic action coming from a homomorphism $\tau\colon N\to \mathbb Z$, and that $\ker \tau$ acts trivially on $\Lambda$. Since $\ker \tau$ is precisely the subset of $N$ acting with fixed points, the subgroup $\ker \tau$ is necessarily normal in the whole group $G$, and so, as before, we must have $\ker \tau=\{1\}$. Thus $N$ is infinite cyclic and acts as a group of translations. Since $N$ is normal in $G$, this implies that $N$ is central in $G$, contradicting the assumption.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop.maximal}]
We first observe the following.
\begin{claim}
For every open interval $I\subset \mathbb{R}$, there exists an element $h\in N$ such that $h.I\cap I\neq \varnothing$ and $h\restriction_I \neq \mathsf{id}$.
\end{claim}
\begin{proof}[Proof of claim]
Consider the subset of points where the condition in the statement fails, namely
\[
A:=\left\{
\xi\in \mathbb{R}\colon \exists\,\text{neighborhood }V\ni \xi\text{ s.t.\ }\forall h\in N\text{ either }h\restriction_{V}=\mathsf{id}\text{ or }h.V\cap V=\varnothing
\right\}.
\]
It is clear from the definition that $A$ is open, and the fact that $N$ is normal in $G$ gives that it is $G$-invariant. As we are assuming that the action is minimal, we conclude that $A$ is either empty or the whole real line. Assume for contradiction that $A=\mathbb{R}$, and note that this implies that every element $h\in N$ either acts trivially or has no fixed point at all. The first possibility is ruled out by the fact that the action $\varphi$ is minimal, so that by H\"older theorem (see \cite[Theorem 6.10]{Ghys}), we have that $\varphi(N)$ is semi-conjugate to a group of translations and in particular $\varphi(N)$ admits a minimal invariant set. From Lemma \ref{l-normal-minimal} we get that this minimal set must be the whole real line which contradicts our hypothesis that $\varphi(N)$ does not act minimally.
\end{proof}
Now, for each interval $I\subset \mathbb{R}$ denote by $\stab^\varphi_N(I)$ the stabilizer of $I$ in $N$, and by $\operatorname{\mathsf{Fix}}^\varphi_I(N)$ the subset of fixed points of $\stab^\varphi_N(I)$ in $I$. Since $\varphi(N)$ does not act minimally, there exists a proper, closed and $\varphi(N)$-invariant subset of the line that we denote by $\Lambda$. Let $U$ be a connected component of $\mathbb{R}\setminus \Lambda$. The claim implies that the stabilizer $\stab^\varphi_N(U)$ is non-trivial, and moreover the subset $\operatorname{\mathsf{Fix}}^\varphi_U(N)$ has empty interior, so that it is strictly contained in $U$.
Hence the subset $\Lambda_1:=\Lambda\cup\left( \bigcup_U \operatorname{\mathsf{Fix}}^\varphi_U(N)\right) $ (where the union runs over the connected components of $\mathbb{R}\setminus \Lambda$) is a proper closed subset of $\mathbb{R}$, which is $\varphi(N)$-invariant (indeed, for $g\in N$ we have $g.\operatorname{\mathsf{Fix}}^\varphi_U(N)=\operatorname{\mathsf{Fix}}^\varphi_{g.U}(N)$) and moreover $\operatorname{\mathsf{Fix}}^\varphi_V(N)=\varnothing$ for every connected component $V$ of $\mathbb{R}\setminus\Lambda_1$.
Consider now the collection $\mathcal{S}$ of all non-empty open bounded intervals $I\subset \mathbb{R}$ such that
\begin{itemize}
\item for every $h\in N$, either $h.I=I$ or $h.I\cap I=\varnothing$, and
\item $\operatorname{\mathsf{Fix}}^\varphi_I(N)=\varnothing$.
\end{itemize}
The collection $\mathcal S$ is non-empty since it contains each connected component of $\mathbb{R}\setminus \Lambda_1$. Also note that, since $N$ is normal, the family $\mathcal{S}$ is $\varphi(G)$-invariant. We will show that $\mathcal S$ is cross-free. To see this, suppose that $I,J\in\mathcal{S}$ are crossed, and let $\xi$ and $\eta$ be the points in the intersections $I\cap \partial J$ and $\partial I\cap J$, respectively. Since $\stab^\varphi_N(J)$ acts on $J$ without fixed points and fixes $\xi$, we can find an element $h\in \stab^\varphi_N(J)$ which moves $\eta$ and thus satisfies $h.I\cap I \neq \varnothing$, but $h.I\neq I$ (see Figure \ref{fig.focal_condition}). This gives the desired contradiction, and
we conclude that $\mathcal S$ is a $\varphi(G)$-invariant cross-free family of intervals. As $\varphi$ is minimal, Proposition \ref{prop.minimalimpliesfocal} implies that it is $\mathbb{R}$-focal.
\end{proof}
\begin{figure}[ht]
\includegraphics[scale=1]{focal_condition-1.pdf}
\caption{End of the proof of Proposition \ref{prop.maximal}.}\label{fig.focal_condition}
\end{figure}
\begin{rem} The converse to Proposition \ref{prop.maximal} is not true: there exist $\mathbb{R}$-focal actions of groups such that all non-trivial normal subgroups act minimally (see \S \ref{ss.Bieri-Strebelfocal} for counterexamples), and even simple groups which admit minimal $\mathbb{R}$-focal actions (such as the commutator subgroup $[F, F]$ of Thompson's group $F$, see \S \ref{s-F-hyperexotic}). However a partial converse to Proposition \ref{prop.maximal} will be given in \S \ref{ssc.simplicial_actions}\end{rem}
\subsection{$\mathbb{R}$-focal actions and alternatives}
\label{ssc.alternatives}
Here we discuss two situations where $\mathbb{R}$-focal actions appear naturally.
\subsubsection{$\mathbb{R}$-focal actions and micro-supported groups} In the next proposition we show that if a micro-supported group acting minimally is not locally moving (see \S\ref{sc.defi_lm}), then its action is necessarily $\mathbb{R}$-focal.
\begin{prop}[Alternative for micro-supported groups]\label{prop.dicotomia} A micro-supported subgroup $G\subseteq \homeo_0(\mathbb{R})$ whose action is minimal, is either locally moving or $\mathbb{R}$-focal.
\end{prop}
For the proof, given a micro-supported group $G\subseteq\homeo(\mathbb{R})$ and an interval $I\subset \mathbb{R}$ (not necessarily bounded) we write $\operatorname{\mathsf{Fix}}_I:=\operatorname{\mathsf{Fix}}(G_I)\cap I$ for the set of fixed points inside $I$ of the rigid stabilizer of $I$. We first need the following observation.
\begin{lem}\label{lem.dicotomicro} Let $G\subseteq\homeo_0(\mathbb{R})$ be a micro-supported subgroup whose action is minimal. Suppose there exist $\xi,\eta\in\mathbb{R}$ such that $\operatorname{\mathsf{Fix}}_{(-\infty,\xi)}=\operatorname{\mathsf{Fix}}_{(\eta,+\infty)}=\varnothing$. Then $G$ is locally moving.
\end{lem}
\begin{proof} Take a bounded open interval $I=(\alpha,\beta)$ and $x\in I$. We need to show that $g(x)\neq x$ for some $g\in G_I$. From the minimality of the action of $G$, after possibly conjugating $G_{(\eta,+\infty)}$ and $G_{(-\infty,\xi)}$, we can assume that $\xi$ and $\eta$ satisfy that
\[
\alpha<\xi<x<\eta<\beta.
\]
Since $G$ is micro-supported, the subgroup $G_c$ of compactly supported elements is non-trivial.
Since the action of $G$ is minimal and $G_c$ is normal, we have that $G_c$ has no fixed points, and thus there exists $h\in G_c$ such that $h(x)\neq x$. Consider the smallest interval $(a,b)$ containing $\supp(h)$.
From the assumption, we can find elements $k_1\in G_{(-\infty,\xi)}$ and $k_2\in G_{(\eta,+\infty)}$ such that $k_1(a)\in (\alpha,\xi)$ and $k_2(b)\in (\eta,b)$. Write $k=k_1k_2$. Then $g=khk^{-1}\in G_I$ and $g(x)\neq x$, as desired.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop.dicotomia}] Suppose that $G$ is not locally moving. After Lemma \ref{lem.dicotomicro}, we can assume that there exists $\xi\in\mathbb{R}$ such that $\operatorname{\mathsf{Fix}}_{(\xi,+\infty)}\neq\varnothing$ (the symmetric case $\operatorname{\mathsf{Fix}}_{(-\infty,\xi)}\neq\varnothing$ can be treated similarly). Therefore, the support of the subgroup $G_{(\xi,+\infty)}$ has a bounded connected component, let us call it $U=(\alpha,\beta)\subseteq (\xi,+\infty)$. We claim that the $G$-orbit of $U$ is cross-free.
Looking for a contradiction, suppose that there is $g\in G$ such that $U$ and $g(U)$ are crossed. Clearly $g(U)$ is a connected component of the support of $g G_{(\xi,+\infty)}g^{-1}=G_{(g(\xi),+\infty)}$. But, up to changing $g$ by its inverse, we can assume that $\xi\leq g(\xi)$, and hence that $G_{(g(\xi),+\infty)}$ is a subgroup of $G_{(\xi,+\infty)}$. In particular we get that $U$ is fixed by every element in $G_{(g(\xi),+\infty)}$ and therefore it is not possible that $U$ is crossed with a component of support of $G_{(g(\xi),+\infty)}$. This contradicts our assumption that $U$ and $g(U)$ are crossed, and so the claim follows. From Proposition \ref{prop.minimalimpliesfocal}, we get that the action of $G$ is $\mathbb{R}$-focal.
To conclude, we need to show that the action of $G$ cannot be simultaneously locally moving and $\mathbb{R}$-focal. For this, note that when $G$ is locally moving the diagonal action of $G$ on the set $X_2:=\{(x,y)\in\mathbb{R}^2:x<y\}$ is minimal, while in the case when the action is $\mathbb{R}$-focal this is not the case. For instance, if $U=(\alpha,\beta)$ is an interval whose $G$-orbit has no crossed elements, then $(\alpha,\beta)$ cannot accumulate on $(\alpha', \beta')$ for any $\alpha'<\alpha<\beta'<\beta$.
\end{proof}
We have the following straightforward application.
\begin{cor}\label{c-lm-criterion} If $G\subseteq \homeo_0(\mathbb{R})$ is a group acting minimally on $\mathbb{R}$ and containing an element of relatively compact support and an element without fixed points, then $G$ is locally moving.
\end{cor}
\begin{proof}
The fact that $G$ acts minimally on $\mathbb{R}$ and contains an element of compact support implies that $G$ is micro-supported by Proposition \ref{p-micro-compact}. The assumption that $G$ contains an element without fixed points excludes the case of $\mathbb{R}$-focal action, as every element should have with fixed points (Lemma \ref{rem.fixedepoints}). By Proposition \ref{prop.dicotomia}, the action is locally moving.
\end{proof}
\subsubsection{Actions of solvable groups} \label{ssc.solvable}
We conclude this section with the following application of Proposition \ref{prop.maximal}, which shows that $\mathbb{R}$-focal actions appear naturally in the context of solvable groups. Its derivation from Proposition \ref{prop.maximal} uses ideas of Rivas and Tessera in \cite{Rivas-Tessera}. By an affine action of $G$ on $\mathbb{R}$ we mean an action by affine transformations, i.e.\ transformations of the form $x\mapsto ax+b$ with $a>0$ and $b\in \mathbb{R}$.
\begin{thm}[Alternative for solvable groups]\label{thm.notaffine} Let $G$ be a finitely generated solvable group and let $\varphi:G\to\homeo_0(\mathbb{R})$ be an action without fixed points. Then either $\varphi$ is semi-conjugate to an affine action, or it is $\mathbb{R}$-focal.
\end{thm}
\begin{proof} If $\varphi$ is semi-conjugate to a cyclic action then the first case holds. So up to semi-conjugacy we can assume that $\varphi$ is minimal; moreover upon replacing $G$ by a quotient we can suppose that it is faithful. Let $G^{(n)}$ denote the derived series of $G$, and consider $k\in\mathbb N$ so that $G^{(k)}\neq\{1\}$ and $G^{(k+1)}=\{1\}$. Then $H:=G^{(k)}$ is an abelian and normal subgroup of $G$. As $H$ is normal, we deduce that $\varphi(H)$ cannot have fixed points (otherwise we would get a fixed point for the action of $G$).
Assume first that $H$ is cyclic contained in the center of $G$. Then $\varphi(H)$ is conjugate to the cyclic group generated by a translation and the action $\varphi$ induces a minimal action on the circle $\mathbb{R}/\varphi(H)$. As $G$ is solvable and thus amenable, it preserves a Borel probability measure on the circle, which must be of total support for the action is minimal; after \cite[Proposition 1.1.1]{Navas-book}, we get that the action is conjugate to a minimal action by rotations (basically, this is the action defined by the rotation number homomorphism $\mathsf{rot}:G\to \mathbb S^1$). Therefore, $\varphi(G)$ is the lift of a group of rotations of the circle and thus conjugate to a group of translations, so that we get that the action is affine in this case.
Assume next that $\varphi(H)$ acts minimally. Take a non-trivial $h\in H$ and note as before that $\varphi(h)$ has no fixed points (otherwise we would get a fixed point for $\varphi(H)$). Thus we obtain a minimal action of $H$ on the circle $\mathbb{R}/\langle \varphi(h)\rangle$, and the argument for the previous case leads to the conclusion that $\varphi(H)$ is conjugate to a minimal group of translations. Then the set of invariant Radon measures for $\varphi(H)$ corresponds to the one-parameter family of positive multiples of the Lebesgue measure, and this family must be preserved by $\varphi(G)$, for $H$ is normal in $G$. By a standard argument, we deduce that $\varphi(G)$ is conjugate to a group of affine transformations (see for instance \cite{Plante}).
If neither of the previous cases hold, by Proposition \ref{prop.maximal} we get that $\varphi$ is $\mathbb{R}$-focal as desired. \qedhere
\end{proof}
Since in a minimal $\mathbb{R}$-focal action every element has fixed points (Lemma \ref{rem.fixedepoints}), we deduce the following result first obtained Guelman and Rivas in \cite{Guelman-Rivas}.
\begin{cor} Let $G$ be a finitely generated solvable group, and let $\varphi:G\to \homeo_0(\mathbb{R})$ be an action such that some element acts without fixed points. Then, the action $\varphi$ is semi-conjugate to an affine action.
\end{cor}
\section{Planar trees and horograded $\mathbb{R}$-focal actions}
\label{sec_focal_trees}
In this section we study in more detail $\mathbb{R}$-focal actions and reinterpret them in terms of actions on planar directed trees, which provides a visual formalism to study them. This will allow to define a natural invariant associated to an $\mathbb{R}$-focal action, called the \emph{focal germ representation}, which bears a lot of information about the action. It is also the natural setting to introduce the tightly related notion of \emph{horograding} of an $\mathbb{R}$-focal action of a group $G$ by another action of $G$, which allows to compute its focal germ representation. This notion will be important later to establish a relation between exotic actions of a class of a locally moving groups to their standard action.
\subsection{Preliminaries on trees}\label{ss.trees_preliminaries} We first make a digression to set some terminology about (real) \emph{trees}.
\subsubsection{Directed trees} Roughly speaking, a real tree is a space $\mathbb T$ obtained by gluing copies of the real line in such a way that no closed loops appear. We are specifically interested in \emph{directed} real trees, which are trees together with a preferred direction to infinity (equivalently a preferred end $\omega\in \partial \mathbb T)$.
Usually real trees are defined as metric spaces, but we adopt the point of view of Favre and Johnson \cite{Fav-Jon}, and introduce the following definition which does not refer to any metric or topology. (The equivalence with the more familiar metric notion will be discussed in \S \ref{subsec metric R-tree} below.)
\begin{dfn}\label{def d-trees}
A (non-metric) countably branching directed tree (hereinafter just \emph{directed tree}) is a partially ordered set (\emph{poset} for short) $(\mathbb T, \triangleleft)$ with the following properties.
\begin{enumerate}[label=(T\arabic*)]
\item \label{i-tree} For every $v\in \mathbb T$ the subset $\{u\in \mathbb T \colon v \trianglelefteq u\}$ is totally ordered and order-isomorphic to a half-line $[0, +\infty)$.
\item \label{i-directed} Every pair of points $v, u\in \mathbb T$ has a smallest common upper bound, denoted $v\wedge u$.
\item \label{i-chains-R} There exist a countable subset $\Sigma \subset \mathbb T$ such that for every distinct $u, v\in \mathbb T$ with $u \trianglelefteq v$ there exists $z\in \Sigma$ such that $u\trianglelefteq z\trianglelefteq v$.
\end{enumerate}
We say that a point $v$ is \emph{below} $u$ (or that $u$ is {\em above} $v$) if $v\neq u$ and $v\trianglelefteq u$, and write $v\triangleleft u$.
\end{dfn}
Condition \ref{i-chains-R} is a technical separability assumption. A first consequence of it is that all totally ordered subsets of $\mathbb T$ are isomorphic to subsets of the real line. More precisely we have the following lemma. (Recall that every totally ordered subset of a poset is contained in a maximal one, by Zorn's lemma.)
\begin{lem}\label{l.rays} Let $(\mathbb T, \triangleleft)$ be a directed tree.
If $\ell\subset \mathbb T$ is a maximal totally ordered subset, then it is order-isomorphic to either $[0, +\infty)$ or to $\mathbb{R}$.
If $\ell_1, \ell_2$ are two maximal totally ordered subsets, then either $\ell_1=\ell_2$ or $\ell_1\cap \ell_2$ is order-isomorphic to $[0, +\infty)$.
\end{lem}
\begin{proof}
First of all, observe that for every $v\in \ell$ we have $\{u\colon v\trianglelefteq u\} \subset \ell$, or otherwise $ \ell\cup \{u\colon v\trianglelefteq u\}$ would be a strictly larger totally ordered set contradicting maximality of $\ell$. Thus if $\ell$ has a minimum $v$, then we must have $\ell=\{v \colon v\trianglelefteq u\}\cong [0, +\infty)$ by \ref{i-tree}.
If not, then \ref{i-chains-R} implies that for every maximal totally ordered subset $\ell\subset \mathbb T$ contains a strictly decreasing sequence $(v_n)$ such that $\ell=\bigcup_n\{u\colon v_n\trianglelefteq u\}$. Since by \ref{i-tree}, every element in the union is isomorphic to $[0, +\infty)$ it follows that $\ell$ is isomorphic to $\mathbb{R}$.
Let $\ell_1$ and $\ell_2$ be two maximal totally ordered subsets. If $\ell_1\neq \ell_2$ then there exist $v_1\in \ell_1$ and $v_2\in \ell_2$ which are not comparable. Then $v_1\wedge v_2$ (which exists by \ref{i-directed}) is the smallest element in $\ell_1\cap \ell_2$ and it follows from \ref{i-tree} that $\ell_1\cap \ell_2=\{w: v_1\wedge v_2\trianglelefteq w\}\cong [0, +\infty)$. \qedhere
\end{proof}
\subsubsection{End-completion and boundary} From Lemma \ref{l.rays} we can introduce a notion of boundary for a directed tree.
\begin{dfn}
Given a directed tree $(\mathbb T,\triangleleft)$,
we define its \emph{end-completion} $(\overline {\mathbb T}, \triangleleft)$ as the poset obtained by adding points to $\mathbb T$ as follows. First we add a point $\omega$, called the \emph{focus}, which is the unique maximal point of $(\overline{\mathbb T}, \triangleleft)$. In addition, for each maximal totally ordered subset $\ell\subset \mathbb T$ without mininimum, we add a point $\xi\in \overline{\mathbb T}$ which satisfies $\xi\triangleleft v$ for every $v\in \ell\cup \{\omega\}$. The subset $\partial \mathbb T=\overline{\mathbb T}\setminus \mathbb T$ is called the \emph{boundary} of $\mathbb T$. We will also write $\partial^*\mathbb T=\partial \mathbb T \setminus \{\omega\}$.
\end{dfn}
We next introduce further terminology associated to the end-completion.
Note that any pair of points $x, y\in \overline{\mathbb T}$ still admits a unique smallest upper bound, which we continue to denote by $x\wedge y$, and we have $x\wedge y\in \mathbb T$ unless $x=\omega$ or $y=\omega$.
Given distinct points $u,v\in \overline{\mathbb T}$ we define the \emph{arc} between them as the subset
\[[u, v]=\left \{z\in\overline{\mathbb T}\colon u\trianglelefteq z\trianglelefteq (v\wedge u) \text{ or } v\trianglelefteq z \trianglelefteq (v\wedge u)\right \}.\]
and set $]u, v[=[u, v]\setminus \{u, v\}$. The subsets $[u, v[$ and $]u, v]$ are defined similarly.
A subset $Y\subseteq \overline{\mathbb T}$ is \emph{path-connected} if $[u,v]\subseteq Y$ for every $u, v\in Y$. Every subset of $\overline{\mathbb T}$ is a disjoint union of maximal path-connected subsets, called its \emph{path-components}.
Given $v\in {\mathbb T}$, we define the set $E_v$ of \emph{directions} at $v$ as the set of path-components of $\overline{\mathbb T}\setminus\{v\}$. The cardinal $|E_v|$ is called the \emph{degree} of $v$. Points of degree $1$ are called \emph{leaves}; these are precisely the minimal elements in $(\mathbb T, \triangleleft)$. We say that $v$ is \emph{regular} if $|E_v|=2$ and a \emph{branching point} if $|E_v|\ge 3$. The set of branching points of $\mathbb T$ will be denoted by $\Br(\mathbb T)$.
For $z\in \overline{\mathbb T}\setminus \{v\}$, we let $e_v(z)\in E_v$ denote the direction containing $z$. If $\omega$ is the focus, we let $E^-_v=E_v\setminus \{e_v(\omega)\}$ denote the set of \emph{directions below} $v$.
Finally, we denote by $U_v\subset \mathbb T$ the subset of points below $v$. The corresponding subset of the boundary $\partial U_v\subset \partial^*\mathbb T$ is called the \emph{shadow} of $v$. See Figure \ref{fig.directed_tree} for an illustration of these definitions.
\begin{figure}
\includegraphics[scale=1]{focal_tree-1.pdf}
\caption{A directed tree $(\mathbb T,\triangleleft)$ with focus $\omega$, and the corresponding defined objects: arc between two points (brown), directions (blue), smallest upper bound (blue), shadow (green).}\label{fig.directed_tree}
\end{figure}
\begin{rem}
Note that condition \ref{i-chains-R} implies that the subset $\Br(\mathbb T)$ is at most countable, and for every $v\in \Br(\mathbb T)$ the set of directions $E_v$ is also countable. Indeed if $\Sigma\subset \mathbb T$ is a countable subset as in \ref{i-chains-R} then every $v\in\Br(\mathbb T)$ has a lower bound in $\Sigma$. It then follows that every direction in $E^-_v$ contains a point in $\Sigma$, and hence $E_v$ is countable. In particular every branching point $v$ can be written as $v=z_1\wedge z_2$ for some $z_1, z_2\in \Sigma$, so that the subset $\Br(\mathbb T)$ is at most countable.
\end{rem}
\begin{rem}There is little loss of generality in assuming that $\mathbb T$ has no leaves. Indeed by removing all leaves we obtain a directed tree $\mathbb T'$ and the removed leaves now correspond to points in $\partial^* \mathbb T'$. For this reason we shall often restrict to directed trees without leaves (see also Remark \ref{r.focal_noleaves}).\end{rem}
\subsubsection{Horogradings}
\begin{dfn} \label{def horograding} Let $(\mathbb T, \triangleleft)$ be a directed tree. An increasing (respectively, decreasing) \emph{horograding} of $\mathbb T$ is an increasing (respectively, decreasing) map $\pi\colon \mathbb T\to\mathbb{R}$ such that for every $u, v\in \mathbb T$ with $u\triangleleft v$ the restriction of $\pi$ to the arc $[u, v]\subset \mathbb T$ is an order-preserving bijection onto the interval $[\pi(u), \pi(v)]$ (respectively $[\pi(v),\pi(u)]$). \end{dfn}
\begin{rem}Assume that $\pi$ is an increasing horograding. If we let $\omega\in \partial \mathbb T$ denote the focus of $\mathbb T$, and set $b=\sup \pi(\mathbb T)\in \mathbb{R}\cup\{+\infty\}$, then the definition implies that for every $v\in \mathbb T$ the restriction of $\pi$ to $[v, \omega[$ is an increasing bijection onto $[\pi(v), b)$. This is in fact an equivalent formulation of the definition. There is an analogous characterization for decreasing horogradings.
\end{rem}
\begin{dfn}\label{dfn.pi_complete} Assume that $\pi:\mathbb{T}\to \mathbb{R}$ is an increasing horograding and write $X=\pi(\mathbb T)$, with $a=\inf X$ and $b=\sup X$. Notice that $\pi$ naturally extends to the end-completion $(\overline{\mathbb{T}},\triangleleft)$, taking values in $\overline{X}=[a,b]$. We denote by $\overline{\pi}$ this extension and define the \emph{$\pi$-complete boundary} as the subset $$\partial^\ast_\pi\mathbb{T}:=\{\xi\in\partial^\ast\mathbb{T}:\overline{\pi}(\xi)=a\}.$$
The analogous definition can be given in the case of decreasing horograding.
\end{dfn}
The following proposition is yet another consequence of condition \ref{i-chains-R}.
\begin{prop}\label{prop existe horograding}
Every directed tree admits a horograding.
\end{prop}
\begin{proof}
Assume first that $\mathbb T$ has no leaves (that is, the poset $(\mathbb T, \triangleleft)$ has no minimal elements), and let $\omega\in \partial \mathbb T$ be its focus.
Let $\Sigma=\{v_n\colon n\in \mathbb N\}\subset \mathbb T$ be a countable collection as in \ref{i-chains-R}, and note that every element of $\mathbb T$ has a lower bound in $\Sigma$, so that $\mathbb T=\bigcup_{n\in \mathbb N} [v_n, \omega[$. We define $\pi$ inductively on each ray $[v_n,\omega[$. We begin by choosing arbitrarily a bijection $\pi\colon [v_0, \omega[\to [0, b)$ for some $b\in \mathbb{R}\cup\{+\infty\}$. Assume that $\pi$ has already been defined on $\bigcup_{j=0}^{n-1}[v_j,\omega[$. If $v_j\triangleleft v_n$ for some $j<n$, then $[v_n, \omega[\subset [v_j, \omega[$ so that $\pi$ has already been defined on $[v_n, \omega[$. Otherwise set
$w_n=\min_{\triangleleft}\{v_n\wedge v_j\colon j< n\}$.
Note that the points $v_n\wedge v_j$ are all contained in the totally ordered subset $[v_n, \omega[$ and thus the minimum is well defined; moreover the point $w_n$ satisfies $v_j\triangleleft w_n$ for some $j<n$ and thus $\pi$ has already been defined on the ray $[w_n, \omega[$. We define $\pi$ on $[v_n, w_n[$ by choosing arbitrarily an order-preserving bijection of $[v_n, w_n[$ onto an interval of the form $[x_n, \pi(w_n))$ for some $x_n<\pi(w_n)$. Proceeding in this way for every $n$ we obtain the desired horograding $\pi\colon \mathbb T\to \mathbb{R}$.
Now if $\mathbb T$ has leaves, let $\mathbb T'$ be the directed tree obtained by removing all leaves. By the previous construction we can find a horograding $\pi\colon \mathbb T'\to \mathbb{R}$, and we can do the construction so that its image is bounded below. Then we can extend $\pi$ to $\mathbb T$ by setting $\pi(v)=\inf \pi(u)$ where the infimum is taken over all $u\in \mathbb T'$ such that $v\triangleleft u$. \qedhere \end{proof}
\subsubsection{Metric and topologies on directed trees} \label{subsec metric R-tree} A \emph{metric $\mathbb{R}$-tree} is a metric space $(\mathbb T, d)$ such that every pair of points $u, v\in \mathbb T$ can be connected by a unique injective continuous arc $\gamma\colon [0, 1]\to \mathbb T$, and this arc can be chosen to be geodesic (i.e.\ an isometric embedding). This notion is well-studied in geometric group theory (though not exclusively); see for instance the survey of Bestvina \cite{Bestvina}. Let us clarify the connection between $\mathbb{R}$-trees and directed trees in the sense of Definition \ref{def d-trees}.
A directed tree as in Definition \ref{def d-trees} can always be endowed with a metric that makes it an $\mathbb{R}$-tree. Indeed assume that $(\mathbb T, \triangleleft)$ is a directed tree, and choose a horograding $\pi\colon \mathbb T\to X$ (see Proposition \ref{prop existe horograding}). We can then define a distance on $\mathbb T$ by the formula
\begin{equation}\label{eq.d_pi}d_\pi(u, v)=|\pi(u\wedge v)-\pi(u)|+|\pi(u\wedge v)-\pi(v)|.\end{equation}
This distance turns $\mathbb T$ into an $\mathbb{R}$-tree; see \cite{Fav-Jon}. We call a distance of this form a \emph{compatible $\mathbb{R}$-tree metric} on $(\mathbb T, \triangleleft)$, associated with the horograding $\pi$.
Conversely assume that $(\mathbb T, d)$ is a separable metric $\mathbb{R}$-tree with no leaves. Let $(\hat{\mathbb T}, d)$ be the metric completion of $\mathbb T$, and set $\partial_f\mathbb T:=\hat{\mathbb T}\setminus \mathbb T$ (the set $\partial_f\mathbb T$ consists of leaves of $\hat{\mathbb T}$). Further let $\partial_\infty {\mathbb T}$ be the \emph{Gromov} (or \emph{visual}) \emph{boundary} of $\hat{\mathbb T}$, namely $\partial_\infty\mathbb T$ is the set of equivalence classes of geodesic rays in $\hat{\mathbb T}$, where a geodesic ray is a subset $\rho\subset \mathbb T$ isometric to $[0, +\infty)$, and two rays $\rho_1, \rho_2$ are equivalent if $\rho_1\cap \rho_2$ is a geodesic ray.
We define the end-boundary of $\mathbb T$ as the set $\partial \mathbb T=\partial_f \mathbb T \sqcup \partial_\infty \mathbb T$. The choice of a point $\omega\in \partial \mathbb T$ defines naturally a partial order $\triangleleft$ on $\mathbb T$, by the condition that $v\triangleleft u$ when $u\in [v, \omega[$, where $[v, \omega[$ is the ray from $v$ to $\omega$.
Moreover the choice of a point $z_0\in \mathbb T$ allows to define a \emph{horofunction} $\pi_{\omega, z_0}\colon \mathbb T \to \mathbb{R}$ centered at $\omega$ (normalized to vanish on $z_0$), by the formula
\[\pi_{\omega, z_0}(v)= d(z_0, v\wedge z_0)-d(v, v\wedge z_0).\]
This function is a horograding in the sense of Definition \ref{def horograding}, and commonly the subsets $\{v\in \mathbb T \colon \pi_{\omega,z_0}(v)=n\}$ are called \emph{horospheres}, see for instance \cite[Chapter II.8]{BridsonHaefliger}.
One can readily check that these two constructions are inverse to each other. Note however that the partition of the boundary $\partial \mathbb T=\partial_f\mathbb T\sqcup \partial_\infty \mathbb T$ appearing above depends on the choice of a compatible $\mathbb{R}$-tree metric and cannot be reconstructed from the poset structure of $(\mathbb T, \triangleleft)$. Notice also that, in the case that $\partial_\infty\mathbb{T}$ is non-empty, it holds that $\partial_\infty\mathbb{T}\cap\partial^\ast\mathbb{T}=\partial^\ast_\pi\mathbb{T}$. That is, up to possibly the focus, the Gromov boundary of $\mathbb{T}$ and the $\pi$-complete boundary coincide.
\begin{rem} We note that every directed tree $(\mathbb T, \triangleleft)$ the set $\mathbb T$ can also be endowed with an intrinsic topology (which does not depend on the choice of a horograding), called the \emph{weak topology} in \cite{Fav-Jon} (or \emph{observers' topology} in Coulbois, Hilion, and Lustig \cite{CHL}). A subbasis of open subsets for the weak topology is given by the set of directions $e_v(w)$ for $v\in \mathbb T$ and $w\in {\mathbb T}$. This topology turns the end-completion $\overline{\mathbb T}=\mathbb T\sqcup \partial \mathbb T$ into a compact space \cite{CHL}.
\end{rem}
\subsubsection{Case of simplicial trees} \label{ssc.simplicial}
A simple special case of directed trees are directed \emph{simplicial} trees. A simplicial tree is a connected graph $\mathbb T$ without cycles. Its boundary $\partial \mathbb T$ is defined as its set of ends (or equivalently as its Gromov boundary with respect to the simplicial metric). Assume that $\mathbb T$ is a simplicial tree of countable degree (that is, the degree at every point is at most countable) and infinite diameter. Then the choice of a point $\omega\in\partial \mathbb T$ turns it into a directed tree, with respect to the partial order $u\triangleleft v$ if $v$ lies on the ray from $u$ to $\omega$ in $\mathbb T$. We will say that a directed tree $(\mathbb T, \triangleleft)$ is \emph{simplicial} if it arises in this way from a simplicial tree. Note that in this case the set of directions $E_v^-$ below a vertex $v$ can be identified with the set of edges at $v$ which are below $v$.
\begin{rem} Note that simplicial trees, admit natural horogradings mapping vertices to integers. Moreover, for such horogradings, the boundaries $\partial^\ast\mathbb{T}$ and $\partial^\ast_\pi\mathbb{T}$ coincide.
\end{rem}
\subsubsection{Planar directed trees}
\begin{dfn}
Let $(\mathbb T, \triangleleft)$ be a directed tree without leaves, and denote by $\omega\in \partial \mathbb T$ its focus.
A \emph{planar order} on $(\mathbb T, \triangleleft)$ is a collection $\prec=\{<^v\colon v\in \Br(\mathbb T) \}$ of total orders defined on the set $E_v^{-}$ for each branching point $v\in \Br(\mathbb T)$. The triple $(\mathbb T, \triangleleft, \prec)$ will be called a \emph{planar directed tree}.
\end{dfn}
A planar order induces a total order on $\partial^\ast \mathbb T=\partial \mathbb T\setminus\{\omega\}$, that we will still denote by $\prec$, as follows. Given two distinct ends $\xi_1,\xi_2\in \partial^* \mathbb T $, write $v=\xi_1 \wedge \xi_2$, and note that this gives $e_v(\xi_1)\neq e_v(\xi_2)$ in $E^-_v$. We set $\xi_1\prec \xi_2$ if $e_v(\xi_1)<^v e_v(\xi_2)$. It is straightforward to check that this defines an order on $\partial^\ast \mathbb T$.
Moreover, we say that the planar order $\prec$ is \emph{proper} if for every $v\in \mathbb T$ the shadow $\partial U_v$ is bounded above and below in $(\partial^*\mathbb T, \prec)$.
\begin{rem}\label{r.shadow_convex}
Note that for every planar order $\prec$ on $(\mathbb T,\triangleleft)$ and $v\in \mathbb T$, the shadow $\partial U_v$ is a convex subset of $(\partial^*\mathbb T,\prec)$.
\end{rem}
\subsubsection{Automorphisms and focal actions} Given a directed tree $(\mathbb T, \triangleleft)$ we denote by $\Aut(\mathbb T, \triangleleft)$ the group of its order-preserving bijections.
We will be particularly interested in the following class of subgroups of automorphisms.
\begin{dfn} \label{d-focal-action-tree}
A subgroup $G\subseteq \Aut(\mathbb T, \triangleleft)$ is \emph{focal} if for every $u, v\in \mathbb T$ there exists an element $g\in G$ such that $v \triangleleft g(u)$.
\end{dfn}
\begin{rem}\label{r.focal_noleaves}
Note that if a directed tree $(\mathbb T,\triangleleft)$ admits a focal action, then it has no leaves. As an equivalent definition, a subgroup $G\subseteq \Aut(\mathbb T, \triangleleft)$ is focal if for every $v\in \mathbb T$ there exists a sequence $(g_n)\subset G$ such that $g_n(v)$ tends to $\omega$ along the ray $[v, \omega[$.
\end{rem}
Given a planar order $\prec$ on $(\mathbb T,\triangleleft)$, we denote by $\Aut(\mathbb T, \triangleleft, \prec)$ the subgroup of $\Aut(\mathbb T,\triangleleft)$ which preserves the planar order $\prec$, meaning that for every element $g\in G$ the corresponding bijections between $E_v^-$ and $E_{g(v)}^-$ induce isomorphisms between $\prec_v$ and $\prec_{g(v)}$.
Note that the induced action of $\Aut(\mathbb T, \triangleleft, \prec)$ on $\partial^*\mathbb T$ preserves the total order $\prec$.
\begin{rem}\label{rem.planarexistence} A subgroup $G\subseteq\Aut(\mathbb T, \triangleleft)$ of automorphisms of a directed tree admits an invariant planar order if and only if for every $v\in \Br(\mathbb T)$ there is an order on $E^-_v$ which is invariant under $\stab_G(v)$. Indeed it is enough to choose such an order for $v$ in a system of representatives of $G$-orbits $B\subseteq\Br(\mathbb T)$, and then extend this collection to all $v\in \Br(\mathbb T)$ uniquely in a $G$-invariant way. \end{rem}
The terminology extends to group actions $\Phi:G\to \Aut(\mathbb T, \triangleleft)$. For instance, we say that the action $\Phi$ is focal if the image $\Phi(G)$ is a focal subgroup.
In the particular case when $(\mathbb T,\triangleleft)$ is simplicial (see \S\ref{ssc.simplicial}), we say that an action $\Phi:G\to \Aut(\mathbb T, \triangleleft)$ is \emph{simplicial} if it arises from a simplicial action of $G$ on $\mathbb T$ preserving the end $\omega\in \partial \mathbb T$.
Assume now that $G$ is a countable group and $\Phi\colon G\to \Aut(\mathbb T, \triangleleft, \prec)$ is an action on a planar directed tree. For every end $\xi\in \partial^*\mathbb T$ let us denote by $\mathcal{O}_\xi\subset \partial^*\mathbb T$ its $G$-orbit, which is thus an at most countable totally ordered set on which $G$ acts by order-preserving bijections. We can therefore consider the dynamical realization $\varphi_\xi\colon G\to \homeo_0(\mathbb{R})$ of the action of $G$ on $(\mathcal{O}_\xi, \prec)$ (see \S \ref{sec.dynreal} for details). We will see in the next subsection (Proposition \ref{prop.focalisminimal}) the dynamical realization $\varphi_{\xi}$ does not depend on $\xi$, up to positive conjugacy.
For this, we will need the following result.
\begin{lem}\label{l-focal-implies-proper}
Let $(\mathbb T, \triangleleft, \prec)$ be a planar directed tree with $|\partial^*\mathbb T|\ge 2$, and $\Phi\colon G \to \Aut(\mathbb T, \triangleleft, \prec)$ be a focal action of a group $G$. Then the following hold.
\begin{enumerate}
\item \label{i-focal-implies-proper} The planar order $\prec$ is proper.
\item \label{i-focal-implies-denselyordererd} For every $\xi \in \partial^*\mathbb T$, the orbit $\mathcal{O}_\xi$ is cofinal (that is, unbounded in both directions) and densely ordered (that is, for every $\eta_1, \eta_2 \in \mathcal{O}_\xi$ with $\eta_1\prec \eta_2$, there exists $\xi_0\in \mathcal{O}_\xi$ with $\eta_1\prec \xi_0\prec \eta_2$).
\end{enumerate}
\end{lem}
\begin{proof} We first prove
\eqref{i-focal-implies-proper}. Since $|\partial^*\mathbb T|\ge 2$, we can find two points $u, v\in \mathbb T$ such that neither is below the other, so that the shadows $\partial U_u$ and $ \partial U_v$ are disjoint and $\prec$-convex (Remark \ref{r.shadow_convex}). Therefore at least one of them is bounded below and at least one of them is bounded above. As the action is focal, for every $z\in \mathbb T$, there exist elements $g,h\in G$ such that $g.\partial U_u=\partial U_{g.u}\supset \partial U_z$ and $h.\partial U_v= \partial U_{h.v}\supset \partial U_z$. It follows that the shadow $\partial U_z$ is bounded above and below, so that $\prec$ is proper.
We next prove \eqref{i-focal-implies-denselyordererd}. First of all, observe that by focality, we have $\mathcal{O}_\xi\cap \partial U_v\neq \varnothing$ for every $v\in \mathbb T$. Indeed it is enough to choose $u$ above $\xi$ and $g\in G$ such that $u\triangleleft g.v$, so that $\xi\in \partial U_{g.v}$, or equivalently $g^{-1}.\xi \in \partial U_v$. This immediately gives that the orbit $\mathcal{O}_\xi$ is cofinal in $(\partial^*\mathbb T,\prec)$. Let us show that it is densely ordered. Assume by contradiction that there is a pair $\xi_1\prec\xi_2$ with no elements of $\mathcal{O}_\xi$ between them, and let $g\in G$ be such that $\xi_2=g.\xi_1$. Since the action is order-preserving, applying $g^{-1}$ we deduce that the point $\xi_0=g^{-1}.\xi_1$ satisfies $\xi_0\prec \xi_1\prec \xi_2$, and there is no $\eta\in \mathcal{O}_\xi$ satisfying $\xi_0\prec \eta \prec \xi_1$ nor $\xi_1\prec \eta \prec \xi_2$. Now choose $v\in \mathbb T$ such that $\xi_1\in \partial U_v$ and $\xi_0, \xi_2\notin \partial U_v$. Since $\partial U_v$ is a $\prec$-convex subset, we deduce that $\partial U_v\cap \mathcal{O}_\xi=\{\xi_1\}$. But clearly we can find $u\in \mathbb T$ such that $|\partial U_u\cap \mathcal{O}_\xi|\ge 2$ (just take $u=\xi_1\wedge \xi_2$). This gives a contradiction since, by focality, there exists $h\in G$ so that $h.\partial U_u\subset \partial U_v$, and therefore $\{h.\xi_1,h.\xi_2\}\subset \partial U_v$. See Figure \ref{fig.focal_proper}.
\end{proof}
\begin{figure}[ht]
\includegraphics[scale=1]{focal_tree-2.pdf}
\caption{Proof of \eqref{i-focal-implies-denselyordererd} in Lemma \ref{l-focal-implies-proper}.}\label{fig.focal_proper}
\end{figure}
A more detailed description of the dynamics of a focal action will be given in \S\ref{s-dynclasselements}.
\subsection{Planar directed trees and $\mathbb{R}$-focal actions}\label{s-planar-trees}
The connection between focal and $\mathbb{R}$-focal actions is clarified by Propositions \ref{prop.focalisminimal} and \ref{p-from-focal-to-trees} below, which allow to make a transition from one to the other.
\begin{prop}[From planar directed trees to $\mathbb{R}$-focal actions] \label{prop.focalisminimal}
Let $\Phi\colon G \to \Aut(\mathbb T, \triangleleft, \prec)$ be a focal action of a countable group $G$ on a planar directed tree $(\mathbb T, \triangleleft, \prec)$ with $|\partial^*\mathbb T| \ge 2$. Then, for $\xi\in \partial^*\mathbb T$, the dynamical realization $\varphi_\xi\colon G\to \homeo_0(\mathbb{R})$ is a minimal $\mathbb{R}$-focal action, which does not depend on the choice of $\xi$ up to positive conjugacy.
\end{prop}
\begin{proof}
Consider points $\zeta_1\prec \xi_1\prec \xi_2 \prec \zeta_2$ of $\mathcal{O}_\xi$. By Lemma \ref{l-focal-implies-proper} we can choose $\eta\in \mathcal O_\xi$ such that $\xi_1\prec \eta \prec \xi_2$. Let $v,w\in \mathbb T$ be such that $\partial U_v$ contains $\{\zeta_1, \xi_1, \eta, \zeta_2, \xi_2\}$ and $\partial U_w$ separates $\eta$ from $\{\zeta_1, \xi_1, \zeta_2, \xi_2\}$ (one can take for instance $v=\zeta_1\wedge\zeta_2$ and $w\in ]\eta,\xi_1\wedge\xi_2[$). By focality, there exists $g\in G$ such that $g.\partial U_v\subset \partial U_w$. Since $\partial U_w$ is convex, this implies that $\xi_1\prec g(\zeta_1) \prec g(\zeta_2)\prec \xi_2$. It follows that $\varphi_\xi$ is minimal from Lemma \ref{lem.minimalitycriteria}.
To prove that the action $\varphi_\xi$ is $\mathbb{R}$-focal, we observe that the collection $\mathcal S=\{\partial U_v\cap \mathcal{O}_\xi \colon v\in \mathbb T\}$ is a CF-cover of $(\mathcal{O}_\xi, \prec)$ which is $\Psi(G)$-invariant. As in Example \ref{subsec.Plantefocal}, we get a CF-cover $\mathcal S_\xi$ of $\mathbb{R}$ which is $\varphi_{\xi}(G)$-invariant, by taking the interior of the closures of elements in $\mathcal S$ under a good embedding $\iota_\xi : \mathcal{O}_\xi\to \mathbb{R}$.
Now, given $\xi,\xi'\in \partial^*\mathbb T$, the map $f:\mathcal S_\xi\to \mathcal S_{\xi'}$ given by
\[\Int \overline{\iota_\xi(\partial U_v\cap \mathcal O_\xi)}\mapsto \Int \overline{\iota_{\xi'}(\partial U_v\cap \mathcal O_{\xi'})}\]
satisfies all conditions in Lemma \ref{l-focal-equivalence}, whence we deduce that $\varphi_{\xi}$ and $\varphi_{\xi'}$ are positively conjugate.
\end{proof}
After Proposition \ref{prop.focalisminimal}, there is no much ambiguity when saying that the action $\varphi_\xi:G\to \homeo_{0}(\mathbb{R})$ is the \emph{dynamical realization} of the focal action $\Phi\colon G \to \Aut(\mathbb T, \triangleleft, \prec)$. Given an action $\varphi\colon G\to \homeo_0(\mathbb{R})$, we will say that $\varphi$ is \emph{represented} by a focal action $\Phi\colon G\to \Aut(\mathbb T, \triangleleft, \prec)$ if $\varphi$ is positively conjugate to the dynamical realization of $\Phi$.
Conversely to Proposition \ref{prop.focalisminimal}, we have the following.
\begin{prop}[From $\mathbb{R}$-focal actions to planar directed trees] \label{p-from-focal-to-trees}
Let $G$ be a countable group and let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be a minimal $\mathbb{R}$-focal action. Then there exists a planar directed tree $(\mathbb T, \triangleleft, \prec)$ and a focal action $\Phi \colon G\to \Aut(\mathbb T, \triangleleft, \prec)$ representing $\varphi$.
\end{prop}
\begin{proof} Let $\mathcal{S}$ be an invariant CF-cover for the $\mathbb{R}$-focal action $\varphi\colon G\to \homeo_0(\mathbb{R})$. We want to define a planar directed tree starting from the poset $(\mathcal{S}, \subset)$. However, for general $\mathcal{S}$, none of the properties \ref{i-tree}--\ref{i-chains-R} for a directed tree is satisfied, and we will need to consider a completion of $(\mathcal{S}, \subset)$.
Let us start with some preliminary considerations on the structure of $(\mathcal{S}, \subset)$.
Firstly, from the cross-free property we have that for every $I\in \mathcal{S}$, the subset $\{J\in \mathcal{S} \colon J\supseteq I \}$ is totally ordered, while from condition \ref{ii.CF}, we see that the collection $\mathcal{S}$ cannot have a maximal element. This is in the direction of condition \ref{i-tree}, although we cannot guarantee that $\{J\in \mathcal{S} \colon J\supseteq I \}$ is order-isomorphic to $[0,+\infty)$.
To ensure a property analogue to \ref{i-directed}, we will assume that $\mathcal{S}$ is closed, in the sense that the set of pairs of points defined by intervals in $\mathcal{S}$ is a closed subset of $\mathbb{R}^2\setminus \mathsf{diag}$. There is no loss of generality in assuming this, as the closure of $\mathcal{S}$ in the above sense is still an invariant CF-cover.
In particular, this implies that any $I, J\in \mathcal{S}$ have a smallest common upper bound in $\mathcal{S}$.
We next observe that minimality of the action ensures that $(\mathcal S,\subset)$ basically has no leaves (and this is a necessary condition to define a planar order). Namely, we have that
every maximal totally ordered subset $\mathcal{C}\subset \mathcal{S}$ must contain intervals of arbitrarily small length.
Indeed, if the length of every interval in $\mathcal C$ is uniformly lower-bounded, the intersection $L=\bigcap_{I\in \mathcal C}I$ is an interval of non-empty interior. However, as we are assuming that $\varphi$ is minimal, then it is proximal (Proposition \ref{prop.minimalimpliesfocal}), so any interval of $\mathcal S$ can be mapped by some element $g\in G$ into $L$, and this contradicts maximality of $\mathcal C$, because of the invariance of $\mathcal S$.
In particular, every maximal totally ordered set $\mathcal{C}$ defines a point $x_\mathcal{C}\in \mathbb{R}$, as the unique point in the intersection:
\begin{equation} \label{e-chain-defines-point} \{x_\mathcal{C}\}:=\bigcap_{I\in \mathcal{C}} \overline{I}.\end{equation}
Note that an element $I\in \mathcal{S}$ belongs to $\mathcal{C}$ if and only if $x_\mathcal{C}\in I$. In particular if we choose a countable collection $(\mathcal{C}_n)$ of maximal totally ordered subsets such that the sequence $(x_{\mathcal{C}_n})$ is dense in $\mathbb{R}$ then $\mathcal{S}=\bigcup_n \mathcal{C}_n$ (by minimality of the $G$-action on $\mathbb{R}$, we may for instance choose the $G$-orbit of some given $\mathcal{C}$). This observation will guarantee condition \ref{i-chains-R}.
Let us now proceed to introduce a good completion. Consider the map $\pi\colon \mathcal{S}\to (0,+\infty)$, that associates to an interval $I$ its length $|I|$. Then $\pi$ is clearly increasing, and maps every maximal totally ordered subset bijectively onto a closed subset of $(0, +\infty)$ (the fact that the image is closed follows from the fact that we are assuming the collection $\mathcal S$ closed).
Call a pair of distinct intervals $(I, J)\in \mathcal{S}^2\setminus \mathsf{diag}$ a \emph{gap} if $I\subset J$ and there exists no $L\in \mathcal{S}\setminus \{I,J\}$ such that $I\subset L \subset J$. Let $\mathcal{G}\subset\mathcal{S}^2\setminus \mathsf{diag}$ be the set of gaps. For each $(I, J)\in \mathcal{G}$ let $L_{(I, J)}$ be a homeomorphic copy of the interval $(\pi(I), \pi(J))$. As a set, we define a directed tree $\mathbb T$ as the disjoint union
\[\mathbb T= \mathcal{S} \sqcup \left(\bigsqcup_{(I, J) \in \mathcal{G}} L_{(I, J)}\right).\]
The partial order $\subset$ on $\mathcal{S}$ extends in a natural way to a partial order $\triangleleft$ on $\mathbb T$, by declaring each $L_{I, J}$ to be an order-isomorphic copy of $(\pi(I), \pi(J))$ lying between $I$ and $J$, and extend the map $\pi\colon \mathcal S\to (0, +\infty)$ to $\mathbb T$ by in the obvious way. After our preliminary discussion, we are sure that the poset $(\mathbb T, \triangleleft)$ is a directed tree.
The action of $G$ on $\mathcal{S}$ extends to $\mathbb T$ by letting each $g\in G$ map $L_{I, J}$ affinely onto $L_{g(I), g(J)}$, and this action is order-preserving. The fact that it is focal follows from the assumption that the action of $G$ on $\mathbb{R}$ is minimal and thus proximal (Proposition \ref{prop.minimalimpliesfocal}).
We next introduce a planar order $\prec$ on $(\mathbb T,\triangleleft)$ which is preserved by the action of $G$. Note first that by construction all $v\in \mathbb T \setminus \mathcal{S}$ have degree 2. Thus every branching point $v\in \mathbb T$ is of the form $v=J\in \mathcal{S}$, and the set $E^-_v$ of components below $v$ is in one-to-one correspondence with gaps of the form $(I, J)$. Let $e_1,e_2\in E^-_v$ be two distinct components below $v=J$ and let $I_1,I_2\in \mathcal S$ be the intervals defining the corresponding gaps with $J$. Note that $I_1$ and $I_2$ are necessarily disjoint, so we can declare $e_1<^{v} e_2$ if $\sup I_1\leq \inf I_2$.
The family $\{<^{v} \colon v\in \Br(\mathbb T)\}$ defines a $G$-invariant planar order $\prec$ on $(\mathbb T,\triangleleft)$.
Finally note that every maximal totally ordered subset $\mathcal{C}\subset \mathbb T$ defines a point $\xi_{\mathcal{\mathcal{C}}}\in \partial^*\mathbb T$, represented by any decreasing proper ray taking values in $\mathcal{C}$. By construction, the $G$-orbit of $\xi_{\mathcal{C}}$ can be identified with the orbit of the point $x_{\mathcal{C}}\in \mathbb{R}$ defined by \eqref{e-chain-defines-point}, in an order-preserving way. Thus $\varphi$ is conjugate to the dynamical realization of the action on $(\mathcal{O}_{\xi_{\mathcal{C}}}, \prec)$. This concludes the proof. \qedhere
\end{proof}
\begin{rem}\label{r.representations_focal}
The proof of Proposition \ref{p-from-focal-to-trees} provides a simple canonical way to associate a planar directed tree with every invariant CF-cover $\mathcal{S}$ of $\mathbb{R}$. However we point out that the planar directed tree satisfying the conclusion of the proposition is not unique, and this general construction yields a tree which is often ``too large''. In practice it is useful to find smaller tree encoding the action having some additional structure (such as a $G$-equivariant horograding, or a $G$-invariant compatible $\mathbb{R}$-tree metric, etc.) In most cases of our interest, a more direct construction having such additional structure will be available.
Let us just observe here that the action of $G$ on the planar directed tree $\mathbb T$ constructed in the proof is continuous with respect to a compatible $\mathbb{R}$-tree metric on $\mathbb T$, which can be chosen to be complete.
Indeed, the map $\pi \colon \mathbb T\to (0, +\infty)$ appearing in the proof is a horograding of $(\mathbb T,\triangleleft)$. In particular it defines a compatible $\mathbb{R}$-tree metric $d_\pi$ on $\mathbb T$ by the expression \eqref{eq.d_pi}, and one can check the action of $G$ on $\mathbb T$ is continuous with respect to the topology defined by this metric. Moreover if we reparametrize $\pi$ by post-composing it by an orientation-preserving homeomorphism $H\colon (0, +\infty)\to \mathbb{R}$, then $\mathbb{R}$-tree metric associated with the new horograding is complete and induces the same topology (so that the $G$-action remains continuous).
\end{rem}
\begin{ex}[Planar directed trees for Plante actions]\label{ex.tree_Plante}
Let us illustrate how to get a focal action on a planar directed tree in the case of a Plante action of a wreath product. As in Example \ref{subsec.Plantefocal}, we consider two countable groups $G$ and $H$ and fix left-invariant orders $ <_G\in\LO(G)$ and $<_H\in \LO(H)$. We also take a good embedding $\iota:G\to \mathbb{R}$. For every $x\in \mathbb{R}$, consider the collection of functions
\[
\mathsf S_x=\left\{s:(x,+\infty)\to H\colon s(y)=1_H\text{ for all but finitely many $y$, in $\iota(G)$}\right\}.
\]
As a set, we define $\mathbb T$ as the disjoint union $\bigsqcup_{x\in \mathbb{R}}\mathsf S_x$. We declare that $s\trianglelefteq t$ if $s\in \mathsf S_x$, $t\in \mathsf S_y$, with $x\le y$ and $s\restriction_{(y,+\infty)}=t$. With this definition it is immediate to see that for every $s\in \mathsf S_x$, the subset $\{u:s\trianglelefteq u\}=\{s\restriction_{(y,+\infty)}:y\ge x\}$ is order-isomorphic to $[x,+\infty)$, giving \ref{i-tree}. To verify \ref{i-directed}, given $s\in \mathsf S_x$ and $t\in \mathsf S_y$, the restriction $s\restriction_{(-\infty,x_*)}$ with $x_*=\max\{z:s(z)\neq t(z)\}$ gives the desired smallest common upper bound. Finally, we have that the collection $\Sigma=\bigsqcup_{x\in \iota(G)}\mathsf S_x$ is countable, as $G$ and $H$ are countable, and this gives \ref{i-chains-R}. Note that $\Sigma$ coincides with the collection of branching points $\Br(\mathbb T)$. For $x\in \iota(G)$ and $s\in \mathsf S_x\subset\Br(\mathbb T)$, we note that $E^-_s$ is in one-to-one correspondence with $H$, the identification being given by the value of $s$ at $x$; this allows to put on $E^-_s$ the total order coming from $<_H$. This defines a planar order $\prec$ on $(\mathbb T,\triangleleft)$.
A horograding $\pi:\mathbb T\to \mathbb{R}$ is simply given by $\pi(s)=x$, where $x\in \mathbb{R}$ is such that $s\in \mathsf S_x$.
In order to describe the boundaries $\partial^*\mathbb T$ and $\partial^\ast_\pi\mathbb T$, set $$\mathsf{S}=\{s:(x,+\infty)\to H:s\upharpoonright_{(y,+\infty)}\in\mathsf{S}_y\text{ for every }y>x\}$$ and notice that the partial order $\triangleleft$ naturally extends to $\mathsf{S}$. Denote by $\mathsf{S}^\ast\subseteq\mathsf{S}$ the subset of the functions which are minimal for the relation $\triangleleft$. That is, an element $s\in\mathsf{S}$ is in $\mathsf{S}^\ast$ if and only if, either $s$ is defined over $\mathbb{R}$ or the support of $s$ is infinite. There is a correspondence between $\mathsf{S}^\ast$ and $\partial^\ast\mathbb{T}$ so that, the function $s:(x,+\infty)\to H$ corresponds to the equivalence class of the ray $y\mapsto s\upharpoonright_{(y,+\infty)}$. Under this correspondence, the $\pi$-complete boundary $\partial^\ast_\pi\mathbb{T}$ is identified with those functions in $\mathsf{S}^\ast$ whose domain is $\mathbb{R}$. In particular, we can see
$\bigoplus_GH$ as a subset of $\partial^*_\pi\mathbb T$ (a function $\s:G\to H$ can be seen as a function $\s:\mathbb{R}\to H$ with support in $\iota(G)$). Notice that the order induced on $\bigoplus_GH$ by $\prec$ under this identification coincides with that introduced in Example \ref{subsec.Plantefocal}.
The wreath product $H\wr G$ acts on $(\mathbb T,\triangleleft,\prec)$. Indeed, every $g\in G$ sends $\mathsf S_x$ to $\mathsf S_{g.x}$ by pre-composition with $g^{-1}$, where the action on $\mathbb{R}$ is the dynamical realization of $<_G$ corresponding to the good embedding $\iota$. If $\s\in \bigoplus_GH$, then we can see it as a function defined on $\mathbb{R}$ as before, and thus $\s$ acts on $\mathbb T$ by pointwise multiplication of functions. It is direct to check that this action preserves the poset structure $\triangleleft$ and the planar order $\prec$.
Note that this action naturally extends to $\mathsf{S}^\ast$, giving the action of $H\wr G$ on $\partial^\ast\mathbb T$ through the correspondence between $\mathsf{S}^\ast$ and $\partial^\ast\mathbb{T}$. Moreover, the $\pi$-complete boundary $\partial^\ast_\pi\mathbb T$ and (the copy of) $\bigoplus_GH$ are invariant subsets for this action. Indeed, the restriction to $\bigoplus_GH$ coincides with the action $\Psi$ (discussed in Example \ref{subsec.Plantefocal}). In other terms, we constructed an order-preserving and equivariant ``embedding'' of the Plante product $\Psi:H\wr G\to\Aut\left (\bigoplus_GH,\prec\right)$ defined in Example \ref{subsec.Plantefocal}, into the restriction to the boundary of an action on a planar directed tree.
Finally, note that the horograding $\pi$ satisfies $\pi(g.s)=g.\pi(s)$ for every $g\in G$ and $s\in \mathbb T$, while it is constant on $\bigoplus_GH$-orbits. In the terminology that will be introduced later in \S\ref{ss.horograding}, the action of $H\wr G$ on $(\mathbb T,\triangleleft)$ is increasingly horograded by the action $j:H\wr G\to\homeo_0(\mathbb{R})$ defined as $j=j_0\circ pr_G$, where $j_0$ is the dynamical realization of $<_G$ and $pr_G:H\wr G\to G$ is the projection to the factor $G$. See Figure \ref{fig.Plante-2}.
\begin{figure}[ht]
\includegraphics[scale=1]{focal_tree-3.pdf}
\caption{The directed tree for the Plante action (Example \ref{ex.tree_Plante}).}\label{fig.Plante-2}
\end{figure}
\end{ex}
\subsection{Focal germ representation and dynamical classification of elements} \label{s-dynclasselements}
In this section we will show that $\mathbb{R}$-focal actions admit a dynamical classification of elements which closely resembles that of isometries of trees into hyperbolic and elliptic elements. Moreover the type of every element can be determined by an invariant called the \emph{focal germ representation}. Let us begin by defining this invariant.
\begin{dfn} Let $(\mathbb T, \triangleleft)$ be a directed tree with focus $\omega\in \partial \mathbb T$. Choose also an increasing horograding $\pi\colon \mathbb T\to \mathbb{R}$, and set $b=\sup \pi(\mathbb T)$. Then for every $v\in \mathbb T$ the ray $[v, \omega[$ can be identified with the interval $[\pi(v), b)$. Let $g\in \Aut(\mathbb T, \triangleleft)$ be an automorphism. Since $g$ maps $[v, \omega[$ to $[g(v), \omega[$, it induces a homeomorphism between the intervals $[\pi(v), b)$ and $[\pi(g.v), b)$. Since for any other $v'$ the element $v \wedge v'$ belongs to $[v,\omega[$, we have that the left germ at $b$ of the homeomorphism induced by $g$ does not depend on the choice of $v\in \mathbb T$.
In this way we obtain a homomorphism from $\Aut(\mathbb T, \triangleleft)$ to the group of (left) germs at $b$:
\[\tau\colon \Aut(\mathbb T, \triangleleft)\rightarrow \Germ(b).\]
We call this homomorphism the \emph{focal germ representation} of $\Aut(\mathbb T, \triangleleft)$. When $\Phi:G\to \Aut(\mathbb T, \triangleleft)$ is a group action, we can consider the composition of $\Phi$ with the focal germ representation. We will still denote by $\tau \colon G\to \Germ(b)$ such composition.
\end{dfn}
\begin{rem}
Note that the focal germ representation $\tau\colon \Aut(\mathbb T, \triangleleft)\rightarrow \Germ(b)$ does not depend on the choice of the horograding $\pi$, up to conjugacy of germs.
\end{rem}
When $\varphi\colon G\to \homeo_0(\mathbb{R})$ is a minimal $\mathbb{R}$-focal action, we say that $\tau\colon G\to \Germ(b)$ is a focal germ representation \emph{associated to} $\varphi$ if it is conjugate to the focal germ representation of a focal action $\Phi\colon G\to \Aut(\mathbb T, \triangleleft)$ which represents $\varphi$. As discussed in Remark \ref{r.representations_focal}, a given $\mathbb{R}$-focal action can be represented $\varphi$ by a focal action on a planar directed tree in several ways, and this may lead to different (i.e.\ non-conjugate) focal germ representations associated to $\varphi$. However, we can introduce the following notion of semi-conjugacy of germs so that the focal germ representation of an $\mathbb{R}$-focal actions is well defined up to it.
\begin{dfn} \label{d-semiconjugacy-germs}
For $i\in\{1,2\}$, let $\tau_i:G\to \Germ(b_i)$ be two germ representations of a group $G$. We say that $\tau_1$ and $\tau_2$ are \emph{semi-conjugate} if there exist $\varepsilon_1,\varepsilon_2>0$ and a non-decreasing map $h:(b_1-\varepsilon_1,b_1)\to (b_2-\varepsilon_2,b_2)$ with $\lim_{t\to b_1}h(t)=b_2$ such that for every $g\in G$ one has $h\tau_1(g)=\tau_2(g) h$ at the level of germs.
\end{dfn}
\begin{lem}\label{l-focal-germ-well-defined}
Let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be a minimal $\mathbb{R}$-focal actions, and for $i\in \{1,2\}$, let $\tau_i\colon G\to \Germ(b_i)$ be a focal germ representation associated to $\varphi$. Then $\tau_1$ and $\tau_2$ are semi-conjugate.
\end{lem}
\begin{proof}
Suppose for simplicity that $b_1=b_2=:b$.
Consider focal actions on planar directed trees $\Phi_1:G\to \Aut(\mathbb T_1,\triangleleft_1,\prec_1)$ and $\Phi_2:G\to \Aut(\mathbb T_2,\triangleleft_2,\prec_2)$ representing $\varphi$.
For $i\in \{1,2\}$, choose an increasing horograding $\pi_i:\mathbb T_i\to (a_i, b)$ so that the focal germ representation associated to $\pi_i$ is $\tau_i$. Let us see how to construct the desired semi-conjugacy. For $i\in \{1,2\}$, fix $\xi_i\in \partial^*\mathbb T_i$ and let $\iota_i:\mathcal O_{\xi_i}\to \mathbb{R}$ be an order-preserving embedding such that the corresponding dynamical realization is exactly the action $\varphi$.
We first introduce a $G$-equivariant map $H:\mathbb{T}_1\to\mathbb{T}_2$, as follows:
for $v\in \mathbb T_1$, the image $H(v)$ is the smallest $w\in \mathbb T_2$ such that $\overline{\iota_2(\partial U_w\cap \mathcal O_{\xi_2})}$ contains $\iota_1(\partial U_v\cap \mathcal O_{\xi_1})$.
The map $H$ is clearly $G$-equivariant and monotone on maximal totally ordered subsets of $(\mathbb T_1,\triangleleft_1)$.
When restricted to a ray converging to the focus $\omega_1$ of $\mathbb{T}_1$, it gives a semi-conjugacy $h$ between $\tau_1$ and $\tau_2$. \qedhere
\end{proof}
The focal germ representation carries a huge amount of information about the dynamics of $\mathbb{R}$-focal actions and actions of planar trees. We now proceed to give a dictionary between the two.
We say that a germ $\gamma\in \Germ(b)$ \emph{has fixed points near $b$} if there is an increasing sequence $(z_n)$ converging to $b$ such that every representative of $\gamma$ fixes $z_n$ for $n$ large enough. A subgroup $\Gamma$ of $\Germ(b)$ has \emph{fixed points near} $b$ if there is such a sequence which works simultaneously for every element of $\Gamma$.
Note that if $\gamma\in \Germ(b)$ has no fixed points near $b$, then every representative $h$ of $\gamma$ satisfies that either $h(x)>x$ or $h(x)<x$ for all $x$ close enough to $b$. In that case we say that $\gamma$ is \emph{positive} or \emph{negative} respectively. Note that if $\tau\colon G\to \Germ(b)$ is a germ representation, then for $g\in G$ the type of $\tau(g)$ according to this classification is invariant under semi-conjugacy of $\tau$.
By extension to the case of the real line, we say that a group $\Gamma$ of automorphisms of an ordered set $(\Omega, \prec)$ is totally bounded if all of its orbits are bounded above and below with respect to the order $\prec$. Similarly an automorphism $g$ is totally bounded if $\langle g\rangle$ is. We say that $g$ is an expanding (respectively contracting) \emph{pseudohomothety} of $(\Omega, \prec)$ if there exists $\alpha \prec \beta$ such that $g^n(\alpha)\to -\infty$ and $g^n(\beta)\to +\infty$ in $(\Omega, \prec)$ as $n\to +\infty$ (respectively as $n\to -\infty)$. Finally we say that $g$ is an (expanding or contracting) homothety if it has a unique fixed point $\xi\in \Omega$ and the previous conclusion holds for every $\alpha \prec \xi \prec \beta$.
\begin{prop}[Dynamical classification of automorphisms of planar directed trees]\label{prop.dynclasselements0}
Let $(\mathbb T, \triangleleft, \prec)$ be a planar directed tree with $|\partial^*\mathbb T|\ge 2$. Let $\tau\colon \Aut(\mathbb T, \triangleleft, \prec)\to \Germ(b)$ be the associated focal germ representation. Then the following hold.
\begin{enumerate}
\item \label{i0-totally-bounded} if $\Gamma \subseteq\Aut(\mathbb T, \triangleleft, \prec) $ is a finitely generated subgroup such that $\tau(\Gamma)$ has fixed points near $b$, then the image of $\Gamma$ in $\Aut(\partial^* \mathbb T, \prec)$ is totally bounded. In particular every element $g\in \Aut(\mathbb T, \triangleleft, \prec)$ such that $\tau(g)$ has fixed points near $b$, gives a totally bounded automorphism of $(\partial^* \mathbb T, \prec)$.
\item If $g\in \Aut(\mathbb T, \triangleleft, \prec)$ is such that $\tau(b)$ has no fixed points near $b$, then $g$ defines a pseudohomothety of $(\partial^* \mathbb T, \prec)$, which is expanding if $\tau(b)$ is positive, and contracting otherwise. If moreover $g$ has no fixed points inside $\mathbb T$, then it is a homothety of $(\partial^*\mathbb T, \prec)$.
\end{enumerate}
\end{prop}
\begin{proof}
Note that by Lemma \ref{l-focal-implies-proper}, the planar order $\prec$ is proper.
Let $\pi\colon \mathbb T\to \mathbb{R}$ be the increasing horograding used to define the focal germ representation. Assume that $(z_n)$ is an increasing sequence converging to $b$ such that every germ in $\tau(\Gamma)$ eventually fixes $z_n$. Fix $v_0\in \mathbb T$ such that $\pi(v_0)=z_0$. Then we can choose an increasing sequence $(v_n)\subset [v_0, \omega[$ such that $\pi(v_n)=z_n$. Since $\Gamma$ is finitely generated, the assumption implies that it fixes $v_n$ for all $n$ large enough. Thus it preserves the sequence of shadows $\partial U_{v_n}$ which form an increasing exhaustion of $\partial^*\mathbb T$ by $\prec$-bounded subsets (because $\prec$ is proper). It follows that every $\Gamma$-orbit in $\partial^*\mathbb T$ is contained in of these subsets, and thus $\Gamma$ is totally bounded. This proves \eqref{i0-totally-bounded}.
Assume now that $g$ is such that $\tau(g)$ has no fixed points near $b$. Without loss of generality we can suppose that $\tau(g)$ is positive. This implies that we can find a point $v\in \mathbb T$ such that $g^n.v$ approaches $\omega$ along the ray $[v, \omega[$ as $n\to \infty$. Thus the sequence of shadows $\partial U_{g^n(v)}=g_n.\partial U_v$ is an increasing exhaustion of $(\partial^*\mathbb T, \prec)$ by $\prec$-convex subsets, which are bounded because $\prec$ is proper. This implies that $g$ is an expanding pseudohomothety.
Assume now that $g$ has no fixed point inside $\mathbb T$. Consider the sequence $\{g^n.v\colon {n\in \mathbb Z}\}$, where $v$ is as above. This is a totally ordered subset of $\mathbb T$, and thus it has an infimum $\xi\in \mathbb T\cup \partial^*\mathbb T$, which must be fixed by $g$ and so $\xi\in \partial^*\mathbb T$. Since $g^n.v$ must approach $\xi$ along the ray $]\xi, \omega[$ as $n\to -\infty$, the intersection of the subsets $g^n.\partial U_v$ for $n\in \mathbb Z$ is reduced to $\{\xi\}$. This implies that $g$ is a homothety with fixed point $\xi$.
\end{proof}
From Proposition \ref{prop.dynclasselements0}, we get the analogous result for $\mathbb{R}$-focal actions.
\begin{cor}[Dynamical classification of elements for $\mathbb{R}$-focal actions] \label{prop.dynclasselements}
Let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be a minimal $\mathbb{R}$-focal action, with associated focal germ representation $\tau\colon G\to \Germ(b)$. Then the following hold.
\begin{enumerate}
\item \label{i-classification-bounded} If $\Gamma \subseteq G$ is a finitely generated subgroup such that $\tau(\Gamma)$ has fixed points near $b$, then $\varphi(\Gamma)$ is totally bounded. In particular if $g\in G$ is an element such that $\tau(g)$ has fixed points near $b$, then $\varphi(g)$ is totally bounded.
\item \label{i-classification-homothety} If $g\in G$ is an element such that $\tau(g)$ has no fixed points near $b$, then $\varphi(g)$ is a pseudohomothety, which is expanding if $\tau(g)$ is positive, and contracting otherwise. \end{enumerate}
\end{cor}
\begin{proof}
Assume that $\varphi$ is represented by a focal action $\Phi:G\to \homeo(\mathbb T, \triangleleft, \prec)$. Note that if $\Phi$ represents $\varphi$, we necessarily have $|\partial^*\mathbb T|\ge 2$. Take $\xi\in \partial^\ast\mathbb T$, and up to conjugating $\varphi$, assume that $\varphi$ is the dynamical realization of the action induced by $\Phi$ on $(\mathcal O_\xi,\prec)$, and let $\iota \colon \mathcal{O}_{\xi}\to \mathbb{R}$ be the associated good embedding. Observe that as $\varphi$ is minimal, the image $\iota(\mathcal O_\xi)$ is dense.
Moreover, Lemma \ref{l-focal-implies-proper} guarantees that $\mathcal{O}_{\xi}$ is cofinal in $(\partial^\ast\mathbb T,\prec)$, whence \eqref{i-classification-bounded} and the first part of \eqref{i-classification-homothety} readily follow from Proposition \ref{prop.dynclasselements0}. \qedhere
\end{proof}
\subsection{Horograded $\mathbb{R}$-focal actions}\label{ss.horograding} After Proposition \ref{prop existe horograding}, every directed tree $(\mathbb T,\triangleleft)$ admits a horograding $\pi:\mathbb T \to \mathbb{R}$ (which, moreover, can be taken to be continuous with respect to some metric, see \S \ref{subsec metric R-tree}). When in addition we have a group action $\Phi:G\to \Aut(\mathbb T,\triangleleft)$, then the compatibility between the horograding and the action becomes relevant. For instance, if $\Phi(G)$ preserves the horospheres defined by $\pi$ (in the sense that $\pi(u)=\pi(v)$ implies that $\pi(\Phi(g)(u))=\pi(\Phi(g)(v))$ for all $g\in G$), then $G$ naturally acts on the image of $\pi$ simply by $g.\pi(v)=\pi(\Phi(g)(v))$. This motivates the following definition, which will play a crucial role in the sequel (see for instance Theorem \ref{t-C-trichotomy}).
\begin{dfn} Let $X=(a, b)$ be an interval, and let $j\colon G\to \homeo_0(X)$ be an action of a group $G$. We say that an action $\Phi\colon G\to \Aut(\mathbb T, \triangleleft)$ on a directed tree is increasingly (respectively, decreasingly) \emph{horograded} by $j$ if there exists an increasing (respectively, decreasing) $G$-equivariant surjective horograding $\pi\colon \mathbb T\to X$.
In addition, we say that a minimal $\mathbb{R}$-focal action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is horograded by $j$ if $\varphi$ can be represented by a focal action $\Phi\colon G\to \Aut(\mathbb T, \triangleleft, \prec)$ on a planar directed tree which is horograded by $j$.
\end{dfn}
The following remark serves as an easy example of the above definition.
\begin{rem} One can consider the more restrictive assumption that a group action $\Phi\colon G\to \Aut(\mathbb T, \triangleleft)$ on a directed tree preserves a compatible $\mathbb{R}$-tree metric on $\mathbb T$, in the sense of \S \ref{subsec metric R-tree}. From the discussion in \S \ref{subsec metric R-tree}, the reader may realize that this occurs if and only if the $\Phi$-action is horograded by an action by translations of $G$ on the line (the horograding is in this case a metric horofunction). Thus the existence of an equivariant horograding to a more general action can be seen as a relaxation of the existence of such an invariant metric.
\end{rem}
\begin{rem} A first class of examples of horograded $\mathbb{R}$-focal actions are the Plante actions of wreath products $H\wr G$ (Example \ref{ex.tree_Plante}) where the horograding was given by the dynamical realization of the order $<_G$ chosen on $G$. We will also see later with Theorem \ref{t-C-trichotomy}, that for a large class of locally moving groups, every exotic action can be horograded by its locally moving action.
\end{rem}
\begin{rem} Notice that every action $\Phi\colon G\to \Aut(\mathbb T, \triangleleft)$ can be extended to an action on the end-completion $(\overline{\mathbb{T}},\triangleleft)$. If in addition $\pi:\mathbb{T}\to X$ is a $G$-equivariant horograding, the $\pi$-complete boundary $\partial^\ast_\pi\overline{\mathbb{T}}$ is always $G$-invariant.
\end{rem}
If a minimal $\mathbb{R}$-focal action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is increasingly horograded by an action $j:G\to\homeo_0((a,b))$, then its associated focal germ representation can be identified by the representation $\tau\colon G\to \Germ(b)$ induced by the action $j$. In particular, in this situation we have the following.
\begin{prop}[Dynamical classification of elements in the horograded case]\label{prop.dynamic-class-horo}
For $X=(a,b)$, let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be a minimal $\mathbb{R}$-focal action which is increasingly horograded by an action $j\colon G\to \homeo_0(X)$. Then the following hold.
\begin{enumerate}[label=(\roman*)]
\item \label{i-totally-bounded} If $\Gamma\subseteq G$ is a finitely generated subgroup such that $j(\Gamma)$ has fixed points arbitrarily close to $b$, then $\varphi(\Gamma)$ is totally bounded.
\item \label{i-pseudo-homothety} If $g\in G$ is an element without fixed points in a neighborhood of $b$ then $\varphi(g)$ is a pseudohomothety which is expanding if $g(x)>x$ for every $x$ sufficiently close to $b$, and contracting otherwise. Moreover if $g$ has no fixed points in $X$ then $\varphi(g)$ is a homothety.
\end{enumerate}
\end{prop}
\begin{proof}
Choose an action on a planar directed tree $\Phi\colon G\to \Aut(\mathbb T, \triangleleft, \prec)$ which represents $\varphi$ and is increasingly horograded by $j$, with horograding $\pi\colon \mathbb T\to X$. Then the focal germ representation associated to $\Phi$ is conjugate to the germ representation $\mathcal{G}_b\colon G\to \Germ(G, b)$. So \ref{i-totally-bounded} and the first sentence in \ref{i-pseudo-homothety} are direct consequences of Corollary \ref{prop.dynclasselements}. Assume now that $g$ has no fixed point in $X$. Then $g$ has no fixed point in $\mathbb T$ (as the image of a fixed point under $\pi$ would be fixed in $X$). Thus by Proposition \ref{prop.dynclasselements0} $\Phi(g)$ is a homothety of $(\partial^*\mathbb T, \prec)$; let $\eta\in \partial^*\mathbb T$ be its unique fixed point. Since $\varphi$ is conjugate to the dynamical realisation of the $G$-action on the orbit $(\mathcal{O}_\eta, \prec)$ (Proposition \ref{prop.focalisminimal}), there is an order-preserving equivariant map $\iota\colon \mathcal{O}_\eta\to \mathbb{R}$ with dense image. Since $\phi(g)$ is a homothety on $(\mathcal{O}_\eta, \prec)$, this implies that $\varphi(g)$ is a homothety with unique fixed point $\iota(\eta)$. \qedhere
\end{proof}
\subsection{$\mathbb{R}$-focal actions arising from simplicial trees}\label{ssc.simplicial_actions}
In view of the correspondence between $\mathbb{R}$-focal action and actions on planar directed trees, perhaps the simplest type of $\mathbb{R}$-focal actions are those that can be represented by an action on a \emph{simplicial} planar directed tree (of countable degree) by simplicial automorphisms (see \S \ref{ssc.simplicial}). This is characterised by the following result, which is a partial converse to Proposition \ref{prop.maximal}. This result will be invoked in \S \ref{s-F-simplicial} when studying $\mathbb{R}$-focal actions of Thompson's group $F$.
\begin{prop} \label{p-focal-simplicial}
Let $G$ be a group not isomorphic to $\mathbb Z^2$ and let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be a faithful minimal action. Then the following are equivalent.
\begin{enumerate}[label=(\roman*)]
\item \label{i-focal-simplicial} $\varphi$ is $\mathbb{R}$-focal and can be represented by a focal action on a planar directed tree $(\mathbb T, \triangleleft, \prec)$, such that $\mathbb T$ is a simplicial tree of countable degree and the action of $G$ on $\mathbb T$ is by simplicial automorphisms.
\item \label{i-coZ} There exists a non-trivial normal subgroup $N\unlhd G$ such that $G/N\cong \mathbb Z$ and $\varphi(N)$ does not act minimally.
\end{enumerate}
\end{prop}
\begin{proof}
Let us prove that \ref{i-focal-simplicial} implies \ref{i-coZ}. Assume that $\varphi$ is the dynamical realization of an action on a planar directed simplicial tree $(\mathbb T, \triangleleft, \prec)$ by simplicial automorphisms. Let $\mathbb T_0\subset \mathbb T$ be the vertex set of $\mathbb T$, as simplicial complex. The focal germ representation associated with this action (see \S\ref{s-dynclasselements}) provides a homomorphism $\tau\colon G\to \mathbb Z$, which does not vanish precisely on the elements which act as hyperbolic isometries on $\mathbb T$ and it is given by the translation length along their axes. Let $N$ be its kernel. Fix also $v_0\in \mathbb T_0$ and consider the horofunction $\pi\colon \mathbb T_0\to \mathbb Z$ given by $\pi(v)=d_\mathbb T(v\wedge v_0, v_0)-d_\mathbb T(v\wedge v_0, v)$ where $d_\mathbb T$ is the simplicial distance. This function is $G$-equivariant (with respect to the action on $\mathbb Z$ given by $\tau$), so that the action of $N$ on $\mathbb T$ must preserve each horosphere $S_n:=\pi^{-1}(n)$, for $n\in \mathbb Z$. For $u, v\in S_n$, the shadows $\partial U_u, \partial U_v\subset \partial^*\mathbb T$ span two disjoint intervals $I_v, I_w\subset \mathbb{R}$. It follows that for every $n\in \mathbb Z$, $\varphi(N)$ preserves the open subset $\bigcup_{v\in S_n}I_v$ and thus it does not act minimally.
Let us now proceed to prove the converse statement.
The assumption that $G\neq \mathbb Z^2$ rules out that $N$ is a cyclic subgroup of the center of $G$. Hence by Proposition \ref{prop.maximal} the action is $\mathbb{R}$-focal. Let $\Lambda \subset \mathbb{R}$ be a proper closed $\varphi(N)$-invariant subset, and as in the proof of Proposition \ref{prop.maximal} we assume that for every connected component $U$ of $\mathbb{R}\setminus \Lambda$ we have $\operatorname{\mathsf{Fix}}^\varphi_U(H)=\varnothing$. Fix such a component $U$. The argument in the proof of Proposition \ref{prop.maximal} shows that the $G$-orbit of $U$ under $G$ defines a CF-cover $\mathcal{S}$ invariant under $\varphi(G)$.
Choose $f\in G$ which projects to a generator of $G/N\cong \mathbb Z$, so that $g=N\rtimes \langle f \rangle$. Since $\varphi$ is $\mathbb{R}$-focal the image of $f$ must have a fixed point $\xi$ (Lemma \ref{rem.fixedepoints}). Since moreover the action is minimal, we can choose $g\in G$ such that $g.\xi\in U$, so that upon replacing $f$ by $gfg^{-1}$ we can assume that $f$ has a fixed point in $U$, so that $f.U\cap U\neq \varnothing$. Note also that $f$ fixes no endpoint $\eta$ of $U$; indeed since $\eta\in \Lambda$ this would imply that for every $g\in G$, writing $g=hf^n$ with $h\in N$, we would have $g.\eta=h.\eta\in \Lambda$, contradicting that the $G$-orbit of $\eta$ is dense. Thus since the intervals $U$ and $f.U$ intersect non-trivially and cannot cross, we must either have or $U\Subset f.U$ or $f.U\Subset U$, and upon replacing $f$ by its inverse we assume that $U\Subset f.U$.
For $n\in \mathbb Z$ write $\mathcal{S}_n=\{hf^n.U \colon h\in N\}$, so that $\mathcal{S}=\bigcup_{n\in \mathbb Z}\mathcal{S}_n$. Note that every $V\in \mathcal{S}_n$ is a connected component of $\mathbb{R}\setminus \Lambda_n$, with $\Lambda_n:=f^n.\Lambda$, so that in particular every $V_1, V_2\in \mathcal{S}_n$ are either equal or disjoint. Moreover, every $V\in \mathcal{S}_n$ is contained in a unique $W\in \mathcal{S}_{n+1}$, indeed $V=hf^n.U\Subset hf^{n+1}.U$. It follows that the set $\mathcal{S}$ is naturally the vertex set of a directed simplicial tree $(\mathbb T, \triangleleft)$. The group $G$ acts on $\mathbb T$ by simplicial automorphisms. Moreover the $G$-action preserves the natural planar order $\prec=\{\prec^W \colon W\in \mathcal{S}\}$ on $\mathbb T$, where two edges $(V_1, W)$ and $(V_2, W)$ with $V_1,V_2\Subset W$ are ordered according to the order in which $V_1$ and $V_2$ appear in $W$. By construction $\varphi$ is conjugate to the dynamical realization of the action on $(\mathbb T, \triangleleft, \prec)$.
\end{proof}
\subsection{Constructing $\mathbb{R}$-focal actions from actions on ultrametric spaces}\label{subsec.ultra} We conclude this section by explaining a way to construct horograded $\mathbb{R}$-focal actions based on the a well-known correspondence between trees and \emph{ultrametric spaces} (see for instance \cite{choucroun,hughes}), which will be usfeul for certain examples. Recall that an ultrametric spaces is a metric space $(\mathcal{Z}, \delta)$ which satisfies the \emph{ultrametric inequality}, that is, for every $\xi_1,\xi_2,\xi_3\in\mathcal{Z}$ it holds that \[\delta(\xi_1, \xi_2)\leq \max\{\delta(\xi_1, \xi_3), \delta(\xi_3, \xi_2)\}.\]
The next definition is slightly more general:
\begin{dfn}\label{def.ultra} Let $X=(a,b)$ be an interval, and let $\mathcal Z$ be a set. A function $\delta:\mathcal{Z}\times\mathcal{Z}\to X\cup\{a\}$ is an \emph{ultrametric kernel} if it is symmetric, it satisfies the ultrametric inequality and one has $\delta(\xi_1,\xi_2)=a$ if and only if $\xi_1=\xi_2$. We say that it is unbounded if $\sup_{\xi_1, \xi_2\in \mathcal{Z}} \delta(\xi_1, \xi_2)=b$.
Given an ultrametric kernel, for $\xi\in\mathcal{Z}$ and $t\in X$, we write $B_\delta(\xi,t)=\{\xi'\in\mathcal{Z}:\delta(\xi,\xi')\leq t\}$. We refer to the sets of this form as \emph{$\delta$-balls}.
\end{dfn}
Equivalently, $\delta$ is an ultrametric kernel if and only if $H \circ \delta$ is an ultrametric distance on $\mathcal Z$ for whenever $H$ is a homeomorphism $H\colon [a, b)\to [0, +\infty)$.
Note that the ultrametric inequality implies that if $(\mathcal{Z}, \delta)$ is a set with a ultrametric kernel, then the $\delta$-balls form a cross-free family of subsets of $\mathcal{Z}$ in the sense of Definition \ref{d-CF-cover}.
\begin{rem}\label{r.ultrametric_horo} Consider a directed tree $(\mathbb{T}, \triangleleft)$ endowed with a horograding map $\pi\colon \mathbb T\to X$, where $X=(a, b)$. Recall that we write $\partial^*_\pi \mathbb T$ for the set of $\pi$-complete ends, i.e.\ the subset of ends $\xi\in \partial^*\mathbb T$ such that $\pi(]\xi, \omega[)=X$. Then the set $\partial^*_\pi \mathbb T$ admits an ultrametric kernel $\delta:\partial^*_\pi \mathbb T\times\partial^*_\pi \mathbb T\to [a, b)$ defined as $\delta(\xi_1,\xi_2)=\pi(\xi_1\wedge\xi_2)$ if $\xi_1\neq\xi_2$, and $\delta(\xi_1,\xi_2)=a$ if $\xi_1=\xi_2$. It is straightforward to check that $\delta$ is an ultrametric kernel. Moreover if $G$ is a group endowed with actions $\Phi\colon G\to \Aut(\mathbb T, \triangleleft)$ and $j\colon G\to \homeo_0(X)$ such that $\pi$ is equivariant, then the map $\delta$ is equivariant in the sense that $\delta(\Phi(g)(\xi_1), \Phi(g)(\xi_2))=j(g)(\delta(\xi_1, \xi_2))$. This remark motivates our previous and next definitions.
\end{rem}
The following construction is a special case of the well-known correspondence between real trees and ultrametric spaces.
\begin{dfn}\label{dfn.ultratree}
Assume that $(\mathcal{Z}, \delta)$ be a space with an unbounded ultrametric kernel $\delta\colon \mathcal{Z} \times \mathcal{Z}\to X$. We define a directed tree $(\mathbb T, \triangleleft)$, called the directed tree \emph{associated with} $(\mathcal{Z}, \delta)$, endowed with a horograding $\pi\colon \mathbb T\to X$ and an injective map $\iota \colon \mathcal{Z}\to \partial^*\mathbb T$ identifying $\mathcal{Z}$ with a subset of $\partial_\pi^*\mathbb T$, as follows.
As a set, the tree $\mathbb T$ is the quotient $\mathbb T=\mathcal{Z}\times X/\sim$ with respect to the equivalence relation
\[(\xi_1, t)\sim (\xi_2, s) \quad \text{if and only if} \quad t=s \text{ and } t\ge \delta(\xi_1, \xi_2) .\]
Note that by the ultrametric inequality, the condition $t\ge \delta(\xi_1,\xi_2)$ is equivalent to $B_\delta(\xi_1, t)=B_\delta(\xi_2, t)$, and from this one immediately sees that the above relation is an equivalence relation.
Denote by $p\colon \mathcal{Z}\times X\to \mathbb T$ the quotient map. The partial order $\triangleleft$ on $\mathbb{T}$ is defined by declaring $u\trianglelefteq v$ if for some (equivalently, for every) representatives of $u$ and $v$, say $u=p(\xi_1, s)$ and $v=p(\xi_2, t)$, it holds that $s\leq t$ and $B_\delta(\xi_1, s)\subset B_\delta(\xi_2, t)$ (equivalently that $s\leq t$ and $\delta(\xi_1, \xi_2)\le t$). With these definitions $(\mathbb T, \triangleleft)$ is a directed tree \cite[\S3.1.6]{Fav-Jon}. The map $\pi\colon \mathbb T\to X$, defined by $\pi(p(\xi_1, t))=t$ is an increasing horograding of $(\mathbb T, \triangleleft)$. Finally the set $\mathcal{Z}$ can be naturally identified with a subset of $\partial^*_\pi\mathbb T$, by the map $i:\mathcal{Z}\to\partial^\ast\mathbb{T}$, mapping each $\xi\in \mathcal{Z}$ to the unique infimum in $\partial^*\mathbb T$ of the ray $\{p(\xi, t)\colon t\in X\}$.
\end{dfn}
We now consider the previous situation in the presence of a group $G$ acting.
When $G$ is a group acting on both $\mathcal Z$ and $X$, by actions $\varphi:G\to\Bij(\mathcal{Z})$ and $j:G\to\homeo_0(X)$ respectively, we say that an ultrametric kernel $\delta:\mathcal Z\times \mathcal Z\to X\cup \{a\}$ is $G$-equivariant if for every $\xi_1,\xi_2\in\mathcal{Z}$ and $g\in G$ one has
\begin{equation}\label{eq:delta_equiv}
\delta\left (\varphi(g)(\xi_1),\varphi(g)(\xi_2)\right )=j(g)\left (\delta(\xi_1,\xi_2)\right ).\end{equation}
In what follows, to simplify notation, we will write as usual $g.\xi$ and $g.t$ instead of $\varphi(g)(\xi)$ and $j(g)(t)$, respectively. We place ourselves in the following setting.
\begin{assumption}\label{a.ultra} Let $G$ be a group, and consider actions $\varphi:G\to \Bij(\mathcal Z)$ on a countable set, and $j:G\to \homeo_{0}(X)$ on an open interval $X=(a,b)$. We let $\delta:\mathcal Z\times \mathcal Z\to X\cup \{a\}$ be an unbounded $G$-equivariant ultrametric kernel.
\end{assumption}
The equivariance relation \eqref{eq:delta_equiv} implies that for every $g\in G$, $\xi\in \mathcal Z$, and $t\in X$ one has
\begin{equation}\label{e-ball-equivariant}
g.B_\delta(\xi,t)=B_\delta(g.\xi,g.t).\end{equation}
In particular, the set of $\delta$-balls is $G$-invariant.
Under Assumption \ref{a.ultra}, we will say that the action of $G$ on $\mathcal{Z}$ \emph{expands $\delta$-balls} if for every ball $B_\delta(\xi, t)$ there exists a sequence of elements $(g_n)\subset G$ such that the sequence of balls $g_n.B_\delta(\xi, t)$ is an increasing exhaustion of $\mathcal{Z}$.
\begin{prop}\label{prop.ultraconstruction} Under Assumption \ref{a.ultra}, let $(\mathbb T, \triangleleft)$ be the directed tree associated with $(\mathcal{Z}, \delta)$. Then there exists an action $\Phi:G\to\Aut(\mathbb{T},\triangleleft)$ increasingly horograded by $j$, and such that the map $\iota\colon \mathcal{Z}\to \partial^*\mathbb T$ is $G$-equivariant. Moreover, the action $\Phi$ is focal if and only if the action of $G$ on $\mathcal{Z}$ expands $\delta$-balls.
\end{prop}
\begin{proof}
The diagonal action (induced by $\varphi$ and $j$) of $G$ on $\mathcal{Z}\times X$ descends to the quotient to an action on the directed tree $(\mathbb{T},\triangleleft)$. Thus, we get an action $\Phi:G\to\Aut(\mathbb{T},\triangleleft)$, given by $\Phi(g)(p(\xi,t))=p(g.\xi,g.t)$. It is evident from the definition of the tree $\mathbb T$ and from \eqref{e-ball-equivariant} that the horograding $\pi\colon \mathbb T\to X$ from Definition \ref{dfn.ultratree} is $G$-equivariant and thus, $\Phi$ is increasingly horograded by $j$. Moreover, notice that the natural map $i:\mathcal{Z}\to\partial^\ast\mathbb{T}$ defined previously is also $G$-equivariant with respect to the actions $\varphi$ and $\Phi$. Finally if the action of $G$ on $\mathcal{Z}$ expands $\delta$-balls then it follows from the construction of the tree $\mathbb T$ that for every $v\in \mathbb T$ there exists a sequence of elements $(g_n)$ such that $g_n.v$ tends to the focus $\omega$ along the ray $[v, \omega[$, which implies that $\Phi$ is a focal action.
\end{proof}
The next step is to introduce an invariant planar order on the tree $\mathbb T$, whenever the set $\mathcal Z$ is endowed with an appropriate total order.
\begin{dfn}\label{dfn.ultraconvex} Given an ultrametric kernel $\delta:\mathcal Z\times \mathcal Z\to X\cup \{a\}$, we say that a total order $<$ on $\mathcal{Z}$ is \emph{$\delta$-convex} if all $\delta$-balls are $<$-convex subsets.
\end{dfn}
\begin{prop}\label{prop.deltaconvex} Under Assumption \ref{a.ultra}, consider the action $\Phi:G\to\Aut(\mathbb{T},\triangleleft)$ on the directed tree associated with $(\mathcal{Z}, \delta)$. Consider also a $\varphi$-invariant total order $<$ on $\mathcal{Z}$.
Then, the order $<$ is induced by a $\Phi$-invariant planar order $\prec$ (via the map $i:\mathcal{Z}\to\partial^\ast\mathbb{T}$ defined in Proposition \ref{prop.ultraconstruction}) if and only if $<$ is $\delta$-convex.
\end{prop}
\begin{proof} Recall that we denote by $p:\mathcal{Z}\times X\to\mathbb{T}$ the quotient projection. Also, we denote by $\partial U_v$ the shadow in $\partial^\ast\mathbb{T}$ of a point $v\in\mathbb{T}$. Then, for every $\xi \in \mathcal Z$ and $t\in X$, we have \begin{equation}\label{eq.ultrashadow} i(B_\delta(\xi,t))=\partial U_{p(\xi,t)}\cap i(\mathcal{Z}).\end{equation}
Assume first that $<$ is induced by a $\Phi$-invariant planar order $\prec$ on $(\mathbb{T},\triangleleft)$ and take a $\delta$-ball $B_\delta(\xi,t)$ in $\mathcal{Z}$. Since shadows of vertices are $\prec$-convex and $<$ is induced by $\prec$, the equality \eqref{eq.ultrashadow} ensures that $B_\delta(\xi,t)$ is $<$-convex, as desired.
Conversely, assume that $<$ is $\delta$-convex. Let $v\in \Br(\mathbb T)$, and let us define an order $\prec^v$ on $E^-_v$. For this, let $e_1, e_2\in E^-_v$ be distinct directions in $E^-_v$. By construction of $(\mathbb{T},\triangleleft)$ there exist $\xi_1,\xi_2\in i(\mathcal{Z})\subseteq\partial^\ast\mathbb{T}$ with $e_v(\xi_i)=e_k$ for $k\in \{1,2\}$ (according to the notation in \S\ref{s-planar-trees}). Notice that in this case $\xi_1\wedge \xi_2=v$. We define $\prec^v$ on $E^-_v$ by setting $e_1\prec^v e_2$ if and only if $\xi_1< \xi_2$. To show that $\prec^v$ is well defined, consider different choices of representatives $\xi'_1$ and $\xi'_2$ for the corresponding directions. By definition, for $k\in \{1,2\}$ we have that $\{\xi_k,\xi'_k\}$ is contained in the $\delta$-ball $B_\delta(\xi_k,\delta(\xi_k,\xi'_k))$, which is $<$-convex by hypothesis. Thus, in order to show that $\prec^v$ is well defined, it is enough to show that $B_\delta(\xi_1,\delta(\xi_1,\xi'_1))\cap B_\delta(\xi_2,\delta(\xi_2,\xi'_2))=\emptyset$. For this, for $k\in \{1,2\}$ write $v_k=\xi_k\wedge \xi_k'$ and notice that, since $v_k\in e_k$ and $e_1\neq e_2$, we have $\partial U_{v_1}\cap \partial U_{v_2}=\emptyset$.
Finally, since for $k\in \{1,2\}$ one has $p(\xi_k,\delta(\xi_k,\xi_k'))=v_k$, the equality \eqref{eq.ultrashadow} implies that $i(B_\delta(\xi_k,\delta(\xi_k,\xi_k')))=\partial U_{v_k}\cap i(\mathcal{Z})$. This shows that $\prec^v$ is well defined. The fact that $<$ is induced by $\prec$ and that $\prec$ is $\Phi$-invariant is clear from the construction and the $\varphi$-invariance of $<$.
\end{proof}
We record the following consequence of the previous discussion.
\begin{cor}\label{cor.ultra} Let $X=(a,b)$ be an interval, $(\mathcal{Z}, <)$ a totally ordered countable set, and $G$ be a group together with two actions actions $\varphi:G\to\Aut(\mathcal{Z},<)$ and $j:G\to\homeo_0(X)$. Assume further that $\delta:\mathcal{Z}\times\mathcal{Z}\to X\cup \{a\}$ is a $G$-equivariant ultrametric kernel such that the order $<$ is $\delta$-convex and the action of $G$ on $\mathcal{Z}$ expands $\delta$-balls. Let $\psi\colon G\to \homeo_0(\mathbb{R})$ be the dynamical realization of $\varphi$. Then $\psi$ is a minimal $\mathbb{R}$-focal action, increasingly horograded by $j$.
\end{cor}
\begin{proof} Applying Proposition \ref{prop.ultraconstruction} we get an action of $G$ on a directed tree $(\mathbb T, \triangleleft)$ and a $G$-equivariant map $i:\mathcal{Z}\to\partial^\ast\mathbb{T}$. On the other hand, since $<$ is $\varphi$-invariant and $\delta$-convex, Proposition \ref{prop.deltaconvex} implies the existence of a $\Phi$-invariant planar order $\prec$ inducing $<$ on $i(\mathcal{Z})$. Thus, by the definition of dynamical realization of a focal action (notice that $\Phi$ is focal by hypothesis) we get that the dynamical realization of $\varphi$ and that of $\Phi$ coincide. Then, Proposition \ref{prop.focalisminimal} implies that the dynamical realization of $\varphi$ (and also of $\Phi$) is a minimal $\mathbb{R}$-focal action. Finally, since $\Phi$ is increasingly horograded by $j$ we conclude that the dynamical realization of $\varphi$ is increasingly horograded by $j$.
\end{proof}
\begin{ex}[Ultrametric kernels for Plante actions]\label{ex.ultra_Plante}
We revisit the example of the Plante product of two left-orders, already treated in Examples \ref{subsec.Plantefocal} and \ref{ex.tree_Plante},
but now with the perspective of ultrametric kernels. Recall that for given countable left-ordered groups $(G,<_G)$ and $(H,<_H)$, we defined their Plante product as an action on the ordered space $\varphi:H\wr G\to\Aut(\bigoplus_GH,\prec)$ of the wreath product $H\wr G = \bigoplus_GH\rtimes G$, where $\bigoplus_GH$ is the set of functions $\s:G\to H$ which are trival at all but finitely many elements of $G$. In the action $\varphi$, the subgroup $G$ acts by pre-composing with left translations, whilst $\bigoplus_GH$ acts by pointwise left-multiplication. We already explained in Example \ref{ex.tree_Plante} how to get a focal action on a planar directed tree $(\mathbb T,\triangleleft,\prec)$ representing the dynamical realization of the Plante product, and we observed at the end that the focal action is increasingly horograded by the dynamical realization of $<_G$. Here we will get the same conclusion, by simply introducing an ultrametric kernel (which actually corresponds to the ultrametric kernel for a horograded focal action defined in Remark \ref{r.ultrametric_horo}).
Consider a good embedding $\iota:G\to\mathbb{R}$ associated with the total order $<_G$. We define an ultrametric kernel $$\delta:\bigoplus_GH\times\bigoplus_GH\to\mathbb{R}\cup \{-\infty\}$$ as follows: when $\s\neq \mathsf t$, we define $\delta(\s,\mathsf t)=\iota(x_*)$ where $x_*=\max_{<_G}\{x\in G:\s(x)\neq \mathsf t(x)\}$, while in the remaining case, we set $\delta(\s,\s)=-\infty$ for every $\s\in\bigoplus_GH$. It is direct to check that $\delta$ is an ultrametric kernel. On the other hand, with arguments similar to those in Example \ref{subsec.Plantefocal} we get that $<$ is a $\delta$-convex total order. This ultrametric kernel $\delta$ is $H\wr G$-equivariant if we consider the action $\varphi$ and $j:H\wr G\to\homeo_0(\mathbb{R})$ defined as $j=j_0\circ pr_G$ where $pr_G:H\wr G\to G$ is the projection and $j_0$ is the dynamical realization of $<_G$ corresponding to the good embedding $\iota$.
We next verify that the action of $H\wr G$ on $\bigoplus_GH$ expands $\delta$-balls. By transitivity of the action, it is enough to consider $\delta$-balls centered at the trivial element $\mathsf e\in \bigoplus_GH$, and we can simply consider the action of $G$, which is exactly the stabilizer of $\mathsf e$. Thus for every $x\in \mathbb{R}$ and $g\in G$, we have that $g.B_\delta(\mathsf e,x)=B_\delta(\mathsf e,g.x)$. As the action of $G$ on $\mathbb{R}$ is given by the dynamical realization of $<_G$, for every $x\in \mathbb{R}$ we can find a sequence of elements $(g_n)\subset G$ such that $g_n.x\to \infty$ as $n\to \infty$. This proves that the action expands $\delta$-balls.
Finally, applying Corollary \ref{cor.ultra} we get that the Plante action of $H\wr G$ associated with $<_G$ and $<_H$ is $\mathbb{R}$-focal, increasingly horograded by $j$.
\end{ex}
\section{Introduction}
\subsection{Background and overview}
The study of group actions on one-manifolds is a classical topic at the interface of dynamical systems, topology, and group theory, which is still under intense development and has been the object of several monographs in the recent years, such as Ghys \cite{Ghys}, Navas \cite{Navas-book}, Clay and Rolfsen \cite{ClayRolfsen}, Deroin, Navas, and the third named author \cite{GOD}, Kim, Koberda, and Mj \cite{KKMj}; see also the surveys by Mann \cite{MannHandbook} and Navas \cite{NavasICM}. The ultimate goal would be, given a group $G$, to be able to describe all possible actions of $G$ on a one-manifold $M$ by homeomorphisms, and as the topology of the manifold is rather elementary one expects that this is more tractable than in higher dimension.
In this framework, there is no much loss of generality of considering the following simplifying assumptions: the manifold $M$
is connected (that is, either the real line or the circle), and the action of the group is \emph{irreducible}, in the sense that it preserves the orientation and has no (global) fixed points. We then let $\homeo_{0}(M)$ be the group of orientation-preserving homeomorphisms of $M$, and write $\Homirr(G,\homeo_{0}(M))$ for the space of irreducible actions of $G$ on $M$. Moreover, to avoid a redundant description, we want to identify two actions if one is obtained from the other with a change of variables (what is called a \emph{conjugacy}); more precisely, from a dynamical perspective, it is natural to study the space $\Homirr(G, \homeo_0(M))$ up to \emph{semi-conjugacy}. The definition of semi-conjugacy in this setting is recalled in Section \ref{s-preliminaries}; here let us simply remind that every action $\varphi\in \Homirr(G, \homeo_0(M))$ of a \emph{finitely generated} group is semi-conjugate either to a \emph{minimal} action (i.e.\ an action all whose orbits are dense) or $\varphi(G)$ has a discrete orbit, in which case $\varphi$ is semi-conjugate to \emph{cyclic} action (i.e.\ an action by integer translations in the case of the real line or by rational rotations in the case of the circle); moreover this minimal or cyclic model is unique up to conjugacy (see e.g.\ Ghys \cite{Ghys}). Thus studying actions of finitely generated groups up to semi-conjugacy is essentially the same as to study its minimal actions up to conjugacy.
The situation is particulary nice when $M$ is a circle, for which more tools are available. In particular the \emph{bounded Euler class} is an complete algebraic invariant for classifying irreducible actions up to semi-conjugacy (see Ghys \cite{Ghys-bounded, Ghys}), which generalizes to arbitrary groups the rotation number for homeomorphisms. For instance, this has been successfully used to understand the space $\Homirr(G, \homeo_0(\mathbb S^1))$ for various discrete subgroups of Lie groups (see for instance Burger and Monod \cite{BurgerMonod}, Matsumoto \cite{MatsumotoSurface}, Mann \cite{MannThesis},
or Mann and Wolff \cite{mann2018rigidity}), mapping class groups (Mann and Wolff \cite{MannWolff2018mcg}) or of Thompson's group $T$ \cite{GhysSergiescu,Ghys}. On the other hand there is no known complete invariant to understand semi-conjugacy classes of actions on $M=\mathbb{R}$ (see the discussion in \S\ref{sub.Deroin-intro}), and here there are fewer groups for which the space $\Homirr(G, \homeo_0(\mathbb{R}))$ is well understood (beyond deciding whether it is empty). Most results concern rather small groups. For example, if $G$ does not contain a free semigroup on two generators (e.g.\ if $G$ has subexponential growth), then every action in $\Homirr(G, \homeo_0(\mathbb{R}))$ is semi-conjugate to an action taking values in the group of translations $(\mathbb{R}, +)$ (see Navas \cite{Navas2010}), and actions up to semi-conjugacy can be completely classified for some classes of solvable groups such as polycyclic groups (Plante \cite{Plante}) or the solvable Baumslag--Soliltar groups $\BS(1, n)$ (see \cite{RivasBS}). Some further classification results for actions on the line can also be obtained by considering lifts of actions on the circle, but the scope of this method is rather limited: for instance one can show that the central lifts of the actions of the whole group $\homeo_{0}(\mathbb S^1)$ (Militon \cite{Militon}), or of Thompson's group $T$ (see \cite[Theorem 8.6]{MatteBonTriestino}) are the unique actions of such groups on the line up to semi-conjugacy.
The main goal of this work is to prove structure theorems for actions on the real line of various classes of groups which arise as sufficiently rich subgroups of the group of homeomorphisms of an interval. Throughout the introduction we fix an open interval $X=(a, b)\subseteq \mathbb{R} $ with $a\in \mathbb{R}\cup\{-\infty\}$ and $b\in \mathbb{R}\cup\{+\infty\}$. We are interested in studying actions on the real line and intervals of groups $G\subset \homeo_0(X)$. The action of $G$ on $X$ will be referred to as \emph{the standard action} of $G$. Of course there would be no loss of generality in assuming $X=\mathbb{R}$, but since we will be dealing with more general actions of $G$ on the real line it is convenient to keep a separate notation for $X$. The precise assumptions we will make on $G$ will depend on the situation, but the most important one is that $G$ is a \emph{locally moving} group. To recall this notion, given an open interval $I\subset X$, we denote by $G_I\subset G$ the subgroup consisting of elements that fix $X\setminus I$ pointwise (called the \emph{rigid stabilizer} of $I$).
\begin{dfn}
Let $X=(a, b)$ be an open interval. We say that a subgroup $G\subset \homeo_0(X)$ is \emph{locally moving} if for every $I\subset X$ the subgroup $G_I$ acts on $I$ without fixed points.
\end{dfn}
The most basic example of a locally moving group is $\homeo_0(X)$ itself. However there are also many finitely generated (and even finitely presented) groups that admit a locally moving action on an interval: a relevant example is {Thompson's group} $F$, as well as many other related groups studied in the literature.
The main question addressed in this work is the following.
\begin{main question}
Let $G\subset \homeo_0(X)$ be a locally moving group of homeomorphisms of an open interval $X$. What are the possible actions of $G$ on the real line, and in particular under which conditions an action $\varphi\in \Homirr(G, \homeo_0(\mathbb{R}))$ must be semi-conjugate to its standard action on $X$?
\end{main question}
Evidence towards rigidity of actions of locally moving groups on the real line comes from a general reconstruction theorem of Rubin \cite{Rubin,Rubin2} holding for groups of homeomorphisms of locally compact spaces. In this (very special) case, Rubin's theorem implies that any group isomorphism between two locally moving groups of homeomorphisms of intervals must be implemented by a topological conjugacy; equivalently for a locally moving group $G\subset \homeo_0(X)$, the standard action on $X$ is the {unique} faithful \emph{locally moving} action of $G$. This result does not allow to draw many conclusions on more general actions of $G$ which may fail to be locally moving (see Lodha \cite{Coherent} for some conditions ensuring this). Some further evidence towards rigidity comes from the fact that the group $\homeo_0(\mathbb{R})$ is known to admit a unique action on the real line up to conjugacy by a result of Militon \cite{Militon} (see also Mann \cite{Mann}), and the same result was recently shown for the group of compactly supported diffeomorphisms of $\mathbb{R}$ by Chen and Mann \cite{ChenMann}. However, for smaller (e.g.\ finitely generated) locally moving groups very little appears to be known, even for well-studied cases such as Thompson's group $F$. In fact it turns out that smaller groups admit a much rich variety of actions on the line than one might guess based on the previous results, and satisfy a combination of rigidity and flexibility phenomena. In particular many of them (but not all!) admit a a vast class of ``exotic'' actions on the real line.
Before proceeding to discuss our main results, let us clarify what we mean (or rather what we \emph{don't} mean) by ``exotic'' action in this context. When $G\subset \homeo_0(X)$ is a locally moving group acting on $X=(a, b)$, it is often possible to obtain new actions of $G$ on the real line by considering actions induced from \emph{proper quotients} of $G$.
To explain this, we denote by $G_c\subset G$ the normal subgroup of $G$ consisting of compactly supported elements (that is those elements that act trivially on a neighborhood of $a$ and $b$), then a well-known simplicity argument shows that the commutator subgroup $[G_c, G_c]$ of $G_c$ is simple and contained in every non-trivial normal subgroup of $G$ (see Proposition \ref{p-micro-normal}). Thus, if $G=[G_c, G_c]$ then $G$ is simple, and otherwise the group $G/[G_c, G_c]$ is the {largest proper quotient} of $G$.
In particular whenever $G$ is finitely generated the associated \emph{groups of germs} $\Germ(G, a)$ and $\Germ(G, b)$ are non-trivial quotients of $G$ which moreover can be faithfully represented inside $\homeo_0(\mathbb{R})$ (see \cite{Mann}). Thus finitely generated locally moving groups admit non-trivial, yet non-faithful, actions on the real line factoring throughout its germs\footnote{Even if one restricts to study faithful actions, it should be kept in mind that a faithful action can be \emph{semi-conjugate} to an action arising from a proper quotient. Indeed, one can start from a non-faithful action of a countable group and perform a ``Denjoy's blow up'' by replacing each point of an orbit by an interval, and then extend the action to these intervals in a faithful way by identifying them equivariantly with $X$. This phenomenon can never happen for actions which are \emph{minimal and faithful}, and we will often restrict to such. }. Non-faithful actions of $G$ correspond to actions of the largest quotient $G/[G_c, G_c]$ and can be studied separately. For example the largest quotient of Thompson's group $F$ coincides with its abelianization $F=F/[F, F]\cong \mathbb Z^2$, whose actions on the real line are always semi-conjugate to an action by translations arising from a homomorphism to $(\mathbb{R}, +)$.
In view of this, when $G\subset \homeo_0(X)$ is locally moving, we will reserve the term \emph{exotic action} for actions $\varphi\in \Homirr(G, \homeo_0(\mathbb{R}))$ which are \emph{not semi-conjugate to the standard action of $G$ on $X$, nor to any action induced from the quotient $G/[G_c, G_c]$}.
\subsection{Actions by $C^1$ diffeomorphisms} We begin with a result for actions of locally moving groups on the line by diffeomorphisms of class $C^1$, which states that such actions are never exotic in the sense clarified above.
\begin{thm}[$C^1$ rigidity of locally moving groups] \label{t-intro-C1}
Let $X$ be an open interval and let $G\subset \homeo_0(X)$ be a locally moving group. Then every irreducible action $\varphi\colon G\to \Diff^1_0(\mathbb{R})$ is semi-conjugate either to the standard action of $G$ on $X$, or to a non-faithful action (induced from an action of the largest quotient $G/[G_c, G_c]$).
\end{thm}
For actions on closed intervals one can rule out the non-faithful case under some mild additional assumption on $G$. Given an interval $X=(a, b)$, we say that a group $G\subset \homeo_0(X)$ has \emph{independent groups of germs at the endpoints} if for every $g_1, g_2\in G$, there is $g\in G$ which coincides with $g_1$ on a neighborhood of $a$ and with $g_2$ on a neighborhood of $b$.
\begin{cor} \label{c-intro-C1} Let $X$ be an open interval and let $G\subset \homeo_0(X)$ be a locally moving group with independent groups of germs at the endpoints. Let $\varphi\colon G\to \Diff^1([0, 1])$ be a faithful action with no fixed point in $(0, 1)$. Then the $\varphi$-action of $G$ on $(0, 1)$ is semi-conjugate to its standard action on $X$.
\end{cor}
The condition in the previous corollary is satisfied by many locally moving groups. In particular Corollary \ref{c-intro-C1} applies to Thompson's group $F$. However an interesting feature of the proof of Theorem \ref{t-intro-C1} is that the group $F$ plays a key role in an intermediate step. In fact this proof combines three ingredients. The first is a general trichotomy for $C^0$ actions of locally moving groups on the line (Theorem \ref{t-lm-trichotomy}), which is the common building ground for all the results in this paper. The second is the fact that every locally moving group over an interval contains many copies of $F$ (Proposition \ref{p-chain}), as follows from an argument based on a presentation of $F$ going back to Brin \cite{Brin} and extensively exploited and popularized by Kim, Koberda, and Lodha \cite{KKL} under the name ``2-chain lemma''. These two results together imply that if a locally moving group $G$ admits an exotic $C^1$ action on the line, then one can find an embedding of $F$ in the $C^1$ centralizer of a diffeomorphism of a compact interval without fixed points in the interior. The third and last step consists in showing that the group $F$ cannot admit such an embedding (this uses the $C^1$ Sackseteder's theorem from \cite{DKN-acta} and an elementary version of {Bonatti's approximate linearization} \cite{BonattiProches,BMNR}, together with algebraic properties of $F$).
We note that, while the abundance of copies of $F$ inside rich groups of homeomorphisms of the line is a long-standing fact \cite{Brin,KKL}, this seems to have been rarely exploited to prove general results about such groups.
\subsection{On existence of exotic actions by homeomorphisms}
The rigidity displayed in Theorem \ref{t-intro-C1} fails in the $C^0$ setting. Perhaps the simplest way to give counterexamples is to consider countable groups of compactly supported homeomorphisms. For such groups, one can always obtain exotic actions via two general constructions presented in \S \ref{s-germ-type} and \S \ref{s-escaping-sequence}, which yield two proofs of the following fact.
\begin{fact} \label{f-intro-exotic-baby} Let $X$ be an open interval, and let $G\subset \homeo_0(X)$ be a countable group of compactly supported homeomorphisms of $X$ acting minimally on $X$. Then $G$ admits irreducible actions on $\mathbb{R}$ which are not semi-conjugate to its standard action on $X$, nor to any non-faithful action of $G$.
\end{fact}
While this observation is formally sufficient to rule out the $C^0$ version of Theorem \ref{t-intro-C1}, it is not fully satisfactory, for instance because a group $G$ as in Fact \ref{f-intro-exotic-baby} cannot be finitely generated. In fact, the proofs from \S \ref{s-germ-type} and \S \ref{s-escaping-sequence} yield actions which admit no non-empty closed invariant set on which the group acts minimally (in particular, the action is \emph{not} semi-conjugate to any minimal action nor to any cyclic action); this phenomenon is somewhat degenerate, and cannot arise for a finitely generated group (see \cite[Proposition 2.1.12]{Navas-book}).
Much more interesting is the fact that many (finitely generated) locally moving groups admit exotic actions which are \emph{minimal and faithful}. Various constructions of such actions of different nature will be provided in Section \ref{sec.examplesRfocal}, and more constructions can be found in Section \ref{s-F} in the special case of Thompson's group $F$. Here we only mention the following existence criteria, based on a construction in \S \ref{ssec.germtype}, which are satisfied by some well-studied groups.
\begin{prop}[Criteria for existence of minimal exotic actions] \label{p-intro-exotic}
For $X=(a, b)$, let $G\subset \homeo_0(X)$ be a finitely generated subgroup. Assume that $G$ acts minimally on $X$ and contains non-trivial elements of relatively compact support in $X$, and that at least one of the following holds.
\begin{enumerate}[label=(\alph*)]
\item The group $G$ is a subgroup of the group of piecewise projective homeomorphisms of $X$.
\item The groups of germs $\Germ(G, b)$ is abelian and its non-trivial elements have no fixed points in a neighborhood of $b$.
\end{enumerate}
Then there exists a faithful minimal action $\varphi\colon G\to \homeo_0(\mathbb{R})$ which is not topologically conjugate to the action of $G$ on $X$ (nor to any non-faithful action of $G$).
\end{prop}
\subsection{Actions on planar real trees and $\mathbb{R}$-focal actions} We now leave briefly aside locally moving groups to introduce a concept that plays a central role in this paper: the notion of $\mathbb{R}$-\emph{focal action}. This will be the main tool to understand exotic actions on the line of a vast class of locally moving groups (see Theorem \ref{t-intro-class-main} below).
In order to define this notion, we say that a collection $\mathcal{S}$ of open bounded real intervals is a \emph{cross-free cover} if it covers $\mathbb{R}$ and every two intervals in $\mathcal{S}$ are either disjoint or one is contained in the other.
\begin{dfn}
Let $G$ be a group. An action $\varphi \colon G\to \homeo_0(\mathbb{R})$ is $\mathbb{R}$-focal if there exists a bounded open interval $I\subset \mathbb{R}$ whose $G$-orbit is a cross-free cover.\end{dfn}
The terminology comes from group actions on trees (and Gromov hyperbolic spaces). In this classical setting, an isometric group action on a tree $\mathbb T$ is called \emph{focal} if it fixes a unique end $\omega\in \partial \mathbb T$ and contains hyperbolic elements (which necessarily admit $\omega$ as an attracting or repelling fixed point). The dynamics of an $\mathbb{R}$-focal action on the line closely resembles the dynamics of the action on the boundary of a focal action on a tree. In fact this is more than an analogy: every $\mathbb{R}$-focal action on the line can be encoded by an action on a tree, except that we need to consider group actions on \emph{real trees} (or $\mathbb{R}$-trees) by homeomorphisms (not necessarily isometric). Let us give an overview of this connection (for more precise definitions and details we refer to Section \ref{sec_focal_trees}).
Recall that a real tree is a metrizable space $\mathbb T$ where any two points can be joined by a unique path, and which admits a compatible metric which makes it geodesic. By a \emph{directed tree} we mean a (separable) real tree $\mathbb T$ together with a preferred end $\omega \in \partial \mathbb T$, called the \emph{focus}\footnote{While this is not exactly the definition of directed tree which we will work with (see Definition \ref{def d-trees}), it can be taken as equivalent definition for the purpose of this introduction (the connection is explained in \S \ref{subsec metric R-tree}).}. If $\mathbb T$ is a directed tree with focus $\omega$, we write $\partial^*\mathbb T:=\partial \mathbb T\setminus\{\omega\}$. An action of a group $G$ on a directed tree $\mathbb T$ (by homeomorphisms) is always required to fix the focus. In this topological setting we will say that such an action is \emph{focal} if for every $v\in \mathbb T$ there exists a sequence $(g_n)\subset G$ such that $(g_n.v)$ approaches $\omega$ along the ray $[v, \omega[$.
By a \emph{planar directed tree} we mean a directed tree $\mathbb T$ endowed with a \emph{planar order}, which is the choice of a linear order on the set of directions below every branching point of $\mathbb T$ (one can think of $\mathbb T$ as embedded in the plane; see \S \ref{s-planar-trees} for a formal definition). Note that in this case the set $\partial^*\mathbb T$ inherits a linear order $\prec$ in a natural way. Assume that $\mathbb T$ is a planar directed tree, and that $\Phi\colon G\to \homeo(\mathbb T)$ is a focal action of a countable group which preserves the planar order. Then $\Phi$ induces an order-preserving action of $G$ on the ordered space $(\partial^*\mathbb T, \prec)$. From this action one can obtain an action $\varphi\in \Homirr(G, \homeo_0(\mathbb{R}))$ on the real line (for instance by considering the Dedekind completion of any orbit in $\partial^*\mathbb T$, see \S \ref{sec.dynreal} and \S \ref{s-planar-trees} for details), which we call the \emph{dynamical realization} of the action of $\Phi$. It turns out that such an action is always minimal and $\mathbb{R}$-focal. In fact, we have the following equivalence, which can be taken as an alternative definition of $\mathbb{R}$-focal actions.
\begin{prop}
Let $G$ be a countable group. An action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is minimal and $\mathbb{R}$-{focal} if and only if it is conjugate to the dynamical realization of a focal action by homeomorphisms of $G$ on some planar directed tree.
\end{prop}
We will say that an $\mathbb{R}$-focal action $\varphi$ can be represented by an action $\Phi\colon G\to \homeo(\mathbb T)$ on a planar directed tree if it is conjugate to the dynamical realization of $\Phi$. Note that in general such an action $\Phi$ representing $\varphi$ is not unique.
Examples of $\mathbb{R}$-focal actions appear naturally in the context of solvable groups. In fact, the notion of $\mathbb{R}$-focal action was largely inspired by an action on the line of the group $\mathbb Z \wr \mathbb Z$ constructed by Plante \cite{Plante} to give an example of action of a solvable group on the line which is not semi-conjugate to any action by affine transformations (i.e.\ transformations of the form $x\mapsto ax+b$). See Example \ref{subsec.Plantefocal} for a generalization of Plante's construction to arbitrary wreath products. For finitely generated solvable groups we obtain the following dichotomy.
\begin{thm} \label{t-intro-solvable}
Let $G$ be a finitely generated solvable group. Then every irreducible action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is either semi-conjugate to an action by affine transformations, or to a minimal $\mathbb{R}$-focal action.
\end{thm}
\begin{rem}This should be compared with the result by Plante, which motivated the construction of the action of $\mathbb Z\wr\mathbb Z$ mentioned above, that every irreducible action of a solvable group of \emph{finite (Pr\"ufer) rank} is semi-conjugate to an affine action (a weaker condition on the group is actually sufficient, see \cite[Theorem 4.4]{Plante} for details).\end{rem}
A distinctive feature of $\mathbb{R}$-focal actions is that the action of individual elements of the group satisfy a dynamical classification which resembles the classification of isometries of trees into elliptic and hyperbolic elements. Namely if $ G\subset \homeo_0(\mathbb{R})$ is a subgroup whose action is $\mathbb{R}$-focal, then every element $g\in G$ satisfies one of the following (see Corollary \ref{prop.dynclasselements}).
\begin{itemize}
\item Either $g$ is \emph{totally bounded}: its set of fixed points accumulates on both $\pm \infty$.
\item Or $g$ is a \emph{pseudohomothety}: it has a non-empty compact set of fixed points $K\subset \mathbb{R}$ and either every $x\notin [\min K, \max K]$ satisfies $|g^n(x)|\to \infty$ as $n\to +\infty$ (in which case we say that $g$ is an \emph{expanding} pseudohomothety), or the same holds as $n\to -\infty$ (in which case we say that $g$ is \emph{contracting}). If $K$ is reduced to a single point, we further say that $g$ is a \emph{homothety}.
\end{itemize}
Moreover the dynamical type of each element can be explicitly determined from the $G$-action on a planar directed tree $\mathbb T$ representing the $\mathbb{R}$-focal action by looking at the local dynamics near the focus $\omega\in \partial \mathbb T$.
We finally discuss another concept: the notion of \emph{horograding} of an $\mathbb{R}$-focal action of a group $G$ by another action of $G$. This will be crucial in the sequel, as it will allow us to establish a relation between exotic actions of various locally moving groups and their standard actions. Assume that $\mathbb T$ is a directed tree with focus $\omega$. An \emph{increasing horograding} of $\mathbb T$ by a real interval $X=(a, b)$ is a map $\pi\colon \mathbb T\to X$ such that for every $v\in \mathbb T$ the ray $[v, \omega[$ is mapped homeomorphically onto the interval $[\pi(v), b)$. This is a non-metric analogue of the classical metric notion of horofunction associated with $\omega$. A \emph{decreasing horograding} is defined analogously but maps $[v, \omega[$ to $(a, \pi(v)]$, and a \emph{horograding} is an increasing or decreasing horograding. If $G$ is a group acting both on $\mathbb T$ and $X$ we say that $\pi$ is a $G$-\emph{horograding} if its is $G$-equivariant. This leads to the following definition.
\begin{dfn}
Assume that $\varphi\colon G\to \homeo_0(\mathbb{R})$ is a minimal $\mathbb{R}$-focal action, and that $j\colon G\to \homeo_0(X)$ is another action of $G$ on some open interval $X$. We say that $\varphi$ can be (increasingly or decreasingly) horograded by $j$ if $\varphi$ can be represented by an action on a planar directed tree $ \Phi\colon G\to\homeo(\mathbb T)$ which admits an (increasing or decreasing) $G$-horograding $\pi\colon \mathbb T\to X$.
\end{dfn}
The existence of such a horograding is a tight relation between $\varphi$ and $j$, which is nevertheless quite different from the notion of semi-conjugacy: here the action of $G$ on $X$ plays the role of a hidden ``extra-dimensional direction'' with respect to the real line on which $\varphi$ is defined. For instance, in the presence of an increasing $G$-horograding, the type of each element in $\varphi$ can be determined from its germ on the rightmost point of $X$ as follows.
\begin{prop}\label{p-intro-focal-classelements}
Let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be a minimal $\mathbb{R}$-focal action, and assume that $\varphi$ can be increasingly horograded by an action $j\colon G\to \homeo_0(X)$ on an interval $X=(a, b)$. Then we have the following alternative.
\begin{enumerate}[label=(\roman*)]
\item If the fixed points of $j(g)$ accumulate on $b$ then $\varphi(g)$ is totally bounded.
\item Else $\varphi(g)$ is a pseudohomothety, which is expanding if $j(g)(x)>x$ for $x$ in a neighborhood of $b$, and contracting otherwise. Moreover if $j(g)$ has no fixed points in $X$, then $\varphi(g)$ is a homothety.
\end{enumerate}
\end{prop}
\subsection{A structure theorem for $C^0$ actions}
With the notion of $\mathbb{R}$-focal action in hand, we now go back to our original problem and state a structure theorem for actions on the line by homeomorphisms of a vast class of locally moving group.
\begin{dfn}[The classes $\mathcal F${}\;and $\mathcal{F}_0${}]
For $X=(a, b)$, let $G\subset \homeo_0(X)$ be a subgroup, and recall that for an interval $I=(x, y)\subseteq X$ we denote by $G_{(x, y)}$ the rigid stabilizer of $I$. Write $G_+:=\bigcup_{x<b} G_{(a, x)}$ and $G_-:=\bigcup_{x>a} G_{(x, b)}$ for the subgroups of elements with trivial germ at $a$ and $b$ respectively. Consider the following conditions.
\begin{enumerate}[label=(\roman*)]
\item \label{i-intro-class-lm} $G$ is locally moving.
\item \label{i-intro-class-fg} There exist two finitely generated subgroups $\Gamma_\pm\subset G_\pm$ and $x, y\in X$ such that $G_{(a, x)}\subset \Gamma_+$ and $G_{(y, b)}\subset \Gamma_-$.
\item \label{i-intro-class-translation} There exists an element of $G$ without fixed points in $X$.
\end{enumerate}
We say that $G$ belongs to the class $\mathcal F${}\;if it satisfies \ref{i-intro-class-lm} and \ref{i-intro-class-fg}, and that it belongs to the class $\mathcal{F}_0${}\;if it satisfies \ref{i-intro-class-lm}, \ref{i-intro-class-fg}, and \ref{i-intro-class-translation}.
\end{dfn}
Note that condition \ref{i-intro-class-fg} trivially holds true provided there exist $x, y\in X$ such that $G_{(a, x)}$ and $G_{(y, b)}$ are finitely generated. In practice this weaker condition will be satisfied in many examples, but \ref{i-intro-class-fg} is more flexible and more convenient to handle.
The class $\mathcal{F}_0${}\;contains many well-studied examples of finitely generated locally moving groups, including Thompson's group $F$, and all Thompson--Brown--Stein groups $F_{n_1,\ldots, n_k}$ \cite{Stein}, the groups of piecewise projective homeomorphisms of Lodha--Moore \cite{LodhaMoore}, and several other Bieri--Strebel groups \cite{BieriStrebel}. It also contains various groups which are far from the setting of groups of piecewise linear or projective homeomorphisms: for instance every countable subgroup of $\homeo_0(X)$ is contained in a finitely generated group belonging to $\mathcal{F}_0${}\;(see Proposition \ref{prop.superF}).
Our main result is a qualitative description for the actions on the line of groups in the class $\mathcal F${}, stating that such actions can be classified into three types of behavior.
\begin{thm}[Main structure theorem for actions of groups in $\mathcal F${}] \label{t-intro-class-main}
Let $X$ be an open interval and let $G\subset \homeo_0(X)$ be a group in the class $\mathcal F${}. Then every irreducible action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is semi-conjugate to an action in one of the following families.
\begin{enumerate}
\item \emph{(Non-faithful)} An action which factors through the largest quotient $G/[G_c, G_c]$.
\item \emph{(Standard)} An action which is conjugate to the standard action of $G$ on $X$.
\item \label{i-intro-C-focal} \emph{(Exotic)} A minimal faithful $\mathbb{R}$-focal action which can be horograded by the standard action of $G$ on $X$.
\end{enumerate}
\end{thm}
Note that many finitely generated groups which belong to the class $\mathcal F${}, or even to the more restricted class $\mathcal{F}_0${}, do indeed admit exotic actions falling in \eqref{i-intro-C-focal}, for instance as a consequence of Proposition \ref{p-intro-exotic} (but not all groups in $\mathcal{F}_0${}\;do, see Theorem \ref{t-intro-no-actions} below). Moreover in some cases there are uncountably many non-semi-conjugate such actions (see for instance Theorem \ref{teo F intro} below for the case of Thompson's group $F$), and the variety and flexibility of constructions suggest that in general it is too complicated to obtain a reasonably explicit description of all semi-conjugacy classes of exotic actions (however the word \emph{explicit} is crucial here, as we shall see in Theorem \ref{t-intro-space} that such a description exists in principle). Nevertheless, the main content of Theorem \ref{t-intro-class-main} is that every exotic action of a group in $\mathcal F${}\;must be tightly related to the standard action of $G$, although not via a semi-conjugacy but at the level of a planar directed tree encoding the exotic ($\mathbb{R}$-focal) action. This relation can be effectively exploited to study such exotic actions (in particular the way in which every individual element of $G$ acts oi an exotic action is determined by its standard action, see Proposition \ref{p-intro-focal-classelements}). This leads to various applications that we describe now.
\subsection{Space of semi-conjugacy classes and local rigidity for groups in $\mathcal{F}_0${}}\label{s-intro-lr}
\label{sub.Deroin-intro}
The main structure theorem (Theorem \ref{t-intro-class-main}) can be used to get an insight on the structure of semi-conjugacy classes inside the space of irreducible actions $\Homirr(G,\homeo_{0}(\mathbb{R}))$ whenever $G$ is a group in $\mathcal{F}_0${}. Recall that the space $\Homirr(G,\homeo_{0}(\mathbb{R}))$ can be endowed with the natural \emph{compact-open topology}, which means that a neighborhood basis of a given action $\varphi\in \Homirr(G,\homeo_{0}(\mathbb{R}))$
is defined by considering for every $\varepsilon>0$, finite subset $S\subset G$, and compact subset $K\subset \mathbb{R}$, the subset of actions
\[
\left\{
\psi\in \Homirr(G,\homeo_{0}(\mathbb{R}))\colon \max_{g\in S}\max_{x\in K} |\varphi(g)(x)-\psi(g)(x)|<\varepsilon
\right\}.
\]
A natural way to understand actions of a group $G$ up to semi-conjugacy would be to study the quotient of the space $\Homirr(G,\homeo_{0}(\mathbb{R}))$ by the semi-conjugacy equivalence relation. Unfortunately this quotient space is rather bad, for instance it is in general not Hausdorff and may even fail to have the structure of a standard measurable space (see Remark \ref{rem non smooth}). In fact the semi-conjugacy equivalence relation need not be \emph{smooth}, that is it need not be possible to classify the semi-conjugacy classes of actions in $\Homirr(G, \homeo_0(\mathbb{R}))$ by a Borel complete invariant (see the discussion in \S \ref{ssc.complexity}).
However, for groups in the class $\mathcal{F}_0${}, Theorem \ref{t-intro-class-main} can be used to show that a large part of the set of semi-conjugacy classes in $\Homirr(G, \homeo_0(G))$ can in fact be parametrized by a well behaved space and it is possible to select a system of representatives of each semi-conjugacy class with nice properties. Under this identification, it turns out that the semi-conjugacy class of the standard action of $G$ on $X$ corresponds to an \emph{isolated point} in this space. This is the content of the following result.
\begin{thm}[The space of semi-conjugacy classes for groups in $\mathcal{F}_0${}] \label{t-intro-space}
Let $X$ be an open interval and let $G\subset \homeo_0(X)$ be a finitely generated group in the class $\mathcal{F}_0${}. Denote by $\sim$ the equivalence relation on $\Homirr(G, \homeo_0(\mathbb{R}))$ given by positive semi-conjugacy, and let $\mathcal{U}\subset \Homirr(G, \homeo_0(\mathbb{R}))$ be the $\sim$-invariant set of irreducible actions that are not semi-conjugate to any action induced from the largest quotient $G/[G_c, G_c]$. Then $\mathcal{U}$ is open and the following hold.
\begin{enumerate}[label=(\roman*)]
\item The quotient space $\mathcal{U}/_\sim$, with the quotient topology, is Polish and locally compact.
\item There exists a continuous section $\sigma\colon\mathcal{U}/_\sim\rightarrow \mathcal{U}$ which is a homeomorphism onto its image, whose image is closed in $\mathcal{U}$ and consists of faithful minimal actions.
\item \label{i-isolated} The semi-conjugacy class of the standard action of $G$ on $X$ is an isolated point in $\mathcal{U}/_\sim$.
\end{enumerate}
\end{thm}
\begin{rem}
In the previous theorem, ruling out the actions induced from the largest quotient $G/[G_c, G_c]$ is necessary, as in this generality it may happen that the semi-conjugacy equivalence relation on irreducible actions of $G/[G_c, G_c]$ might not be smooth, and \textit{a fortiori} this may not be true for $G$. However the theorem implies that the semi-conjugacy equivalence relation on the set $\Homirr(G, \homeo_0(\mathbb{R}))$ is smooth provided the same holds for irreducible actions of $G/[G_c, G_c]$ (see Corollary \ref{c-class-borel}, or Theorem \ref{t-F-classification} in the case of Thompson's group $F$).
\end{rem}
The main dynamical significance of the previous result is that it implies a {local rigidity} result. We say that an irreducible action $\varphi \in \Homirr(G,\homeo_{0}(\mathbb{R}))$ of a group $G$ is \emph{locally rigid} if there exists a neighborhood $\mathcal U\subset \Homirr(G,\homeo_{0}(\mathbb{R}))$ of $\varphi$ such that every $\psi\in \mathcal U$ is semi-conjugate to $\varphi$. Otherwise, we say that the action of $\varphi$ is \emph{flexible}. Theorem \ref{t-intro-space} has the following direct consequence.
\begin{cor}[Local rigidity of the standard action for groups in $\mathcal{F}_0${}]\label{c-intro-lr}
Let $X$ be an open interval and let $G\subset \homeo_0(X)$ be a finitely generated group in the class $\mathcal{F}_0${}. Then the natural action of $G$ on $X$ is locally rigid.
\end{cor}
Corollary \ref{c-intro-lr} provides a vast class of locally rigid actions on the real line. For instance, by Proposition \ref{prop.superF}, it implies that every action on the real line of a countable group $G$ can be extended to a locally rigid action of some finitely generated overgroup.
A well-developed approach to local rigidity of group actions on the line is through the space $\LO(G)$ of left-invariant orders on $G$, which is a totally disconnected compact space. Every such order gives rise to an action in $\Homirr(G, \homeo_0(\mathbb{R}))$, and isolated points in $\LO(G)$ produce locally rigid actions (see Mann and the third author \cite[Theorem 3.11]{MannRivas}). This can be used to show that some groups, for instance braid groups (see Dubrovin and Dubrovina \cite{DubrovinDubrovina}, or the monograph by Dehornoy, Dynnikov, Rolfsen, and Wiest \cite{OrderingBraids}), do have locally rigid actions. However the converse to this criterion does not hold, and this approach has been more fruitful in the opposite direction, namely for showing that a group has no isolated order from flexibility of the dynamical realization (see for instance the works by Navas \cite{Navas2010}, or by Alonso, and the first and third named authors \cite{ABR,AlonsoBrum}, as well as by Malicet, Mann, and the last two authors \cite{MMRT}). One difficulty underlying this approach is that it is usually not easy to determine when two orders in $\LO(G)$ give rise to semi-conjugate actions.
To prove Theorem \ref{t-intro-space}, we take a different approach to local rigidity, based on a compact space suggested by Deroin \cite{Deroin} as a dynamical substitute of the space of left-orders. One way to construct this space is based on work by Deroin, Kleptsyn, Navas, and Parwani \cite{DKNP} on symmetric random walks on $\homeo_{0}(\mathbb{R})$. Given a probability measure $\mu$ on $G$ whose support is finite, symmetric, and generates $G$, one defines the \emph{Deroin space} $\Der_\mu(G)$ as the subspace of $\Homirr(G,\homeo_{0}(\mathbb{R}))$ of (normalized) \emph{harmonic actions}, that is, actions of $G$ for which the Lebesgue measure is $\mu$-stationary (see \S \ref{ssubsec.Deroin} for details). The space $\Der_\mu(G)$ is compact and Hausdorff, with a natural topological flow $\Phi:\mathbb{R}\times \Der_\mu(G)\to \Der_\mu(G)$ defined on it by the conjugation action of the group of translations $(\mathbb{R}, +)$, and has the property that two actions in $\Der_\mu(G)$ are (positively semi-)conjugate if and only if the are on the same $\Phi$-orbit. It was shown in \cite{DKNP} that every action in $\Homirr(G, \homeo_0(\mathbb{R}))$ can be semi-conjugated to one inside $\Der_\mu(G)$ by a probabilistic argument. In fact, we shall show in Section \ref{s.Deroin} that this can be done in such a way that the new action depends continuously on the original one (see Theorem \ref{t.retraction_Deroin}). This continuous dependence implies a criterion for local rigidity of minimal actions within the space $\Der_\mu(G)$ (see Corollary \ref{c-rigidity-Deroin}). The proof is based on an alternative description of $\Der_\mu(G)$ a quotient of the space of left-invariant \emph{preorders} on $G$ (Theorem \ref{thm.homeoderoin}), which also implies that $\Der_\mu(G)$ does not depend on the choice of the probability measure $\mu$ up to homeomorphism.
With these tools in hand, to prove Theorem \ref{t-intro-space} we study the shape of orbits of the translation flow $\Phi$ on $\Der_\mu(G)$ when $G$ is a group in $\mathcal{F}_0${}. Faithful actions correspond to an open $\Phi$-invariant subset of $\Der_\mu(G)$. Using Theorem \ref{t-intro-class-main} and the properties of $\mathbb{R}$-focal actions we show that the restriction of the flow $\Phi$ to this open subset has the simplest possible type of orbit structure: there is a closed transversal $\mathcal{S}$ intersecting every $\Phi$-orbit, and $\Phi$ is conjugate to the vertical translation on $\mathcal{S}\times \mathbb{R}$, so that its space of orbits can be identified with $\mathcal{S}$. This analysis yields Theorem \ref{t-intro-space}. (See Figure \ref{fig-intro-F} for a schematic representation for Thompson's group $F$.)
\begin{figure}[ht]
\includegraphics[scale=1]{Deroin_F-2.pdf}
\caption{\small The Deroin space of Thompson's group $F$. The outer red circle parametrizes the non-faithful actions (of $F^{ab}\cong \mathbb Z^2$) and it is pointwise fixed by the flow $\Phi$. The remaining $\Phi$-orbits are the faithful actions. The purple and blue orbits correspond to the standard action and to its conjugate by the reflection, and are transversely isolated (this gives local rigidity). The pencils of green and yellow orbits are the $\mathbb{R}$-focal actions: both pencils contain uncountably many orbits, admit a compact transversal to the flow, and the shown convergence to limit points is uniform. See Figure \ref{fig-F-Der} and \S \ref{s-F-Deroin} for details. }\label{fig-intro-F}
\end{figure}
\subsection{A criterion for non-orientation-preserving groups}
We also observe that the situation is quite different if we leave the setting of orientation-preserving actions. Indeed, in this case we have the following result which follows from Corollary \ref{c-intro-lr} (or can be more directly obtained in the course of its proof). The proof is given in \S \ref{ssec non-orientation-preserving actions}. We denote by $\homeo(X)$ the group of all (not necessarily orientation-preserving) homeomorphisms of an interval $X$.
\begin{cor}[Global rigidity in the non-orientation-preserving case]
Let $X=(a, b)$ and let $G\subset \homeo(X)$ be a subgroup such that $G\subsetneq \homeo_0(X)$ and $G\cap \homeo_0(X)$ is in the class $\mathcal F${}. Then every faithful minimal action $\varphi\colon G\to \homeo(\mathbb{R})$ is conjugate to the standard action on $X$.
\end{cor}
\subsection{Some concrete groups in the class $\mathcal{F}_0${}}
We have seen how Theorem \ref{t-intro-space} yields a global picture of actions on the line for groups in the class $\mathcal{F}_0${}. Nevertheless when $G$ is a group in the class $\mathcal F${}\;the rigidity or abundance of its exotic actions turns out to depend subtly on $G$. This difference is visible already among groups of piecewise linear homeomorphisms.
Given an open interval $X=(a, b)$ we denote by $\PL(X)$ the group of orientation-preserving PL homeomorphisms of $X$, with finitely many discontinuity points for the derivatives. It turns out that this class of groups exhibits a surprising mixture of rigidity and flexibility properties and many illustrative examples of applications of our results arise as subgroups of $\PL(X)$. Note that Proposition \ref{p-intro-exotic} implies that every finitely generated locally moving group $G\subset \PL(X)$ admits a minimal faithful exotic action on the real line.
The most famous example of group of PL homeomorphisms is Thompson's group $F$. Recall that $F$ is defined as the subgroup of $\PL((0, 1))$ whose slopes are powers of 2 and whose constant terms and breakpoints are in the ring $\mathbb Z[1/2]$ of dyadic rationals. The group $F$ belongs to the class $\mathcal{F}_0${}\;and satisfies the assumptions of most of the results in this paper. The reader can find in Section \ref{s-F} a detailed account of our results in this case.
In particular every faithful action $\varphi\colon F\to \Diff^1_0([0, 1])$ without fixed points in $(0,1)$ is semi-conjugate to the standard action (Corollary \ref{c-intro-C1}), every irreducible exotic action $\varphi \colon F\to \homeo_0(\mathbb{R})$ is $\mathbb{R}$-focal and horograded by the standard action of $F$ on $(0,1)$ (Theorem \ref{t-intro-class-main}), and the standard action of $F$ on $(0,1)$ is locally rigid (Corollary \ref{c-intro-lr}).
Despite these rigidity results, it turns out that the group $F$ admits a rich and complicated universe of minimal exotic actions, and our work leaves some interesting questions open: in particular we do not know whether $F$ admits exotic actions that are locally rigid. In Section \ref{s-F} we will discuss several different constructions of minimal $\mathbb{R}$-focal actions of $F$, which for instance leads to the following.
\begin{thm}
\label{teo F intro}
Thompson's group $F$ admits uncountably many actions on the line which are faithful, minimal, $\mathbb{R}$-focal and pairwise non-semi-conjugate. Moreover, there are uncountably many such actions whose restrictions to the commutator subgroup $[F, F]$ remain minimal and are pairwise non-semi-conjugate.
\end{thm}
This abundance of $\mathbb{R}$-focal actions can be explained in part by the fact that the group $F$ admits many focal actions on \emph{simplicial} trees by simplicial automorphisms (the Bass--Serre tree associated with any non-trivial splitting as an ascending HNN extension \cite[Proposition 9.2.5]{Geoghegan}). Some of these actions preserve a planar order on the tree, and many $\mathbb{R}$-focal actions of $F$ arise in this way (a characterization will be given in \S \ref{s-F-simplicial}). However, we show in \S \ref{s-F-hyperexotic} that the group $F$ also admits minimal $\mathbb{R}$-focal actions for which an associated planar directed tree cannot be chosen to be simplicial, or even to carry an invariant $\mathbb{R}$-tree metric. In fact the second claim in Theorem \ref{teo F intro} is directly related to this phenomenon. We refer to Section \ref{s-F} for further discussion on $\mathbb{R}$-focal actions of $F$.
In contrast, this abundance of exotic actions of the group $F$ already fails for some tightly related groups of PL homeomorphisms. Given a real number $\lambda>1$, we denote by $G(X; \lambda)$ the group of all PL homeomorphisms of $X$ where all derivatives are integers powers of $\lambda$ and all constant terms and breakpoints belong to $\mathbb Z[\lambda, \lambda^{-1}]$. Note that the group $F$ is equal to $G((0,1); 2)$. When $X=\mathbb{R}$, the group $G(\mathbb{R}; \lambda)$ belongs to $\mathcal{F}_0${}\;provided $\lambda$ is algebraic, thus satisfies Theorem \ref{t-intro-class-main}. However, in striking difference with the case of $F$, we have the following.
\begin{thm}[PL groups with finitely many exotic actions]\label{t-intro-Glambda}
Let $\lambda>1$ be an algebraic real number. Then the group $G=G(\mathbb{R}; \lambda)$ admits exactly three minimal faithful actions $\varphi\colon G\to \homeo_0(\mathbb{R})$ on the real line up to (semi-)conjugacy, namely its standard action and two minimal $\mathbb{R}$-focal actions (which can be horograded by its standard action).
\end{thm}
Note that this shows that the two minimal exotic actions of $G(\mathbb{R}; \lambda)$ are locally rigid as well; in particular the standard action need not be the unique locally rigid action for a group in $\mathcal{F}_0${}.
Theorem \ref{t-intro-Glambda} is in fact a special case of Theorem \ref{t-BBS} which applies to a more general class of \emph{Bieri--Strebel groups} of the form $G(\mathbb{R}; A, \Lambda)$, which are groups of PL homeomorphisms defined by restricting derivatives to belong to a multiplicative subgroup $\Lambda\subset \mathbb{R}^*_+$ and constant terms and breakpoints to belong to a $\mathbb Z[\Lambda]$-submodule $A\subset \mathbb{R}$ (see \S \ref{sc.BieriStrebel}). The two exotic actions of the group $G(\mathbb{R}; \lambda)$ arise as a special case of a construction of exotic actions of Bieri--Strebel groups explained in \S \ref{ss.Bieri-Strebelfocal}.
Building on the proof of Theorem \ref{t-intro-Glambda}, in \S \ref{s-no-actions} we also construct a finitely generated locally moving group $G$ having no exotic actions at all. This group is obtained by perturbing the group $G(\mathbb{R}; 2)$ locally in a controlled manner.
\begin{thm}[A finitely generated locally moving group with no exotic actions]\label{t-intro-no-actions}
There exists a finitely generated subgroup $G\subset \homeo_0(\mathbb{R})$ in the class $\mathcal{F}_0${}, such that every faithful minimal action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is conjugate to the standard action.
\end{thm}
\subsection{Further consequences} We conclude this introduction by mentioning some additional results obtained along the way, which allow to recover and strenghten some previously known results in the unified setting developed here.
\subsubsection{Very large locally moving groups}
As mentioned at the beginning of the introduction, two rigidity results were known for some very large locally moving groups: namely a result of Militon shows that the group $\homeo_c(\mathbb{R})$ of compactly supported homeomorphisms of $\mathbb{R}$ admits a unique irreducible action on $\mathbb{R}$ up to conjugacy, and a recent result of Chen and Mann implies the same property for the group $\Diff^r_c(\mathbb{R})$ of compactly supported diffeomorphisms in regularity $r\neq 2$. While the main focus of this paper is on countable groups, our method also allows to obtain a criterion for uniqueness of the action of a family of uncountable locally moving groups, which recovers and generalizes those results. In order to do this, we need the following relative version of a group property first considered by Schreier (see \cite[Problem 111]{MR3242261}).
\begin{dfn}\label{d-Schreier}
Let $G$ be a group and let $H$ be a subgroup of $G$. We say that the pair $(G, H)$ has \emph{relative Schreier property} if every countable subset of $H$ is contained in a finitely generated subgroup of $G$.\footnote{The choice of \emph{relative Schreier property} is inspired by the work of Le Roux and Mann \cite{LeRouxMann}. Apparently there is no standard name in the literature, and such choice may be ambiguous, as in other works \emph{Schreier property} is used for different properties.}
\end{dfn}
The following result is a special case of what we show in Section \ref{s-uncountable}.
\begin{thm}[Uniqueness of action for some uncountable locally moving groups]\label{t-intro-uncountable}
Let $G\subset \homeo_c(\mathbb{R})$ be a perfect subgroup of compactly supported homeomorphisms. Suppose that for every bounded open interval $I\subset \mathbb{R}$ the following hold:
\begin{enumerate}
\item the $G_I$-orbit of every $x\in I$ is uncountable;
\item the pair $(G, G_I)$ has relative Schreier property
\end{enumerate}
Then every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ without fixed points is conjugate to the natural action of $G$.
\end{thm}
\subsubsection{Comparison with the case of groups acting on the circle}
One may wonder whether a similar theory can be developed for locally moving groups acting on the \emph{circle} rather than the real line. However this problem turns out to be considerably simpler and less rich, as in this setting locally moving groups essentially do not admit exotic actions at all. This phenomenon is already well-known in some cases. In particular a result of Matsumoto states that the group $\homeo_0(\mathbb{S}^1)$ admits only one action on $\mathbb{S}^1$ up to conjugacy, and an unpublished result of Ghys (announced in \cite{Ghys}) shows that every action of Thompson's group $T$ on $\mathbb{S}^1$ is semi-conjugate to its standard action. The original proofs of these results are both based on the computation of the second bounded cohomology of these groups, and on its relation with actions on the circle established by Ghys \cite{Ghys-bounded} (a tool that does not have an analogue for actions on real line). For Thompson's group $T$ another proof is provided in \cite{LBMB-sub}, which does not use the knowledge of bounded cohomology of $T$, but relies in part on the nature of $T$ as a group of piecewise linear homeomorphisms.
In fact, in Section \ref{s-circle} we observe that elementary considerations based on commutations suffice to obtain a more general criterion (Theorem \ref{t-circle}), stating that if $G$ is a group of homeomorphisms of an arbitrary compact space $X$ satisfying suitable assumptions, then every minimal faithful action of $G$ on the circle must factor onto its action on $X$. A special case of Theorem \ref{t-circle} (with $X=\mathbb{S}^1$) yields the following.
\begin{thm}\label{t-intro-circle}
Let $G\subset \homeo_0(\mathbb{S}^1)$ be a locally moving subgroup.
Then every faithful minimal action $\varphi\colon G\to \homeo_0(\mathbb{S}^1)$ factors onto the standard action of $G$ on $\mathbb{S}^1$.
\end{thm}
In fact, under some mild additional assumption on $G$ one obtains that the action $\varphi$ is conjugate to (possibly a lift of) the standard action on $\mathbb{S}^1$ via a self-cover of $\mathbb{S}^1$. We refer to Section \ref{s-circle} for more precise results and further discussion.
\subsubsection{An application to non-smoothability}
One possible direction of application of Theorem \ref{t-intro-C1} is to use it as a tool to show that certain locally moving group acting on the line cannot be isomorphic to groups of diffeomorphisms (of suitable regularity). Results in this spirit were obtained by Bonatti, Lodha, and the fourth named author in \cite{BLT} for various groups of piecewise linear and projective homeomorphisms. Here we consider the Thompson--Brown--Stein groups $F_{n_1,\cdots, n_k}$, which are a natural generalization of Thompson's group $F$ associated with any choice of integers $n_1,\ldots ,n_k$ whose logarithms are linearly independent over $\mathbb Q$ (see Definition \ref{d-Thompson-Stein}). It was shown in \cite{BLT} that if $k\ge 2$, then the standard action of $F_{n_1,\ldots, n_k}$ on $(0, 1)$ cannot be conjugate to an action by $C^2$ diffeomorphisms.
With our methods we obtain the following strengthening of that result.
\begin{thm}
When $k\ge 2$, any Thompson--Brown--Stein group of the form $F_{n_1,\ldots, n_k}$ admits no faithful action on the real line by diffeomorphisms of class $C^{1+\alpha}$ for any $\alpha>0$.
\end{thm}
In this direction we also mention a recent result of Kim, Koberda, and third named author \cite{KKR}, who used our Theorem \ref{t-intro-C1} to show that the free product $F\ast \mathbb Z$ of Thompson's group and the integers admits no faithful action on $[0,1]$ by $C^1$ diffeomorphisms.
\subsection{Outline of the paper}
This paper is structured as follows. Sections \ref{s-preliminaries} and \ref{s.Deroin} contain the required general theory on group actions on the line. After recalling some elementary preliminaries and notations in Section \ref{s-preliminaries}, we focus in Section \ref{s.Deroin} on the Deroin space $\Der_\mu(G)$ of $\mu$-harmonic actions on the line of a finitely generated group $G$. The main purpose of that section is to prove that there exists a continuous retraction from $\Homirr(G, \homeo_0(\mathbb{R}))$ to the space $\Der_\mu(G)$ and a related local rigidity criterion (Corollary \ref{c-rigidity-Deroin}), which will be used in the proofs of Theorem \ref{t-intro-space} and Corollary \ref{c-intro-lr} (see the discussion in \S \ref{s-intro-lr}). These results appear to be new, although the arguments in the section are mostly based on elementary facts on left-invariant preorders.
The remaining sections can be roughly divided into four parts. Part I contains the central skeleton of the study locally moving groups on which the remaining parts are built, while Parts II--IV are essentially independent from each other and can be read after reading Part I (modulo some minor dependence between them, mostly limited to examples or secondary results).
\medskip
\begin{enumerate}[label= Part \Roman*]
\item (Sections \ref{s-lm-1}--\ref{s-lm-2}). \textbf{General theory on locally moving groups acting on the line}. After introducing notations and terminology on locally moving groups of homeomorphisms of intervals and proving some of their basic properties in Section \ref{s-lm-1}, we begin in Section \ref{s-lm-2} the study of their actions on the real line. The main result of this part is Theorem \ref{t-lm-trichotomy}, which establishes a dynamical trichotomy for actions on the line of a locally moving group $G$ of homeomorphisms, a crucial common ingredient for all the other results in this paper. This result imposes a restrictive technical condition on all exotic actions of $G$, which roughly speaking says that in every exotic action of $G$ there is ``domination'' of certain subgroups of $G$ over other. After proving this, we give in \S \ref{s-lm-first-consequences} some of its simpler applications, such as a rigidity result for actions with nice combinatorial properties (including actions by piecewise analytic transformations). We conclude the section by giving some first examples of exotic actions of locally moving groups in \S \ref{ss.exoticactions} (which allow to prove Fact \ref{f-intro-exotic-baby}).
\item (Section \ref{s-differentiable}). \textbf{Actions by $C^1$ diffeomorphisms.} This part consists of Section \ref{s-differentiable} alone, where we study $C^1$ actions of locally moving group and complete the proofs of Theorem \ref{t-intro-C1} and Corollary \ref{c-intro-C1}.
\item (Sections \ref{sec.focalgeneral}--\ref{s-few-actions}). \textbf{$\mathbb{R}$-focal actions and topological dynamics of exotic actions.}
Here we develop the notion of $\mathbb{R}$-focal action and apply it to prove our main results for actions by homeomorphisms of locally moving groups in the class $\mathcal F${}. In Section \ref{sec.focalgeneral} we introduce $\mathbb{R}$-focal actions of groups on the line and establish their main properties. In Section \ref{sec_focal_trees} we illustrate the connection of $\mathbb{R}$-focal action with focal actions on trees. (Note that Sections \ref{sec.focalgeneral}--\ref{sec_focal_trees} are not directly concerned with locally moving groups, and could also be read directly after reading Section \ref{s-preliminaries}.)
In Section \ref{sec.examplesRfocal} we give various constructions and existence criteria for exotic minimal $\mathbb{R}$-focal actions of locally moving groups. Section \ref{sec.locandfoc} contains our main results on groups in the classes $\mathcal{F}_0${}$\subset$$\mathcal F${}. We begin by proving our main structure theorem (Theorem \ref{t-intro-class-main}) for actions of groups in $\mathcal F${}, and then use it to study the Deroin space $\Der_\mu(G)$ of a group in $\mathcal{F}_0${}. As a corollary of this analysis and of the results in Section \ref{s.Deroin} we obtain Theorem \ref{t-intro-space} and the local rigidity of the standard action for groups in $\mathcal{F}_0${}\;(Corollary \ref{c-intro-lr}). In Section \ref{s-F} we focus on the example of Thompson's group $F$: we illustrate our main rigidity results in this special case, and use them to analyze its Deroin space $\Der_\mu(F)$ (see Figure \ref{fig-F-Der}); then we proceed to give various constructions of minimal exotic actions of the group $F$ and discuss some of their properties. In Section \ref{s-few-actions} we provide examples of groups in the class $\mathcal{F}_0${}\;whose $\mathbb{R}$-focal actions are much more rigid: we classify the $\mathbb{R}$-focal actions of a class of Bieri--Strebel groups of PL homeomorphisms (in particular we prove Theorem \ref{t-intro-Glambda}) and construct a finitely generated locally moving group without exotic actions (Theorem \ref{t-intro-no-actions}).
\item (Sections \ref{s-uncountable}--\ref{s-circle}) \textbf{Additional results}. In this part we prove some additional results which allow to recover and generalize some previously known results from the literature in the framework of this paper. In Section \ref{s-uncountable} we prove results on actions of some very large (uncountable) locally moving groups and prove Theorem \ref{t-intro-uncountable}. In Section \ref{s-circle} we prove a result for actions on the circle, which implies Theorem \ref{t-intro-circle}: this section is essentially self-contained and independent on the other results in this paper (it only relies on Proposition \ref{p-centralizer-fix}).
\end{enumerate}
\subsection*{Acknowledgements}
We are grateful to Sebasti\'an Hurtado, Bertrand Deroin and Maxime Wolff for fruitful discussions about the space of harmonic actions and to Maxime Wolff also for discussions regarding the Plante product. We also acknowledge Todor Tsankov for useful clarifications about Borel reducibility of equivalence relations which motivated the discussion in \S \ref{ssc.complexity}, and Yash Lodha for many discussions related to various locally moving groups of homeomorphisms of the line.
J.B was supported by CONICYT via FONDECYT post-doctorate fellowship 3190719. Part of this work was done during a visit of N.M.B. to the University of Santiago de Chile funded by FONDECYT 1181548.
N.M.B. and M.T. were partially supported by the project ANR Gromeov (ANR-19-CE40-0007). M.T. was also partially supported by
the project ANER Agroupes (AAP 2019 R\'egion Bourgogne--Franche--Comt\'e), and his host department IMB receives support from the EIPHI Graduate School (ANR-17-EURE-0002). C.R. was partially supported by FONDECYT 1181548 and FONDECYT 1210155.
\section{Micro-supported and locally moving groups}\label{sec.locallymgeneral} \label{s-lm-1}
\subsection{Definitions}\label{sc.defi_lm}
Throughout this section (and mostly in the rest of the paper), we let $X=(a, b)$ be an open interval, with endpoints $a, b\in \mathbb{R}\cup\{\pm \infty\}$. Recall that for a subgroup $G\subseteq \homeo_0(X)$ and a subinterval $I\subset X$, we denote by $G_I$ the subgroup of $G$ consisting of elements that fix pointwise $X\setminus I$.
\begin{dfn}\label{d.lm}
A subgroup $G\subseteq \homeo_0(X)$ is \emph{micro-supported} if for every non-empty subinterval $I\subset X$ the subgroup $G_I$ is non-trivial. We also say that $G$ is \emph{locally moving} if for every open subinterval $I\subset X$ the subgroup $G_I$ acts on $I$ without fixed points.
\end{dfn}
Given open subintervals $I$ and $J$ we write $I\Subset J$ if $I$ is relatively compact in $J$.
For $G\subseteq \homeo_0(X)$, we denote by $G_c$ the normal subgroup of elements with relatively compact support in $X$, that is, $G_c=\bigcup_{I\Subset X} G_I$.
%
We also let $\Germ(G, a)$ and $\Germ(G, b)$ be the \emph{groups of germs} of elements of $G$ at the endpoints of $X$. Recall that the germ of an element $g\in G$ at $a$ is the equivalence class of $g$ under the equivalence relation that identifies two elements $g_1, g_2\in G$ if they coincide on some interval of the form $(a, x)$, with $x\in X$. The germ of $g$ at $b$ is defined similarly. We denote by $\mathcal{G}_a\colon G\to \Germ(G, a)$ and $\mathcal{G}_b\colon G\to \Germ(G, b)$ the two natural germ homomorphisms and their kernels by $G_-$ and $ G_+$, respectively. Note that
\[G_-=\bigcup_{x\in X} G_{(x, b)} \quad\text{and} \quad G_+=\bigcup_{x\in X} G_{(a, x)}.\]
When $G$ acts minimally, the micro-supported condition is equivalent to the non-triviality of the subgroup $G_c$:
\begin{prop} \label{p-micro-compact}
For $X=(a,b)$, assume that the subgroup $G\subseteq \homeo_0(X)$ acts minimally on $X$. Then $G$ is micro-supported if and only if it contains a non-trivial element with relatively compact support.
\end{prop}
\begin{proof}
The forward implication is obvious. Conversely, assume that there exists a relatively compact subinterval $I \Subset X$ for which $G_I\neq \{\mathsf{id}\}$. If follows that the centralizer of $G_I$ in $\homeo_0(X)$ must fix the infimum of the support of every non-trivial element of $G_I$. Then by Theorem \ref{t-centralizer}, the action of $G$ on $X$ is proximal. Therefore for every non-empty open subinterval $J\subset X$ there exists $g\in G$ such that $g(I)\subset J$, which implies that the group $G_J$ is non-trivial.
\end{proof}
Let us summarize some basic observations on the locally moving condition in the following lemma.
\begin{lem} \label{l-rigid}
For $X=(a,b)$, let $G\subseteq \homeo_0(X)$ be locally moving. Then the following hold for every non-empty open subinterval $I\subset X$.
\begin{enumerate}
\item \label{i-rigid-minimal} The subgroup $G_I$ acts minimally on $I$. In particular $G$ acts minimally on $X$.
\item \label{i-derived} The derived subgroup $[G_I,G_I]$ also acts without fixed points on $I$.
\end{enumerate}
\end{lem}
\begin{proof}
Write $I=(c,d)$ for a non-empty open subinterval. Fix $x, y\in I$ and assume, say, that $x<y$. Since the subgroup $G_{(c, y)} \subseteq G_I$ has no fixed point in $(c, y)$, there exist elements $g\in G_{(c, y)}$ such that $g(x)$ is arbitrarily close to $y$. Thus the $G_I$-orbit of $x$ accumulates at $y$. By a symmetric argument, the same holds if $y<x$. Since $x$ and $y$ are arbitrary, this shows that every $G_I$-orbit in $I$ is dense in $I$. Finally if $[G_I,G_I]$ admits fixed points, its set of fixed points is closed and $G_I$-invariant and thus by minimality of the action of $G_I$ we deduce that $[G_I,G_I]$ is trivial, and therefore that $G_I$ is abelian and conjugate to a group of translations. This is not possible since the action of $G_I$ on $I$ is micro-supported. \qedhere
\end{proof}
\subsection{Structure of normal subgroups} The following proposition shows that locally moving groups are close to be simple. This follows from well-known arguments, that we repeat here for completeness.
\begin{prop}[Structure of normal subgroups] \label{p-micro-normal}
For $X=(a,b)$, let $G\subseteq \homeo_0(X)$ be a micro-supported subgroup whose action is minimal.
Then every non-trivial normal subgroup of $G$ contains $[G_c,G_c]$. Moreover, if $[G_c,G_c]$ acts minimally, then it is simple.
In particular, when $G$ is locally moving, then $[G_c, G_c]$ is simple and contained in every non-trivial normal subgroup of $G$.
\end{prop}
The proof uses the following classical observation on normal subgroups of homeomorphisms, sometimes known as the ``double-commutator lemma''. With this formulation it is \cite[Lemma 4.1]{Nek-fp}
\begin{lem}\label{l.doublecomm}
Let $H$ be a group of homeomorphisms of a Hausdorff space $Z$, and $N$ be a non-trivial group of homeomorphisms of $Z$ normalized by $H$. Then there exists a non-empty open subset $U\subset Z$ such that $N$ contains $[H_U, H_U]$, where $H_U$ is the pointwise fixator of $Z\setminus U$.
\end{lem}
\begin{proof}[Proof of Proposition \ref{p-micro-normal}]
Suppose that $N$ is a non-trivial normal subgroup of $G$. Then by Lemma \ref{l.doublecomm}, $N$ contains $[G_I, G_I]$ for some non-empty open subinterval $I\subset X$. Take now $g\in [G_c, G_c]$. Then $g\in [G_J, G_J]$ for some non-empty open subinterval $J\Subset X$. Note that the centralizer of $G$ in $\homeo_{0}(X)$ is trivial, for the action is micro-supported (see the proof of Proposition \ref{p-micro-compact}).
Then by Theorem \ref{t-centralizer}, the action is proximal, so that we can find $h\in G$ such that $h(J)\subset I$, so that $hgh^{-1}\in [G_I, G_I]\subseteq N$ and since $N$ is normal we have $g\in N$. Since $g$ is arbitrary, we have $[G_c, G_c]\subseteq N$.
Note that this implies in particular that $[G_c, G_c]$ is perfect, since its commutator subgroup is normal in $G$ and thus must coincide with $[G_c, G_c]$.
Assume now that $[G_c, G_c]$ acts minimally on $X$. Then it is micro-supported (Proposition \ref{p-micro-compact}), and by the previous part every non-trivial normal subgroup of $[G_c, G_c]$ must contain the derived subgroup of $[G_c, G_c]$. Since $[G_c, G_c]$ is perfect, this implies that it is simple.
\end{proof}
It follows that when $G\subseteq \homeo_0(X)$ is micro-supported and acts minimally (in particular, when $G$ is locally moving), the quotient group $\overline{G}:=G/[G_c, G_c]$ is the largest proper quotient of $G$, and thus plays an important role. Note that the (\textit{a priori} smaller) quotient $G/G_c$ has a natural dynamical interpretation, namely it is naturally a subgroup of the product of groups of germs $\Germ(G, a)\times \Germ(G, b)$. The largest quotient $\overline{G}$ is an extension of $G/G_c$ with abelian kernel:
\[1\to G_c/[G_c, G_c] \longrightarrow \overline{G}\longrightarrow G/G_c\longrightarrow 1.\]
The abelian group $G_c/[G_c, G_c]$ can be difficult to identify in general. However for some relevant examples of locally moving groups it is known that the group $G_c$ is perfect, so that $\overline{G}=G/G_c$.
\begin{ex}\label{e-BN}
Consider the Brin--Navas group, which was introduced independently by Brin \cite{Brin} and Navas \cite{NavasAmenable}, and further studied by Bleak \cite{Bleak} who showed that $B$ is contained in any non-solvable subgroup of $\PL([0,1])$.
The group $B$ has the following presentation (see \cite{bleak2016determining} and Proposition \ref{lem.presentation}):
\[
B=\left \langle f,w_n\, (n\in \mathbb Z)\, \middle\vert\, fw_nf^{-1}=w_{n+1}\,\forall n\in \mathbb Z,\,[w_i,w_n^mw_jw_n^{-m}]=1\,\forall n>i,j,\,\forall m\in \mathbb Z\setminus \{0\}\right \rangle.
\]
That is, the group $B$ is defined as an HNN extension of the group generated by the $w_n$, $n\in \mathbb Z$, and this group is a bi-infinitely iterated wreath product of $\mathbb Z$. Following the notation in \cite{Bleak}, we write $(\wr \mathbb Z\wr)^\infty$ for the subgroup generated by the $w_n$ in $B$, so that $B=(\wr \mathbb Z\wr)^\infty\rtimes \mathbb Z$. A minimal micro-supported action on $(0,1)$ of $B$ is realized in the group $\PL([0,1])$ of piecewise linear homeomorphisms, choosing generators (see \cite{Bleak} and Figure~\ref{f-BN})
\[
f(x)=\left\{\begin{array}{lr}
\tfrac14 x & x\in [0,\tfrac14],\\[.5em]
x-\tfrac3{16} & x\in [\tfrac14,\tfrac7{16}],\\[.5em]
4x-\tfrac32 & x\in [\tfrac7{16},\tfrac{9}{16}],\\[.5em]
x+\tfrac3{16} & x\in [\tfrac9{16},\tfrac34],\\[.5em]
\tfrac14 x+\tfrac{3}4 & x\in [\tfrac34,1],
\end{array}\right.
\quad\quad
w_0(x)=\left\{\begin{array}{lr}
x & x\in[0,\tfrac7{16}],\\[.5em]
2x-\tfrac{7}{16} & x\in[\tfrac7{16},\tfrac{15}{32}],\\[.5em]
x+\tfrac1{32} & x\in[\tfrac{15}{32},\tfrac12],\\[.5em]
\tfrac12 x+\tfrac9{32} & x\in[\tfrac12,\tfrac9{16}],\\[.5em]
x & x\in[\tfrac9{16},1].
\end{array}\right.
\]
In this case, the subgroup $B_c$ is the normal subgroup $(\wr \mathbb Z\wr)^\infty$, and $B_c/[B_c,B_c]\cong \mathbb Z^\infty$, so that the largest proper quotient $\overline{B}\cong \mathbb Z^\infty\rtimes \mathbb Z=\mathbb Z\wr\mathbb Z$ is the lamplighter group.
Observe that the bi-infinite wreath product $B_c$ does not act minimally on $(0,1)$.
\begin{figure}[ht]
\includegraphics[scale=1]{Brin-Navas.pdf}
\caption{PL realization of the Brin--Navas group with minimal micro-supported action.}\label{f-BN}
\end{figure}
\end{ex}
\begin{ex}\label{ex:Brown}
A rich source of examples of micro-supported, and actually locally moving groups, are the Bieri--Strebel groups introduced with the Definition \ref{d.BieriStrebel}. Recall that these are defined as the groups of piecewise linear homeomorphisms of an interval with prescribed breakpoints and slopes. For a quite simple class of examples, fix $n\ge 2$ and consider the group $G=F_n$ of all piecewise linear homeomorphisms of $[0,1]$ such that all derivatives are powers of $n$ and the breakpoints are in the ring $A=\mathbb Z[1/n]$. When $n=2$, this is Thompson's group $F$. The subgroup $G_c$ of compactly supported elements consists exactly of elements which have derivative $1$ at the endpoints of $[0,1]$, so that $G/G_c\cong \mathbb Z^2$. However, the abelianization of $G$ is larger for $n\ge 3$, as $G^{ab}\cong \mathbb Z^n$. This is determined by the ``homomorphism into the slope group'' $\nu$ discussed in \cite[\S C11]{BieriStrebel}. The homomorphism $\nu$ is obtained by gathering together the homomorphisms $\nu_{\Omega}:G\to \mathbb Z$, where $\Omega$ is a $G$-orbit of breakpoints (there are exactly $n-1$ many of them), which are defined by considering the total jump of derivatives at points in the orbit:
\[
\nu_{\Omega}(f)=\log_n\prod_{a\in \Omega}\frac{D^+f(a)}{D^-f(a)},\quad f\in G.
\]
\end{ex}
\subsection{Subgroups isomorphic to Thompson's group}
Thompson's group $F$ plays a special role among locally moving groups, due to the following proposition, which will be crucial in our proof of the $C^1$ rigidity of locally moving groups (Theorem \ref{t-lm-C1}).
\begin{prop}\label{p-chain}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be locally moving. Then $G$ contains a subgroup isomorphic to Thompson's group $F$.
\end{prop}
Its proof is based on the following variant of the ``2-chain lemma'' of Kim, Koberda, and Lodha \cite{KKL}. The key idea can be traced to Brin \cite{Brin}, and has been also largely developed in \cite{FastPingPong}.
It is based on the following two properties: every non-trivial quotient of $F$ is abelian, and the finite presentation of $F$,
\begin{equation}\label{eq.presF}
F=\left \langle a,b\,\middle\vert\, [a,(ba)b(ba)^{-1}]=[a,(ba)^2b(ba)^{-2}]=1\right \rangle,
\end{equation}
where the two relations have in fact an interesting dynamical interpretation.
\begin{lem}[Noisy 2-chain lemma] \label{l-2chain}
Take two homeomorphisms $f,g\in \homeo_0(\mathbb{R})$, write $d=\sup \supp(f)$ and $c=\inf \supp(g)$, and assume the following:
\begin{enumerate}
\item\label{i:noisy2c} $c<d$;
\item\label{ii:noisy2c} $c\notin \operatorname{\mathsf{Fix}}(f)$ and $d\notin\operatorname{\mathsf{Fix}}(g)$;
\item\label{iii:noisy2c} $d$ and $f(c)$ are in the same connected component of $\supp(g)$.
\end{enumerate}
Then $\langle f,g\rangle$ contains a subgroup isomorphic to Thompson's $F$
\end{lem}
\begin{figure}[ht]
\includegraphics[scale=1]{2chain-3.pdf}
\caption{Proof on the noisy 2-chain lemma (Lemma \ref{l-2chain}).}\label{f.noisy2chain}
\end{figure}
\begin{proof}
After the assumptions, there exists $N\in \mathbb Z$ such that $g^Nf(c)>d$. Thus we also have $(g^Nf)^2(c)=g^{2N}f(c)>d$. Hence, for $i\in\{1,2\}$, we have that the subset
\[(g^Nf)^i\left (\supp(g)\right )=\supp\left ( (g^Nf)^ig(g^Nf)^{-i}\right )=\supp\left ( (g^Nf)^ig^N(g^Nf)^{-i}\right )\]
is disjoint from $\supp(f)$ (see Figure \ref{f.noisy2chain}). We deduce that the elements $a=f$ and $b=g^N$ satisfy the two relations in the presentation \eqref{eq.presF}, and thus $\langle f,g^N\rangle$ is isomorphic to a quotient of $F$. As the supports $\supp(f)$ and $\supp(g^N)=\supp(g)$ overlap, we deduce that the subgroup $\langle f,g^N\rangle$ is non-abelian and thus isomorphic to $F$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{p-chain}]
Take $f\in G_c$. Using Lemma \ref{l-rigid}, it is not difficult to find an element $g$, conjugate to $f$, such that conditions (\ref{i:noisy2c}--\ref{ii:noisy2c}) in Lemma \ref{l-2chain} are satisfied by $f$ and $g$. Write $c=\inf \supp(g)$ and $d=\sup \supp(f)$. Up to replace $f$ by its inverse, we can assume $f(c)>c$. If condition \eqref{iii:noisy2c} in Lemma \ref{l-2chain} is not satisfied, we use Lemma \ref{l-rigid} again to find an element $h\in G_{(c, d)}$ such that $h(f(c))$ belongs to the same connected component of $\supp(g)$ as $d$.
Replace $f$ by the conjugate $hfh^{-1}$ and now property \eqref{iii:noisy2c} is also satisfied.
\qedhere
\end{proof}
\section{Dynamical trichotomy for actions of locally moving groups on the line} \label{s-lm-2}
\subsection{The trichotomy}
Let $X=(a, b)$ be an open interval. The goal of this section is to establish a first result on the possible actions $\varphi\colon G\to \homeo_0(\mathbb{R})$, when $G \subseteq \homeo_0(X)$ is locally moving, showing that such actions fall into one of three kinds. This trichotomy will be the starting point of most of our results.
In the sequel we will often be dealing with two different actions of the same group $G$, namely its standard action on $X$ and different, given action of $G$ on $\mathbb{R}$. Recall that to avoid confusion we fix the following notation throughout the paper.
\begin{notation}
Let $G\subseteq \homeo_0(X)$ be a subgroup and let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be an action on the real line. For $g\in G$ and $x\in X$ we use the notation $g(x)$ to refer to the standard action on $X$, while for $\xi\in \mathbb{R}$ we will write $g.\xi:=\varphi(g)(\xi)$. We also write $\operatorname{\mathsf{Fix}}^\varphi(H)$ for the set of fixed points of a subgroup $H\subseteq G$ with respect to the action $\varphi$, and $\suppphi(H)=\mathbb{R}\setminus \operatorname{\mathsf{Fix}}^\varphi(H)$.
Also, we will often write $K=G_c$ and $N=[G_c,G_c]$, although this will be systematically recalled.
\end{notation}
Let us also introduce the following terminology, which will simplify many statements.
\begin{dfn} Let $G$ be a group and let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be an action on the real line. Let $A, B\subseteq G$ be subgroups. We say that $A$ is \emph{totally bounded} if $\operatorname{\mathsf{Fix}}^\varphi(A)$ accumulates at both $\pm \infty$ (equivalently, if the $\varphi(A)$-orbit of every $\xi\in \mathbb{R}$ is bounded).
We say that $A$ is \emph{dominated by commuting elements} if its $\varphi$-image has fixed points, while the image of its centralizer $\varphi(C_G(A))$ has none. If moreover $B\subseteq G$ is a subgroup such that the $\varphi$-image of $C_B(A)$ has no fixed point, we say that $A$ is dominated by commuting elements \emph{within $B$}. Note that since $C_B(A)$ preserves the subset $\operatorname{\mathsf{Fix}}^\varphi(A)$ and acts without fixed points, this implies that $A$ is totally bounded.
Finally we say that $A$ is \emph{locally} dominated by commuting elements (within $B$) if all finitely generated subgroups of $A$ are dominated by commuting elements (within $B$).
\end{dfn}
Recall that given a subgroup $G\subseteq \homeo_0(X)$, with $X=(a, b)$, we denote by $G_-$ and $G_+$ the kernels of the two germs homomorphisms $\mathcal{G}_a\colon G\to \Germ(G, a)$ and $\
\mathcal{G}_b\colon G\to \Germ(G, b)$, respectively. Recall also that we let $G_c$ be the subgroup of compactly supported elements in $G$, and that $[G_c, G_c]$ is simple and is the smallest normal subgroup of $G$ (Proposition \ref{p-micro-normal}).
The following theorem says that all ``exotic'' actions of $G$ have an abundance of subgroups dominated by commuting elements.
\begin{thm}[Dynamical trichotomy for locally moving groups] \label{t-lm-trichotomy}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be locally moving. Then every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ without fixed points satisfies one of the following statements.
\begin{enumerate}[label=(\roman*)]
\item \emph{(Induced from a quotient)} It is semi-conjugate to an action that factors through the largest quotient $G/[G_c, G_c]$.
\item \emph{(Standard)} It is semi-conjugate to the standard action of $G$ on $X$.
\item \emph{(Exotic)} One of the subgroups $G_+$ and $G_-$ (possibly both) is locally dominated by commuting elements within $[G_c, G_c]$.
\end{enumerate}
\end{thm}
Theorem \ref{t-lm-trichotomy} leverages the abundance of commutation in a locally moving group. This is quite natural to exploit, and can be compared with some rigidity results for actions of certain large groups on one-manifolds, which take advantage of commutation for actions with some regularity, for instance by diffeomorphisms in class $C^2$ (see for example Ghys and Sergiescu \cite[Theorem K]{GhysSergiescu} for Thompson's group $T$, or the results of Mann \cite{Mann-homdiffeo} and the alternative proofs by Matsumoto \cite{Matsumoto}). Indeed recall that Kopell's lemma states that if $f, g$ are two $C^2$ diffeomormisms of a compact interval and if $f$ has no fixed point in the interior, then neither does $g$.
This gives a powerful tool to reconstruct the standard action by keeping track of supports of elements.
In contrast, for $C^0$ homeomorphisms it is much easier to commute, so that mere commutation is not sufficient to reconstruct support of elements and get rigidity (and indeed exotic actions in the sense of Theorem \ref{t-lm-trichotomy} exist in abundance). The conclusion of Theorem \ref{t-lm-trichotomy} describes precisely this failure of rigidity.
We now prove Theorem \ref{t-lm-trichotomy}. We begin with a useful observation based on Theorem \ref{t-centralizer}.
\begin{prop}[Actions of direct products] \label{p-centralizer-fix}
Let $M\in\{\mathbb{R},\mathbb{S}^1\}$.
Let $\Gamma_1$ and $\Gamma_2$ be two groups, and if $M=\mathbb{R}$ assume that $\Gamma_1$ is finitely generated. Then for every action $\varphi\colon \Gamma_1\times \Gamma_2\to \homeo_0(M)$, there exists $i\in\{1,2\}$ such that the image of $[\Gamma_i, \Gamma_i]$ has fixed points.
\end{prop}
\begin{proof}
Note that we identify here $\Gamma_1$ with the subgroup $\Gamma_1\times \{1\}$, and similarly for $\Gamma_2$.
Suppose that $\varphi(\Gamma_1)$ admits no fixed point (otherwise, the conclusion is true). If the action of $\Gamma_1$ admits a closed discrete orbit, then the action on this discrete orbit factors through a cyclic quotient of $\Gamma_1$ and so $[\Gamma_1, \Gamma_1]$ fixes it pointwise, and the conclusion holds true. If not, let $\Lambda\subset M$ be the unique minimal invariant set for $\varphi(\Gamma_1)$, which exists when $M=\mathbb{R}$ since we assume $\Gamma_1$ finitely generated. Then $\Lambda$ is preserved by $\varphi(\Gamma_2)$ and the action of $\Gamma_1$ is semi-conjugate to a minimal action, obtained by collapsing the connected components of $M\setminus \Lambda$. In the case $M=\mathbb{R}$, we apply Theorem \ref{t-centralizer} to this minimal action. If this action is by translations then again $[\Gamma_1, \Gamma_1]$ acts trivially on $\Lambda$. Otherwise its centralizer is trivial or cyclic, so that $[\Gamma_2, \Gamma_2]$ fixes $\Lambda$ pointwise. When $M=\mathbb S^1$, we argue similarly using Theorem \ref{t-Margulis}, which gives that either the action is conjugate to an action by rotations (in which case $[\Gamma_1,\Gamma_1]$ acts trivially), or the centralizer of $\varphi(\Gamma_1)$ is finite cyclic (in which case $[\Gamma_2,\Gamma_2]$ fixes every point of $\Lambda$). \end{proof}
\begin{rem}
Note that here we will only use this for $M=\mathbb{R}$ (but an application of the case $M=\mathbb S^1$ will be given in Section \ref{s-circle}). In this case, the assumption that $\Gamma_1$ be finitely generated cannot be dropped, as shown by the following example. Let $(H, \prec)$ be any left-ordered non-abelian countable group. Let $G=\bigoplus_{n\in \mathbb N}H$ be the infinite restricted product, i.e.\ the group of all sequences $(h_n)$ in $H$ such that $h_n=1$ for all but finitely many $n$, with pointwise multiplication. Consider on $G$ the lexicographic order given by $(h_n)\prec (h'_n)$ if $h_m \prec h'_m$ for $m=\max\{n\in \mathbb N \colon h_n\neq h'_n\}$, and let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be the dynamical realization of this order. Let $\Gamma_1\subseteq G $ be the subgroup consisting of all sequences $(h_n)$ such that $h_n=1$ for $n$ even, and $\Gamma_2$ be the subgroup of sequences such that $h_n=1$ for $n$ odd, so that $G=\Gamma_1\times \Gamma_2$. Then it is easy to see that neither the image of $[\Gamma_1, \Gamma_1]$ nor of $[\Gamma_2, \Gamma_2]$ have fixed points.
\end{rem}
\begin{rem}
A special case of Proposition \ref{p-centralizer-fix} appears in \cite[Proposition 3.1]{Matsumoto} (for $M=\mathbb S^1$ and assuming $\Gamma_i$ are simple).
\end{rem}
Proposition \ref{p-centralizer-fix} implies the following in the setting of micro-supported groups.
\begin{cor}\label{c-micro-fixed}
For $X=(a, b)$, let $G \subseteq \homeo_0(X)$ be a subgroup acting without fixed points on $X$. Let $\Gamma\subseteq G_c$ be a finitely generated subgroup. Then for every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ the image $\varphi([\Gamma, \Gamma])$ has fixed points.
\end{cor}
\begin{proof}
Let $I\Subset X$ be a relatively compact subinterval such that $\Gamma \subseteq G_I$. Since the action of $G$ has no fixed points, we can find $g\in G$ such that $g(I)\cap I=\varnothing$. Then $g\Gamma g^{1} \subseteq G_{g(I)}$ commutes with $\Gamma$, so that Proposition \ref{p-centralizer-fix} applied to the subgroups $\Gamma_1:=\Gamma$ and $\Gamma_2:=g\Gamma g^{-1}$ implies in either case that $\varphi([\Gamma, \Gamma])$ has fixed points. \end{proof}
\begin{rem}
Again, finite generation is essential: it is not true that for every interval $I\subset X$, the $\varphi$-image of $[G_I, G_I]$ must have fixed points. Counterexamples can be found in \S \ref{s-example-PL-Q}.
\end{rem}
The next lemma is an improvement of the previous corollary, and will be used repeatedly.
\begin{lem}\label{l-at-least-one-side}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be a subgroup acting without fixed points, and set $K=G_c$. Let $I$ and $J$ be non-empty disjoint open subintervals of $X$. Let $\Gamma\subseteq G_I$ be a finitely generated subgroup. Then for every action $\varphi\colon G\to \homeo_0(\mathbb{R})$, at least one of the following holds.
\begin{enumerate}[label=(\roman*)]
\item \label{i-gamma-fixes} $\Gamma$ is totally bounded (with respect to the action $\varphi$).\item \label{i-K-fixes} The image $\varphi([K_J, K_J])$ has fixed points.
\end{enumerate}
\end{lem}
\begin{proof} We assume that \ref{i-gamma-fixes} does not hold and prove that \ref{i-K-fixes} must hold. First of all we note that if $\operatorname{\mathsf{Fix}}^\varphi(\Gamma)$ is non-empty, but does not accumulate at both $\pm\infty$, then its supremum (or infimum) must be fixed by $\varphi(G_J)\supseteq \varphi([K_J, K_J])$, since the two subgroups commute. Thus we can suppose that $\operatorname{\mathsf{Fix}}^\varphi(\Gamma)=\varnothing$. In this case, since $\Gamma$ is finitely generated, there exists a compact interval $L$ which intersects all $\varphi(\Gamma)$-orbits. Let $\mathcal{S}$ be the collection of finitely generated subgroups of $K_{J}$. For $\Delta\in \mathcal{S}$, the subset $T_\Delta:=\operatorname{\mathsf{Fix}}^\varphi([\Delta, \Delta])$ is non-empty by Corollary \ref{c-micro-fixed}, and it is $\varphi(\Gamma)$-invariant. Thus, we have $T_\Delta\cap L\neq \varnothing$. Moreover for $\Delta_1, \ldots, \Delta_n\in \mathcal{S}$ we have $T_{\Delta_1}\cap \cdots \cap T_{\Delta_n}\supset T_{\langle \Delta_1, \ldots, \Delta_n\rangle}\neq \varnothing$, so that by compactness of $L$ we have that the subset $T:=\bigcap_{\Delta\in \mathcal{S}}T_\Delta$ is non-empty. Since $[K_{J}, K_{J}]=\bigcup_{\Delta\in \mathcal{S}}[\Delta, \Delta]$, every point of $T$ is fixed by $\varphi([K_{J}, K_{J}])$. This proves the lemma. \qedhere
\end{proof}
We now need to address a minor technical point. Let $G\subseteq \homeo_0(X)$ be locally moving, and set $K=G_c$ and $N=[K, K]$. Let $I\subset X$ be a subinterval. We can then consider the two subgroups $N_I$ and $[K_I, K_I]$. A little thinking gives that $[K_I, K_I]\subseteq N_I$, but in general this inclusion is strict. However the following lemma allows to think of it as an equality for some practical purposes.
\begin{lem} \label{l-rist-nesting}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be locally moving. Let $I\subset J\subset X$ be open nested subintervals with no common endpoint in $X\setminus \{a, b\}$.
With the notation as above, we have $[K_I, K_I]\subseteq N_I\subseteq [K_J, K_J]$.
\end{lem}
\begin{proof} Only the second inclusion needs to be justified.
Let us give the proof in the case where $I=(a, x)$ and $J=(a, y)$ for some $y>x$ (the case where both endpoints of $I$ are in the interior of $X$ is analogous). Take $g\in N_I$ and let us show that $g\in [K_J, K_J]$. Since $g\in N$, it can be written as a product of commutators involving finitely many elements of $G_c$, which all belong to $K_{(a, z)}$ for some $z<b$ large enough. Thus we have $g\in [K_{(a, z)}, K_{(a, z)}]$ for some $z<b$ large enough, and we can suppose that $z>y$ since otherwise $g\in [K_J, K_J]$. Since the group $G_{(x, b)}$ acts without fixed points on $(x, b)$, there exists $h\in G_{(x, b)}$ such that $h(z)\in (z, y)$. But $h$ commutes with $g$, so we have $g=hgh^{-1}\in [K_{(a, h(z))}, K_{(a, h(z))} ]\subseteq [K_{J},K_J]$. \qedhere
\end{proof}
We now give a criterion for an action $\varphi\colon G \to \homeo_0(\mathbb{R})$ to be semi-conjugate either to the standard action, or to an action induced from a proper quotient.
\begin{prop} \label{p-lm-reconstruction}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be locally moving, and write $N=[G_c, G_c]$.
Let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be an action without fixed points. Suppose that there exist $x,y\in X$ such that both images $\varphi(N_{(a, x)})$ and $\varphi(N_{(y, b)})$ admit fixed points. Then $\varphi$ is semi-conjugate either to the standard action on $X$, or to an action which factors through the quotient $G/N$.
\end{prop}
\begin{proof}
First of all, observe that if $\varphi(N)$ admits fixed points, then $\operatorname{\mathsf{Fix}}^\varphi(N)$ is a closed $\varphi(G)$-invariant subset of $\mathbb{R}$ which accumulates at both $\pm \infty$, and the action on it factors through $G/N$. In this case we deduce that $\varphi$ is semi-conjugate to an action of $G/N$. Thus we will assume that $\varphi(N)$ has no fixed point, and
show that in this case the action must be semi-conjugate to the standard action of $G$ on $X$.
Note that since there exists $x$ such that $N_{(a, x)}$ has fixed points for $\varphi$ then this is true for every $x\in X$, since a subgroup $N_{(a, x)}$ is always conjugate into every other $N_{(a, y)}$, for $y\in X$. The same holds true for the subgroups of the form $N_{(x, b)}$. In particular, for every $x\in X$, the images $\varphi(N_{(a, x)})$ and $\varphi(N_{(x, b)})$ both admit fixed points, and since they commute they admit common fixed points. Thus for every $x\in X$ the $\varphi$-image of the subgroup $H_x:=\langle N_{(a, x)}, N_{(x, b)}\rangle$ admits fixed points. The idea is to construct a semi-conjugacy $q\colon X\to \mathbb{R}$ by setting $q(x)=\inf \operatorname{\mathsf{Fix}}^\varphi(H_x)$. We need to check that such map $q$ is well-defined (i.e.\ that the subsets $\operatorname{\mathsf{Fix}}^\varphi(H_x)$ are bounded) and monotone. Let us first prove the following claim.
\begin{claimnum} \label{claim-bounded}
Let $x$ and $y$ be two distinct points of $X$. Then we have either $\sup \operatorname{\mathsf{Fix}}^\varphi(H_x)< \inf \operatorname{\mathsf{Fix}}^\varphi(H_y)$ or viceversa.
\end{claimnum}
\begin{proof}[Proof of claim]
If the conclusion does not hold, then upon exchanging the roles of $x$ and $y$ if needed, we can find two distinct points $\xi, \eta \in \operatorname{\mathsf{Fix}}^\varphi(H_x)$ and $\delta\in \operatorname{\mathsf{Fix}}^\varphi(H_y)$ such that $\xi \le \delta \le \eta$. Assume first that $x<y$. Recall that we are supposing that $\varphi(N)$ has no fixed point, so we can choose $g\in N$ such that $g.\xi>\eta$. Let $z>x$ be a point of $X$ such that $g\in N_{(a, z)}$. By Lemma \ref{l-rigid}, the action of $N_{(x, b)}$ on $(x, b)$ has no fixed point, so we can find $h\in N_{(x, b)}$ such that $h(z)\in (x, y)$. Note that $h\in H_x$, so that $\varphi(h)$ fixes both $\xi$ and $\eta$. On the other hand the element $k=hgh^{-1}$ belongs to $N_{(a, y)}$, thus to $H_y$, and therefore $\varphi(k)$ fixes $\delta$. Thus, writing $g=h^{-1}kh$, we have
\[g.\xi=h^{-1}k.\xi \le h^{-1}k.\delta=h^{-1}.\delta \le h^{-1}.\eta=\eta,\]
contradicting that $g.\xi>\eta$ by the choice of $g$. The case $y<x$ is treated analogously.
\end{proof}
After Claim \ref{claim-bounded}, the map $q\colon X\to \mathbb{R}$, $x\mapsto\inf\operatorname{\mathsf{Fix}}^\varphi(H_x)$ is well-defined and injective. We next have to verify that it is monotone.
\begin{claimnum}
Let $x_1< x_2 < x_3$ be point of $X$. Then either $q(x_1) <q(x_2)< q(x_3)$ or $q(x_1)>q(x_2)>q(x_3)$.
\end{claimnum}
\begin{proof}[Proof of claim]
The arguments are similar to the proof of the previous claim.
For $i\in\{1,2,3\}$, set $\xi_i=q(x_i)$ and note that by the previous claim the points $\xi_1, \xi_2, \xi_3$ are pairwise distinct.
We divide the proof into cases according to their relative position. We will detail the case $\xi_1<\xi_2$ (the case $\xi_1>\xi_2$ being totally analogous); for this, we will assume for contradiction that $\xi_3<\xi_2$ and
we further split into two subcases depending on the position of $\xi_3$.
\begin{case} We have $\xi_1 <\xi_3 <\xi_2$. \end{case}
\noindent In this case, we choose an element $g\in N$ such that $g.\xi_1 > \xi_2$. Let $y>x_2$ be a point of $X$ such that $g\in N_{(a, y)}$. Let $h\in N_{(x_2, b)}$ be such that $h(y)\in (x_2, x_3)$. Note that $h\in H_{x_2}$ and since $(x_2, b)\subset (x_1, b)$ we also have $h\in H_{x_1}$, so that $\varphi(h)$ fixes $\xi_2$ and $\xi_1$. On the other hand the element $k=hgh^{-1}$ belongs to $N_{(a, h(y))} $, and since $(a, h(y))\subset (a, x_3)$ we have $k\in H_{x_3}$ so that $\varphi(k)$ fixes $\xi_3$. Writing $g=h^{-1}kh$, we have
\[g.\xi_1=h^{-1}k.\xi_1 < h^{-1}k.\xi_3=h^{-1}.\xi_3 < h^{-1}.\xi_2=\xi_2,\]
contradicting that $g.\xi_3>\xi_2$.
\begin{case} We have $\xi_3< \xi_1<\xi_2$. \end{case}
\noindent In this case, choose an element $g\in N$ such that $g.\xi_2<\xi_3$. Let $y<x_2$ be a point of $X$ such that $g\in N_{(y, b)}$. Let $h\in N_{(a, x_2)}$ be such that $h(y)\in (x_1, x_2)$.
Note that $h\in H_{x_2}$ and since $(a, x_2)\subset (a,x_3)$ we also have $h\in H_{x_3}$, so that $\varphi(h)$ fixes $\xi_2$ and $\xi_3$. On the other hand the element $k=hgh^{-1}$ belongs to $N_{(h(y),b)} $, and since $(h(y),b)\subset (x_1,b)$ we have $k\in H_{x_1}$ so that $\varphi(k)$ fixes $\xi_1$. Writing $g=h^{-1}kh$, we have
\[g.\xi_2=h^{-1}k.\xi_2 > h^{-1}k.\xi_1= h^{-1}.\xi_1 > h^{-1}.\xi_3=\xi_3,\]
contradicting that $g.\xi_2<\xi_3$.
Thus the unique possibility is that $\xi_1<\xi_2<\xi_3$, as desired.
\end{proof}
The claim implies that the map $q\colon X\to \mathbb{R}$ is monotone (increasing or decreasing). Moreover, it is clearly equivariant by construction:
\[
g.q(x)=g.\inf \operatorname{\mathsf{Fix}}^\varphi(H_x) =\inf g. \operatorname{\mathsf{Fix}}^\varphi(H_x)=\inf \operatorname{\mathsf{Fix}}^\varphi(H_{g(x)})=q(g(x)) .
\]
Therefore the map $q$ establishes a semi-conjugacy between $\varphi$ and the standard action of $G$ on $X$.
\end{proof}
We get to the following statement, which is a more explicit, but slightly more technical, version of Theorem \ref{t-lm-trichotomy}. Here we keep the notation $N=[G_c,G_c]$.
\begin{prop} \label{p-lm-trichotomy}
For $X=(a,b)$, let $G\subseteq \homeo_0(X)$ be locally moving. Then every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ without fixed points satisfies one of the following statements.
\begin{enumerate}[label=(\roman*)]
\item \emph{(Induced from a quotient)} $\varphi$ is semi-conjugate to an action that factors through the largest quotient $G/N$.
\item \emph{(Standard)} $\varphi$ is semi-conjugate to the standard action of $G$ on $X$.
\item\emph{(Exotic)} One of the following holds for every point $x\in X$:
\begin{enumerate}[label=(iii.\alph*)]
\item \label{i-domination-right} \emph{(Right-hand side domination)}
every finitely generated subgroup of $G_+$ is totally bounded, while $N_{(x, b)}$ acts without fixed points;
\item \label{i-domination-left} \emph{(Left-hand side domination)}
every finitely generated subgroup of $G_-$ is totally bounded, while $N_{(a, x)}$ acts without fixed points.
\end{enumerate}
\end{enumerate}
\end{prop}
\begin{proof}[Proof of Theorem \ref{t-lm-trichotomy} from Proposition \ref{p-lm-trichotomy}]
Let $\varphi\colon G\to \homeo_0(\mathbb{R})$ and assume that it corresponds to the exotic case in Proposition \ref{p-lm-trichotomy}, say in case \ref{i-domination-right}. Since every finitely generated subgroup of $\Gamma\subseteq G_+$ is contained in $G_{(a, x)}$ for some $x$, and it commutes with $N_{(x, b)}\subseteq N$, it follows that $G_+$ is locally dominated by commuting elements within $N$. Case \ref{i-domination-left} is analogous. \qedhere
\end{proof}
\begin{proof}[Proof of Proposition \ref{p-lm-trichotomy}]
Let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be an action without fixed points, which is not semi-conjugate to the standard action, nor to any action of $G/N$. Then, by Proposition \ref{p-lm-reconstruction}, there exists $y\in X$ such that either $\varphi(N_{(y, b)})$ or $\varphi(N_{(a, y)})$ has no fixed point. Assume that the first case holds. Note that this implies that $\varphi(N_{(x, b)})$ has no fixed point for every $x\in X$, since $N_{(x, b)}$ is conjugate into $N_{(y, b)}$. Moreover, by Lemma \ref{l-rist-nesting} this also implies that the image $\varphi\left ([K_{(x, b)}, K_{(x, b)}]\right )$ has no fixed point for every $x\in X$, where $K=G_c$. Then Lemma \ref{l-at-least-one-side} implies that every finitely generated subgroup $\Gamma \subseteq G_{(a, x)}$ is totally bounded. This shows that $\varphi$ falls in case \ref{i-domination-right} of Proposition \ref{p-lm-trichotomy}. The case where $\varphi(N_{(a, y)})$ has no fixed point is analogous and leads to case \ref{i-domination-left}.
\end{proof}
\subsection{First consequences of the trichotomy}
\label{s-lm-first-consequences}
Let us describe some elementary consequences of Theorem \ref{t-lm-trichotomy}.
First we record for later use the following observation.
\begin{lem}\label{l.exotic_proximal}
For $X=(a,b)$, let $G\subseteq\homeo_0(X)$ be locally moving, and let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be a minimal faithful action. Then the action $\varphi$ is proximal.
\end{lem}
\begin{proof}
The standard action is proximal (cf.\ the proof of Proposition \ref{p-micro-compact}).
So we assume to be in case \ref{i-domination-right} of Proposition \ref{p-lm-trichotomy} (case \ref{i-domination-left} being analogous). After Theorem \ref{t-centralizer}, we
assume by contradiction that the action has a non-trivial centralizer, which is necessarily an infinite cyclic group generated by an element without fixed points $\tau$, which we can assume satisfies $\tau(x)>x$ for every $x\in X$. Fix $x\in X$ and write $I=[x,\tau(x)]$, which is a fundamental domain for $\tau$.
Let $\Gamma\subset G_+$ be a finitely generated subgroup, and note that as it admits fixed points, it must admit fixed points in the fundamental domain $I$. As $I$ is compact, the intersection of the subsets $\operatorname{\mathsf{Fix}}^\varphi(\Gamma)\cap I$, when $\Gamma$ runs over the finitely generated subgroups of $G_+$, is non-empty, and actually coincides with $\operatorname{\mathsf{Fix}}^\varphi(G_+)\cap I$. In particular, this implies that $G_+$, and thus $G_c$, admits fixed points. As $N_{(x,b)}=[G_c,G_c]_{(x,b)}\subset G_c$, we reach a contradiction.
\end{proof}
We next describe sets of fixed points for exotic actions.
\begin{cor}\label{c-lm-property-exotic}
For $X=(a,b)$, let $G\subseteq\homeo_0(X)$ be locally moving, and let $\varphi\colon G\to \homeo_0(\mathbb{R})$ be an action which is not semi-conjugate to the standard action of $G$ on $X$, nor to any action of a proper quotient. Then the following hold.
\begin{enumerate}[label=(\roman*)]
\item \label{i-lm-support} For every $g\in G$ the support $\supp^\varphi(g)$ accumulates at both $\pm \infty$.
\item \label{i-lm-fixed} Every finitely generated subgroup $\Gamma \subseteq G_c$ is totally bounded with respect to $\varphi$ and its support accumulates at both $\pm \infty$, moreover the boundary $\partial \operatorname{\mathsf{Fix}}^\varphi(\Gamma)$ of its set of fixed points is non-discrete.
\end{enumerate}
\end{cor}
\begin{proof}
Set again $N=[G_c, G_c]$. We first prove \ref{i-lm-support}, which only requires Proposition \ref{p-lm-reconstruction}. Assume by contradiction that $g\in G$ is such that $\supp^\varphi(g)$ is upper-bounded (the case where it is lower-bounded is analogous). Then the germ homomorphism $\mathcal{G}_{+\infty}\circ \varphi \colon G\to \Germ(\varphi(G), +\infty)$ is not injective, and by Proposition \ref{p-micro-normal} its kernel contains $N$. Thus for $h\in N$ we have that $\supp^\varphi(h)$ is also upper-bounded, and the point $\xi=\sup \supp^\varphi(h)$ must be fixed by the centralizer of $h$. Since the centralizer of $h$ contains $N_{(a, x)}$ and $N_{(y, b)}$ for suitable $x, y\in X$, Proposition \ref{p-lm-reconstruction} gives that $\varphi$ must be semi-conjugate to the standard action on $X$ or to an action of $G/N$, which is a contradiction.
Let us now prove \ref{i-lm-fixed}. We apply Theorem \ref{t-lm-trichotomy}, and $\varphi$ must be in the exotic case. Since $G_c=G_+\cap G_-$, in both subdivisions of such case, we have that every finitely generated subgroup $\Gamma \subseteq G_c$ is dominated by commuting elements (within $N$), and thus totally bounded. The fact that its support accumulates at both $\pm\infty$ is a consequence of part \ref{i-lm-support}, applied to any element in a finite generating subset of $\Gamma$.
It remains to prove that $\partial \operatorname{\mathsf{Fix}}^\varphi(\Gamma)$ is non-discrete. For this, let $I=(\xi_1, \xi_2)$ be a connected component of $\supp^\varphi(\Gamma)$. Since the image of $C_N(\Gamma)$ has no fixed point, we can choose $h\in C_N(\Gamma)$ such that $h.\xi_1>\xi_2$. But since $h\in N\subseteq G_c$, the element $h$ is also totally bounded. Thus $h^n.\xi_1$ must converge as $n\to +\infty$ towards a fixed point $\zeta\in \operatorname{\mathsf{Fix}}^\varphi(h)$. Since $\partial \operatorname{\mathsf{Fix}}^\varphi(\Gamma)$ is a closed $\varphi(h)$-invariant set, we have that $h^n.\xi_1$ is a sequence in $\partial \operatorname{\mathsf{Fix}}^\varphi(\Gamma)$ which converges non-trivially to $\zeta\in \partial \operatorname{\mathsf{Fix}}^\varphi(\Gamma) $, proving the statement. \qedhere
\end{proof}
A relevant special case of the second part of Corollary \ref{c-lm-property-exotic} arises when $\Gamma=\langle g\rangle$ is generated by a single element $g\in G_c$. Indeed it implies that if $\varphi\colon G\to \homeo_0(\mathbb{R})$ is an exotic action, the set of fixed points of every element $g\in G_c$ is unbounded, with unbounded complement, and has non-discrete boundary. This can be seen as a first rigidity result of combinatorial nature, saying that actions with particularly ``nice'' structure of fixed points are semi-conjugate to the standard action or to an action of the largest quotient. For instance, recall from the introduction that two locally moving actions on the line of the same group $G$ must be conjugate (this is customarily deduced from the much more general results of Rubin \cite{Rubin,Rubin2}). The first statement in Corollary \ref{c-lm-property-exotic} recovers this result in the following slightly more general form.
\begin{cor}\label{cor.unique}
For $X=(a,b)$, let $G\subseteq \homeo_{0}(\mathbb{R})$ be locally moving. Let $\varphi:G\to \homeo_{0}(\mathbb{R})$ be an action without fixed point whose image contain elements of relatively compact support. Then $\varphi$ is semi-conjugate to the standard action of $G$ on $X$.
\end{cor}
As another application, let us consider actions by \emph{piecewise real-analytic homeomorphisms}, in the following sense.
\begin{dfn} \label{d-pa}Let $\operatorname{\mathsf{PDiff}}^\omega_0([0, 1])$ be the group of all homeomorphisms $f$ of $[0, 1]$ such that there exist finitely many points
\[0=a_0<a_1<\cdots<a_n<a_{n+1}=1\]
such that for every $i\in\{0,\ldots,n\}$, the restriction $f\restriction_{[a_i, a_{i+1}]}$ coincides with an analytic diffeomorphism defined on some open neighborhood of $[a_i, a_{i+1}]$ in the complex plane (in particular $f$ should be analytic on a neighborhood of the endpoints $0$ and $1$).
Given an open interval $(\alpha, \beta)$ (possibly unbounded) we also define the larger group $\operatorname{\mathsf{PDiff}}^\omega_{loc}((\alpha, \beta))$ of homeomorphisms $f$ such that there exists an increasing sequence $(a_n)_{n\in \mathbb Z}$, with $\lim_{n\to - \infty}a_n=\alpha$ and $\lim_{n\to + \infty}a_n=\beta$, such that for every $i\in \mathbb Z$, the restriction $f\restriction_{[a_i, a_{i+1}]}$ coincides with an orientation-preserving diffeomorphism which is analytic on a neighborhood of $[a_i, a_{i+1}]$ in the complex plane (with no condition at $\alpha$ and $\beta$). Note that this includes the more classical setting of piecewise projective homeomorphisms with a discrete set of breakpoints.
\end{dfn}
Since the set of fixed points of any analytic diffeomorphism is isolated, for every element $g\in \operatorname{\mathsf{PDiff}}^\omega_{loc}((\alpha, \beta))$ the boundary $\partial \operatorname{\mathsf{Fix}}(g)$ is a discrete subset of $(\alpha, \beta)$. From Corollary \ref{c-lm-property-exotic} we deduce the following.
\begin{cor}[Rigidity of actions by piecewise analytic homeomorphisms] \label{c-lm-PA}
For $X=(a,b)$, let $G\subseteq \homeo_0(X)$ be locally moving. The following hold.
\begin{enumerate}[label=(\alph*)]
\item \label{i-pa-compact} For every faithful action $\varphi\colon G\to \operatorname{\mathsf{PDiff}}^\omega_0([0, 1])$ without fixed points in the interior $(0, 1)$, the $\varphi$-action of $G$ on $(0, 1)$ is semi-conjugate to its standard action on $X$.
\item For every interval $(\alpha,\beta)$ and faithful action $\varphi\colon G\to \operatorname{\mathsf{PDiff}}^\omega_{loc}((\alpha, \beta))$ without fixed points, one of the following holds:
\begin{enumerate}[label=(b.\arabic*)]
\item either $\varphi$ is semi-conjugate to the standard action of $G$ on $X$,
\item or $\varphi(G)$ admits a closed discrete orbit $O\subset (\alpha, \beta)$, so that $\varphi$ is semi-conjugate to a cyclic action. Moreover, in this case the $\varphi$-image of $N:=[G_c, G_c]$ fixes $O$ pointwise, and its action on each connected component of $\supp^\varphi(N)$ is semi-conjugate to its standard action on $X$.
\end{enumerate}
\end{enumerate}
\end{cor}
\begin{proof}
Let $\varphi\colon G\to \operatorname{\mathsf{PDiff}}^\omega_0([0, 1])$ be a faithful action, without fixed points in $(0, 1)$. Since for every element $g\in \operatorname{\mathsf{PDiff}}^\omega_0([0, 1])$ the set $\partial \operatorname{\mathsf{Fix}}(g)$ is discrete, Corollary \ref{c-lm-property-exotic} implies that the $\varphi$-action on $(0, 1)$ must be semi-conjugate either to the standard action of $G$ on $X$ or to an action of $G/N$, with $N=[G_c, G_c]$. Assume by contradiction that the second case holds. Then $\varphi(N)$ admits fixed points in $(0, 1)$ and since $\operatorname{\mathsf{Fix}}^\varphi(N)$ is $\varphi(G)$-invariant it must accumulate at both $0$ and $1$; moreover its complement $\supp^\varphi(N)=(0, 1)\setminus \operatorname{\mathsf{Fix}}^\varphi(N)$ is non-empty and also accumulates at both $0$ and $1$ by $G$-invariance, so that every neighborhood of $0$, respectively $1$, contains connected components of $\supp^\varphi
(N)$. Since $N$ is a simple group (Proposition \ref{p-micro-normal}), its action on each connected component of $\supp^\varphi(N)$ must be faithful. Thus for every $g\in N$ the set $\partial \operatorname{\mathsf{Fix}}^\varphi(g)$ must accumulate at both $0$ and $1$, and thus cannot be discrete. This is the desired contradiction, and it proves part \ref{i-pa-compact}.
Let now $\varphi\colon G\to \operatorname{\mathsf{PDiff}}^\omega_{loc}((\alpha, \beta))$ be a faithful action. Again, by Corollary \ref{c-lm-property-exotic}, $\varphi$ is semi-conjugate to the standard action or to an action of $G/N$, and we only need to analyze the second case. A similar reasoning as in the compact case, relying on the simplicity of $N$, shows that $\partial \operatorname{\mathsf{Fix}}^\varphi(N)$ must be discrete. Since the latter is a $\varphi(G)$-invariant set, the $G$-action on it must factor through an epimorphism $\tau\colon G\to \mathbb Z$, and $\varphi$ is semi-conjugate to a cyclic action. Finally, the action of $N$ on each connected component of its support must be faithful and semi-conjugate to its standard action on $X$, by part \ref{i-pa-compact} for actions on compact intervals. \qedhere
\end{proof}
\begin{rem}
Corollary \ref{c-lm-PA} can be compared with the work of Lodha \cite{Coherent}, who develops a way to study embeddings between certain groups of PL transformations, such as Thompson--Brown--Stein groups $F_{n_1,\cdots, n_k}$, and proves non-embedding results between them. His approach is based on Rubin's theorem; namely he identifies a class of group actions that he calls \emph{coherent} and shows that such actions are automatically locally moving. The coherent condition is simpler to check than the locally moving condition in some situations. In particular he shows that certain embeddings between groups of PL homeomorphisms must give rise to coherent actions (up to semi-conjugacy), and thus, by Rubin's theorem, to a semi-conjugation to the standard action.
\end{rem}
\subsection{First examples of exotic actions}\label{ss.exoticactions} In this subsection we give some simple examples of actions of locally moving groups which are exotic according to the classification in Proposition \ref{p-lm-trichotomy}. The first two examples show two general mechanisms giving order-preserving actions of the group $\homeo_c(\mathbb{R})$ of compactly supported homeomorphisms of $\mathbb{R}$ on ordered spaces. Thus for countable subgroups $G\subseteq \homeo_c(\mathbb{R})$, we can consider the dynamical realization of these actions and obtain actions on $\mathbb{R}$, which turn to be exotic.
The role of our third example is to explain that it is crucial to consider finitely generated subgroups in the conclusion of Proposition \ref{p-lm-trichotomy}. Indeed we will construct a locally moving subgroup $G\subseteq \homeo_0(X)$ and an exotic action $\varphi\colon G\to \homeo_0(\mathbb{R})$ such that, for every interval $I\Subset X$, the group $[G_I, G_I]$ acts without fixed points (recall that all its finitely generated subgroups must admit fixed points, by Corollary \ref{c-micro-fixed}).
The reader should be warned that all examples of exotic actions discussed in this subsection admit no minimal invariant set. This is a somewhat degenerate phenomenon; for instance it can never arise for finitely generated groups (indeed all groups that we consider here are not finitely generated, and this is essential to the constructions). It is more interesting and less clear how to construct examples of exotic \emph{minimal} actions of locally moving groups. Various such examples will be given later (see Section \ref{sec.examplesRfocal} and \S\S \ref{s-F-plethora1}--\ref{s-F-hyperexotic}).
\subsubsection{Orders of germ type} \label{s-germ-type}
In this paragraph we build exotic actions on the line that are obtained as dynamical realizations of some left-invariant preorder on compactly supported locally moving groups. Our preorders here are inspired by the well known construction of bi-invariant orders on the group of orientation-preserving piecewise linear homeomorphisms of an interval (see \cite{CFP,NavasRivas}).
Let $G\subset\homeo_c(\mathbb{R})$ be a countable locally moving group. Note that $G=G_c$, so that $N=[G,G]$ is the minimal proper normal subgroup of $G$. For $x\in\mathbb{R}$, we denote by $\Germ(G, x)$ the group of germs of $\mathsf{Stab}_G(x)$ at $x$, and for $g\in G$ we let $\mathcal{G}_x(g)$ be its germ. With abuse of notation, we will denote by $\mathsf{id}$ the trivial germ, without reference to the base point.
Recall that groups of germs of interval homeomorphisms are left-orderable (see \cite{GOD}) and therefore, for each $x\in\mathbb{R}$ there exists a left-invariant order $<^{(x)}$ on $\Germ\left (G_{(-\infty,x)},x\right )$. Choose a collection of such orders $\{<^{(x)}\colon x\in \mathbb{R}\}$.
Therefore, we can define $$P=\left \{g\in G:\mathcal{G}_{p_g}(g)>^{(p_g)}\mathsf{id}\right \}$$ where $p_g:=\sup \{x\in X \colon g(x) \neq x\}$. It is straightforward to check that $P$ is a semigroup disjoint from $P^{-1}$, and that it defines a partition $G=P\sqcup \{1\}\sqcup P^{-1}$. Thus, $P$ is the positive cone of a left-invariant order $\prec\in\LO(G)$ (see Remark \ref{r.cones}). Denote by $\varphi:G\to\homeo_0(\mathbb{R})$ its dynamical realization. We want to show that $\varphi$ is an exotic action (according to the classification in Proposition \ref{p-lm-trichotomy}).
First we will show that $\varphi$ is not semi-conjugate to any action induced from a quotient. For this purpose, we claim that the group $\varphi(N)$ acts on $\mathbb{R}$ without fixed points. Indeed to show this it is enough to show that the orbit of $\mathsf{id}$ under the action of $N$ is unbounded above and below in the ordered space $(G, \prec)$. For this, fix an element $h\in G$ with $h\succ \mathsf{id}$. Since $N$ is normal in $G$, the subset $\{p_g\colon g\in N\}\subseteq X$ is $G$-invariant, so that by minimality there exists $g\in N$ with $p_g>p_h$. Upon replacing $g$ with $g^{-1}$ suppose that $\mathcal{G}_{p_g}(g)>^{(p_g)} \mathsf{id}$. Then, since $p_{gh^{-1}}=p_g$ and thus
\[\mathcal{G}_{p_{gh^{-1}}}(gh^{-1})=\mathcal{G}_{p_g}(g) >^{(p_g)} \mathsf{id},\]
we have $g\cdot \mathsf{id}=g\succ h$.
Similarly, for every $h\prec \mathsf{id}$, we can find an element $g'\in N$ such that $g'\cdot \mathsf{id} \prec h$. This shows that the $N$-orbit of $\mathsf{id}$ is unbounded in both directions, as desired. In particular, $\varphi$ is not semi-conjugate to any action that factors through $G/N$. Finally, in order to show that $\varphi$ is not semi-conjugate to the action of $G$, note that the previous argument can be improved to show that for every $x\in \mathbb{R}$, the image $\varphi(N_{(x,+\infty)})$ acts without fixed points: it is enough to take $g\in N$ as above with $p_g>\max\{x,p_h\}$, and as $G$ is locally moving, we can also assume that $\supp(g)\subseteq (x,p_g)$. From this construction we get the following.
\begin{prop} Every countable locally moving group $G\subseteq \homeo_c(\mathbb{R})$ of compactly supported homeomorphisms admits a faithful exotic action on the line.\qed
\end{prop}
By varying on the choice of orders on the groups of germs, one in fact gets an uncountable family of non-semi-conjugate actions. In fact it is not difficult to show the following.
\begin{lem} \label{l-germ-distinguish}
Let $G\subset \homeo_c(\mathbb{R})$ be countable and locally moving. For $i\in \{1,2\}$, let $\mathcal{F}_i=\{<_i^{(x)}\colon x\in \mathbb{R}\}$ be a collection of left-orders on $\Germ\left (G_{(-\infty,x)},x\right )$. Let $\prec_i$ be the associated orders on $G$, and let $\varphi_i$ be their dynamical realizations. Then $\varphi_1$ and $\varphi_2$ are positively semi-conjugate if and only if there exists $y\in \mathbb{R}$ such that $<_1^{(x)}=<_2^{(x)}$ for all $x\geq y$.
\end{lem}
\begin{proof}
First assume that there exists $y\in \mathbb{R}$ as in the statement. Note that the group $G_{(-\infty, y)}$ is convex with respect to both orders $\prec_1, \prec_2$, and that both induce the same order on the coset space $G/G_{(\infty, y)}$. In other words, the quotient preorders of $\prec_1$ and $ \prec_2$ with respect to this convex subgroup coincide, and thus $\prec_1$ and $\prec_2$ admit a common dominating preorder. It follows that $\prec_1$ and $\prec_2$ are equivalent preorders, so that their dynamical realizations are positively semi-conjugate (see Lemma \ref{lem.domsemicon}).
Conversely assume that $\varphi_1$ and $\varphi_2$ are positively semi-conjugate, and let $h\colon \mathbb{R}\to \mathbb{R}$ be a monotone increasing semi-conjugacy from $\varphi_1$ to $\varphi_2$. For $i\in\{1,2\}$, let $\iota_i\colon G\to \mathbb{R} $ be the good embedding associated with $\prec_i$ (Definition \ref{dfn.goodbehaved}), which we assume both to satisfy $\iota_i(\mathsf{id})=0$. Also, for $x\in \mathbb{R}$, denote by $I_{i, x}\subset \mathbb{R}$ the interior of the convex hull of $\iota_i(G_{(-\infty, x)})$. Note that for fixed $i$ the intervals $I_{i, x}$ define an increasing exhaustion of $\mathbb{R}$ as $x\to +\infty$. Since the subgroup $G_{(-\infty, x)}$ is $\prec_i$-convex, we have that for every $g\notin G_{(-\infty, x)}$, the element $\varphi_i(g)$ must map $I_{i, x}$ disjointly from itself, and either to the right or to the left according to the sign of $g$ with respect to the order $\prec_i$. Now choose $y\in \mathbb{R}$ large enough so that $I_{2, y}$ contains $h(0)$. Then if $g\in G$ is such that $p_g>y$ and $g\succ_2 \mathsf{id}$, we have $\varphi_2(g)(h(0))>h(0)$, so by equivariance of $h$ also $\varphi_1(g)(0)>0$, i.e.\ $g\succ_1 \mathsf{id}$. This implies that $<_1^{(x)}=<_2^{(x)}$ for $x>y$. \qedhere
\end{proof}
Recall from the discussion in \S \ref{ssc.complexity} that one way to get insight on the variety of actions on the line of a given group is to study the semi-conjugacy equivalence relation on $\Homirr(G, \homeo_0(\mathbb{R}))$ up to Borel reducibility. The best case scenario would be when this equivalence relation is smooth. However the previous statement implies that this is never the case for locally moving groups of compactly supported homeomorphisms.
\begin{cor}
Let $G\subset \homeo_c(\mathbb{R})$ be countable and locally moving. Then the semi-conjugacy equivalence relation on $\Homirr(G, \homeo_0(\mathbb{R}))$ is not smooth.
\end{cor}
\begin{proof}
Recall from \S \ref{ssc.complexity} that we denote by $\mathcal{E}_0$ the equivalence relation on one-sided binary sequences $\{0, 1\}^\mathbb N$, where $(x_n)$ and $(y_n)$ are equivalent if $x_n=y_n$ for all but finitely many $n$, and that $\mathcal{E}_0$ is not smooth. From Lemma \ref{l-germ-distinguish} it is easy to construct a family of actions of $G$ indexed by $\{0, 1\}^\mathbb N$ (in a Borel way), which are positively semi-conjugate if and only if the two sequences are $\mathcal{E}_0$ equivalent. For instance start from any collection of orders $\mathcal{F}=\{<^{(x)}, x\in \mathbb{R}\}$ on $\Germ(G_{(-\infty, x)}, x)$. For every $\omega\in \{0, 1\}^\mathbb N$, define a new family $\mathcal{F}_\omega$, where for $x\in [n, n+1)$ we leave the order $<^{(x)}$ unchanged if $\omega(n)=0$ and consider the opposite order if $\omega(n)=1$. This together with Lemma \ref{l-germ-distinguish} shows the desired claim.
\end{proof}
This applies for instance to the commutator $[F, F]$ of Thompson's group $F$, and should be compared with Theorem \ref{t-intro-space} for locally moving groups in the class $\mathcal{F}_0${}\; (see also Theorem \ref{t-F-classification} for Thompson's group $F$).
\subsubsection{Escaping sequences} \label{s-escaping-sequence} Let $G\subseteq \homeo_c(\mathbb{R})$ be a locally moving group of compactly supported homeomorphisms. The action of $G$ on $\mathbb{R}$ defines a diagonal action of $G$ on the space of sequences $\mathbb{R}^\mathbb N$. For the second construction, we fix a sequence $\omega_0\in \mathbb{R}^\mathbb N$ of distinct points in $\mathbb{R}$, where we write $\omega_0(n)$ for the $n$-th term, and we suppose that $\omega_0$ is \emph{escaping}, in the sense that every compact subset of $\mathbb{R}$ contains only finitely many terms of the sequence (in other terms, it has no accumulation point). Then we write $\mathsf{S}=G.\omega_0$ for the $G$-orbit of the sequence $\omega_0$. Since elements of $G$ have compact support, and the sequence $\omega_0$ is escaping, it holds that every $\omega \in \mathsf{S}$ agrees with $\omega_0$ on all but finitely many entries. Thus, for $\omega,\omega'\in\mathsf{S}$, we can declare that $\omega \prec \omega'$ if for the largest entry on which they disagree (say $n$), we have that $\omega(n)<\omega'(n)$. Note as $G$ is locally moving, the diagonal action of $G$ on $\mathsf{S}$ is faithful. Moreover, it preserves the total order $\prec$.
Now suppose that $\mathsf{S}$ is countable (this holds, for instance, when $G$ is a countable group) and consider the dynamical realization $\varphi:G\to \homeo_0(\mathbb{R})$ of the diagonal action of $G$ on $(\mathsf{S},\prec)$. We claim that $\varphi$ is an exotic action. To see this, consider two sequences $\omega\prec \omega'$ in $\mathsf S$ and let
$n\in \mathbb N$ be such that the $k$-th entries of both $\omega$ and $\omega'$ agree with the $k$-th entry of $\omega_0$ for all $k\geq n$. So, if $g\in G$ satisfies $g(\omega_0(n))\neq \omega_0(n)$, then we have \[
\min\{g.\omega_0,g^{-1}.\omega_0\}\prec\omega\prec\omega'\prec\max\{g.\omega_0,g^{-1}.\omega_0\}.\]
This implies, on the one hand, that the action of $N=[G,G]=[G_c,G_c]$ (which is a subgroup without fixed points) on $\mathsf{S}$ is cofinal (i.e.\ every $N$-orbit is unbounded in both directions), and hence $\varphi(N)$ acts without fixed points. In particular $\varphi(G)$ is not semi-conjugate to any action coming from a factor of $G$ (see Proposition \ref{p-micro-normal}). On the other hand, since $\omega_0$ is escaping, for any compact interval $I\subset \mathbb{R}$, the action of $G_{\mathbb{R}\setminus I}=\{g\in G: I\subseteq \operatorname{\mathsf{Fix}}(g)\}$ on $\mathsf{S}$ is cofinal as well, so $\varphi(G_{\mathbb{R}\setminus I})$ has no fixed points. In particular, $\varphi(G)$ is not semi-conjugate to the standard action.
We also remark that this construction can be used to show the following (to be compared with Corollary \ref{c-intro-lr}).
\begin{prop}
Let $G\subset \homeo_c(\mathbb{R})$ be a locally moving group of compactly supported homeomorphisms. Then the standard action of $G$ on $\mathbb{R}$ is not locally rigid.
\end{prop}
\begin{proof}
Given an escaping sequence $\omega$, denote by $\preceq_\omega$ the preorder induced on $G$ by the action on the orbit $\mathsf{S}_\omega$, with $g\preceq_\omega h$ if $g.\omega\preceq h.\omega$. Denote also by $\prec_0$ the preorder on $G$ induced by its standard action and the point $0\in \mathbb{R}$, by $g\prec_0 h$ if $g(0)\leq h(0)$. Choose a sequence $\omega_n\in \mathbb{R}^\mathbb N$ of escaping sequences such that $\omega_n(0)=0$ and $|\omega_n(j)|>n$ if $j>0$. Then for every finite subset $S\subset G$ and $n$ large enough we have that every $s\in S$ satisfies $s\prec_{\omega_n} \mathsf{id}$ if and only if $s\prec_0 \mathsf{id}$. Thus the sequence of preorders $(\prec_{\omega_n})$ converges to $\prec_0$ in $\LPO(G)$. Thus by an easy adaptation to preorders of the argument in \cite[Proposition 3.3]{MannRivas}, this implies that the sequence of actions associated to $\omega_n$ as above can be positively conjugated to a sequence $(\varphi_n)$ which converges to the standard action of $G$ in the compact open topology. \qedhere
\end{proof}
\subsubsection{An example where the rigid stabilizers have no fixed points}\label{s-example-PL-Q} We write $\PL_{\mathbb Q}((0,1))$ for the Bieri--Strebel group $G((0,1);\mathbb Q,\mathbb Q_{>0} )$ of piecewise linear homeomorphisms that preserve $\mathbb Q\cap (0,1)$. For a prime number $q\in \mathbb N$, consider the corresponding $q$-adic order $\nu_q:\mathbb Q_{>0}\to \mathbb Z$ (these are the functions such that $r=\prod_{q\text{ prime}}q^{\nu_q(r)}$ for every $r\in \mathbb Q_{>0}$). We then write $\pi_q(r)=q^{\nu_q(r)}$, which corresponds to the power of the prime $q$ appearing in the factorization of $r\in \mathbb Q_{>0}$. Given an element $f\in\PL_{\mathbb Q}((0,1))$, we define $D^-_{q}f:=\pi_q\circ D^-f$. Observe that the chain rule works for $D_q^-$, namely \begin{equation}\label{ecu.chain} D_q^-(f g)(x)=D_q^-f(g(x))\, D_q^-f(x)\quad\text{for every }x\in (0,1)\text{ and }f,g\in\PL_{\mathbb Q}((0,1)).\end{equation}
Every prime number $q\in \mathbb N$ defines a preorder on $\PL_\mathbb Q((0,1))$ in the following way. We consider the subset $H_q\subset \PL_\mathbb Q((0,1))$ defined by
\[
H_q=\left \{f\in \PL_\mathbb Q((0,1))\colon D_q^-f(x)=1\quad\text{for every }x\in (0,1)\right \},
\]
and then the subset
\[
P_q=\left \{f\in\PL_\mathbb Q((0,1))\setminus H_q:D^-_qf(x_f)>1\right \},
\]
where for $f\in\PL_\mathbb Q((0,1))$, we set $x_f=\max\{x\in (0,1]:D^-_qf(x)\neq 1\}$. We have the following.
\begin{prop}\label{prop.isacone} For every prime $q\in \mathbb N$, the subset $P_q$ is the positive cone of a preorder $\preceq_q$ of $\PL_\mathbb Q((0,1))$.
\end{prop}
\begin{proof} The chain rule \eqref{ecu.chain} easily gives that $P_q$ is a semigroup, $\PL_\mathbb Q((0,1))=P_q\sqcup H_q\sqcup P^{-1}_q$, $H_q$ is a subgroup and $H_qP_qH_q\subseteq P_q$. The result then follows from Remark \ref{r.cones}.
\end{proof}
We are ready to construct our action. For this, given $f\in \PL_\mathbb Q((0,1))$ consider the subset $E_f=\{q:f\notin H_q\}$, which is the collection of primes appearing in the factorization of the left derivative $D^-f(x)$, for some $x\in (0,1)$. Note that $E_f$ is a finite set, so we can consider the prime $p_f:=\max E_f$ (where we are considering the standard ordering of primes). Then, define $$P:=\{f\in \PL_\mathbb Q((0,1)): \mathsf{id} \prec_{p_f} f\},$$ where $\preceq_{p_f}$ is the preorder on $\PL_\mathbb Q((0,1))$ from Proposition \ref{prop.isacone}.
\begin{prop} The subset $P$ is the positive cone of an order $\prec$ on $\PL_\mathbb Q((0,1))$. Furthermore, if $\varphi\colon G\to \homeo_0(\mathbb{R})$ is the dynamical realization of $\prec$, then for every non-empty subinterval $I\subseteq (0,1)$ the $\varphi$-image of $[G_I, G_I]$ acts without fixed points.
\end{prop}
\begin{proof} Write $G=\PL_\mathbb Q((0,1))$. For a non-trivial element $f\in G$, the chain rule \eqref{ecu.chain} applied to the inverse function gives that $E_f=E_{f^{-1}}$ and that $f\in P$ if and only if $f^{-1}\notin P$. Thus, we have $ \PL_\mathbb Q((0,1))=P\sqcup\{1\}\sqcup P^{-1}$. Also note that if $f,g\in P$, it holds that $p_{fg}=\max\{p_f,p_g\}$ and that $\mathsf{id}\preceq_{p_{fg}} f$ and $\mathsf{id}\preceq_{p_{fg}} g$, one inequality being strict. Thus, $fg\in P$ showing that $P$ is a positive cone.
Now let $I\subseteq (0,1)$ be a non-empty subinterval, let $G_I$ be its rigid stabilizer inside $G$ and take $f\in G$. Since $E_f$ is finite, there is $g\in [G_I, G_I]$ such that $p_g> \max E_f$, so in particular $\min\{g,g^{-1}\} \preceq f \preceq \max\{g,g^{-1}\}$. This shows that the $[G_I,G_I]$-orbit of $\mathsf{id}$ in $(G,\prec)$ is unbounded in both directions, and thus $\operatorname{\mathsf{Fix}}^\varphi([G_I, G_I])=\emptyset$, as desired. \end{proof}
Analogous constructions will be presented in \S \ref{ss.Bieri-Strebelfocal} for general Bieri--Strebel groups.
\section{Notation and preliminaries}\label{s-preliminaries}
\subsection{Actions on the line}
In this work we are mainly concerned with orientation-preserving actions on the real line, that is, homomorphisms $\varphi:G\to \homeo_0(\mathbb{R})$.
We will almost always be interested in actions without (global) fixed points, which will sometimes be called \emph{irreducible} for short. Note that every action $\varphi:G\to \homeo_0(\mathbb{R})$ can be described in terms of irreducible actions, just considering all restrictions
\[
\dfcn{\varphi_J}{G}{\homeo_{0}(J)\cong \homeo_{0}(\mathbb{R})}{g}{\varphi(g)\restriction_J},
\]
of the action $\varphi$ to minimal $\varphi(G)$-invariant open intervals $J\subset \mathbb{R}$. We write $\Hom(G,\homeo_{0}(\mathbb{R}))$ for the space of order-preserving actions of the group $G$, endowed with the compact open topology, and $\Homirr(G,\homeo_{0}(\mathbb{R}))\subseteq \Hom(G,\homeo_{0}(\mathbb{R}))$ for the subspace of actions without fixed points.
\subsubsection{Some notation on actions}
Given $f\in \homeo(\mathbb{R})$, we write $\operatorname{\mathsf{Fix}}(f)=\{x\in \mathbb{R} \colon f(x)=x\}$ for the set of fixed points and $\supp(f)=\mathbb{R}\setminus \operatorname{\mathsf{Fix}}(f)$ for its \emph{support}. Note that by definition $\supp(f)$ is an open set (we do \emph{not} take its closure unless specified). For $G\subset \homeo_0(\mathbb{R})$ and $x\in \mathbb{R}$ we write $\stab_G(x)$ for the stabilizer of $x$. We denote by $\homeo_c(\mathbb{R})$ the group of homeomorphisms whose support is relatively compact. Given an action $\varphi\colon G\to \homeo(\mathbb{R})$ and $g\in G$ we set $\operatorname{\mathsf{Fix}}^\varphi(g)=\operatorname{\mathsf{Fix}}(\varphi(g))$ and $\suppphi(g)=\supp(\varphi(g))$, and $\stab^\varphi_G(x)$ for the stabilizer. When there is no confusion we write $g.x$ instead of $\varphi(g)(x)$. The notation $g(x)$ will be reserved to the case when $G$ is naturally given as a subgroup $G\subset \homeo_0(\mathbb{R})$ to refer to its standard action (but never to another action $\varphi\colon G\to \homeo_0(\mathbb{R})$).
For $x\in \mathbb{R}\cup\{\pm \infty\}$ we denote by $\Germ(x)$ the group of germs of homeomorphisms that fix $x$. Recall that this is defined as the group of equivalence classes of homeomorphisms $f\in \homeo_0(\mathbb{R})$ that fix $x$, where two such homeomorphisms are identified if they coincide on a neighborhood of $x$. By considering only one-sided neighborhoods, one gets two groups $\Germ_-(x)$ and $\Germ_+(x)$, the groups of right germs and the group of left germs respectively, such that $\Germ(x)=\Germ_-(x)\times \Germ_+(x)$. If $G$ is a group of homeomorphisms that fixes $x$, we denote by $\Germ(G, x)$ the group of germs induced by elements of $G$, and by $\mathcal{G}_x\colon G\to \Germ(G, x)$ the associated homomorphism. Similarly we have groups of one-sided germs $\Germ_\pm(G, x)$. Note however that when $G$ acts non-trivially only on one side of $G$ (e.g.\ if $x$ is an endpoint of its support, or if $x=\pm \infty$), we omit the sign $\pm$ as this is clear from the context.
\subsubsection{Commuting actions}
A particular case of what said above is that every homeomorphism $g\in \homeo_{0}(\mathbb{R})$ is basically determined by its set of fixed points $\operatorname{\mathsf{Fix}}(g)$ and by how it acts on every connected component of its support $\supp(g)=\mathbb{R}\setminus \operatorname{\mathsf{Fix}}(g)$. Therefore, it is fundamental to understand the set of fixed points of a given element, or at least to be able to say whether it is empty or not. For this, a very useful relation is that for an element $g\in \homeo_{0}(\mathbb{R})$ and a subgroup $H\subseteq \homeo_0(\mathbb{R})$, one has $g(\operatorname{\mathsf{Fix}}(H))=\operatorname{\mathsf{Fix}}(gHg^{-1})$. In particular, when $G\subseteq \homeo_0(\mathbb{R})$ and $H\lhd G$ is a normal subgroup, then for every $g\in G$ one has $g(\operatorname{\mathsf{Fix}}(H))=\operatorname{\mathsf{Fix}}(H)$. This holds in particular for commuting subgroups, for which we have the following observation, which is easily obtained using that the set of fixed point is a closed subset.
\begin{lem} \label{l.fixed_commuting}
Consider two commuting subgroups $H_1$ and $H_2$ of $G\subseteq \homeo_{0}(\mathbb{R})$ (that is, $[h_1,h_2]=\mathsf{id}$ for every $h_1\in H_1$ and $h_2\in H_2$). Suppose that both $\operatorname{\mathsf{Fix}}(H_1)$ and $\operatorname{\mathsf{Fix}}(H_2)$ are non-empty. Then $H_1$ and $H_2$ admit a common fixed point.
\end{lem}
This lemma will be used in the text without explicit reference.
\subsubsection{Semi-conjugacy}
It is customary in the field to consider actions up to \emph{semi-conjugacy}. This means that not only we do not really take care of the choice of coordinate on $\mathbb{R}$ (which corresponds to the classical notion of \emph{conjugacy}), but we want to consider only the interesting part of the dynamics of the action. This was first formalized by Ghys in his work on bounded Euler class, but the definition has been unanimously fixed only recently. We follow here \cite[Definition 2.1]{KKMj}, although we allow order-reversing semi-conjugacies. For the statement we will say that a map $h:\mathbb{R}\to \mathbb{R}$ is \emph{proper} if its image $h(\mathbb{R})$ is unbounded in both directions of the line.
\begin{dfn}
Let $\varphi,\psi:G\to \homeo(\mathbb{R})$ be two actions of a group $G$ on the real line. We say that $\varphi$ and $\psi$ are \emph{semi-conjugate} if there exists a proper monotone map $h:\mathbb{R}\to\mathbb{R}$ such that
\begin{equation}\label{eq:semiconj}
h\varphi(g)=\psi(g)h\quad \text{for every }g\in G.
\end{equation}
Such a map $h$ is called a \emph{semi-conjugacy} between $\varphi$ and $\psi$. Note that if $\varphi$ and $\psi$ are irreducible, the requirement that the map $h$ be proper follows automatically from the equivariance \eqref{eq:semiconj} and thus can be dropped.
%
When $h\in\homeo(\mathbb{R})$, we say that $h$ is a \emph{conjugacy} between $\varphi$ and $\psi$, in which case we say that $h$ is a \emph{conjugacy}.
%
When $h$ is nondecreasing, we say that $\varphi$ and $\psi$ are \emph{positively} semi-conjugate.
\end{dfn}
\begin{rem} Both conjugacy and semi-conjugacy are equivalence relations (for conjugacies this is obvious, for semi-conjugacies the reader can check \cite[Lemma 2.2]{KKMj} or \cite[Proposition 1.1.16]{GOD}). Notice that in the semi-conjugacy case, we do not require that $h$ be continuous; indeed, being continuously semi-conjugate is not even a symmetric relation. \end{rem}
Rarely (essentially only in Section \ref{s-circle}) we will need the analogous notion of semi-conjugacy for actions on the circle, which is defined as follows.
\begin{dfn} Let $\varphi, \psi\colon G\to \homeo_0(\mathbb{S}^1)$ be two actions on the circle. They are semi-conjugate if and only if there exists a group $\tilde{G}$ which is a central extension of $G$ of the form $1\to C\to \tilde{G}\to G\to 1$, with $C\cong \mathbb Z$, and two semi-conjugate actions $\tilde{\varphi}, \tilde{\psi}\colon \tilde{G}\to \homeo_0(\mathbb{R})$ which both map $C$ to the group $\mathbb Z$ of integer translations, and which descend to the quotient respectively to the actions $\varphi$ and $\psi$ of $G=\tilde{G}/C$ on $\mathbb{S}^1=\mathbb{R}/\mathbb Z$.
\end{dfn}
Given an action $\varphi:G\to \homeo(\mathbb{R})$, one can consider the \emph{reversed action} $\widehat\varphi:G\to \homeo(\mathbb{R})$, defined by conjugating $\varphi$ by the order-reversing isometry $x\mapsto -x$. After our definition, the actions $\varphi$ and $\widehat{\varphi}$ are conjugate.
Given a monotone map $h:\mathbb{R} \to \mathbb{R}$, we denote by $\gap(h)\subset \mathbb{R}$ the open subset of points at which $h$ is locally constant. We also write $\core(h)=\mathbb{R}\setminus \gap(h)$. Note that when $h:\mathbb{R}\to \mathbb{R}$ is a semi-conjugacy between two actions $\varphi,\psi:G\to \homeo(\mathbb{R})$ without fixed points (in the sense that the equivariance \eqref{eq:semiconj} holds), then $\core(h)$ is a closed $\varphi(G)$-invariant subset and $h(\core(h))=h(\mathbb{R})$.
The following folklore result is a sort of converse to this observation, and it describes the inverse operation of Denjoy's blow up of orbits (compare for instance \cite[Theorem 2.2]{KKMj}).
\begin{thm}\label{prop.basica}
Let $\varphi:G\to\homeo_0(\mathbb{R})$ be an action without fixed points and let $F\subset \mathbb{R}$ be a non-empty closed $\varphi(G)$-invariant subset. Then there exists an action $\varphi_F:G\to \homeo_0(\mathbb{R})$ and a continuous positive semi-conjugacy $h:\mathbb{R}\to \mathbb{R}$ between $\varphi$ and $\varphi_F$ such that $\core(h)=F$.
\end{thm}
We follow with an easy application of Theorem \ref{prop.basica} that will be repeatedly used in the article when discussing actions coming from quotients.
\begin{cor}\label{cor.normalsemicon} Let $\varphi:G\to \homeo_0(\mathbb{R})$ be an action without fixed points, and let $N\lhd G$ be a normal subgroup. Then $\varphi$ is semi-conjugate to an action of the quotient $G/N$ if and only if $\operatorname{\mathsf{Fix}}^\varphi(N)\neq\varnothing$.\end{cor}
\begin{proof} Notice that, since $N$ is normal, the subset $F=\operatorname{\mathsf{Fix}}^\varphi(N)$ is closed and $\varphi(G)$-invariant. Assume that $F\neq\varnothing$; we consider the action $\varphi_F$ given by Theorem \ref{prop.basica}. Clearly we have $N\subseteq \ker \varphi_F$. The other implication is trivial.
\end{proof}
The following lemma basically states that the semi-conjugacy class is determined by the action on one orbit.
\begin{lem}\label{lem.semiconjugacy} Consider two actions $\varphi,\psi:G\to\homeo_0(\mathbb{R})$ without fixed points. Let $\Omega\subset \mathbb{R}$ be a non-empty $\varphi(G)$-invariant subset and $j_0:\Omega\to\mathbb{R}$ a monotone map which is equivariant in the sense that it satisfies
\[
\psi(g) j_0=j_0\varphi(g)\restriction_\Omega\quad\text{for every } g\in G.\]
Then, $j_0$ can be extended to a semi-conjugacy $j:\mathbb{R}\to \mathbb{R}$ between $\varphi$ and $\psi$.
\end{lem}
\begin{proof} Consider the map $j:\mathbb{R}\to \mathbb{R}$ defined by \[j(x)=\sup\{j_0(y):y\in \Omega\text{ and }y\leq x\}.\]
Since $j_0$ is monotone, we get that $j$ is an extension of $j_0$. Also note, it is direct from the definition of $j$ that it is monotone and equivariant and therefore it defines a semi-conjugacy as desired.
\end{proof}
\subsubsection{Canonical model}
We next introduce a family of canonical representatives for the semi-conjugacy relation when $G$ is finitely generated. Such representatives are well defined up to conjugacy. Later, in \S \ref{sec.harmder}, we will define a (much less redundant) family of representatives for the semi-conjugacy relation consisting on normalized $\mu$-harmonic actions.
\begin{dfn}
Let $\varphi:G\to\homeo_0(\mathbb{R})$ be an action, we say that a non-empty subset $\Lambda\subset \mathbb{R}$ is a \emph{minimal invariant set} for $\varphi(G)$ if it is closed, $\varphi(G)$-invariant and contains no proper, closed and $\varphi(G)$-invariant subsets. When $\Lambda=\mathbb{R}$, we say that the action $\varphi$ is \emph{minimal}.
When $\Lambda$ is a proper perfect subset we say that $\Lambda$ is an \emph{exceptional minimal invariant set}.
On the other hand, when the image of an action $\varphi(G)$ is generated by a homeomorphism without fixed points, we say that the action $\varphi$ is \emph{cyclic}.
\end{dfn}
\begin{rem}\label{r.maynotexist}
Note that minimal invariant sets may not exist for general group actions. An archetypical example is given by an action of the group $\mathbb Z^\infty =\bigoplus_\mathbb Z \mathbb Z$ in which each canonical generator has fixed points but there is no fixed point for the action (this happens, for instance, in the dynamical realization of the lexicographic ordering of $\mathbb Z^\infty$). However, when the acting group is finitely generated there is always a minimal invariant set for the action; see e.g.\ \cite[Proposition 2.1.12]{Navas-book}.
\end{rem}
\begin{rem}\label{r.semi_is_continuous}
Let $h:\mathbb{R}\to \mathbb{R}$ be a semi-conjugacy between two actions $\varphi,\psi:G\to \homeo_0(\mathbb{R})$, in the sense that \eqref{eq:semiconj} holds.
It is immediate to show that when $\psi$ is minimal, the semi-conjugacy $h$ is continuous, and that when $\varphi$ is also minimal, then $h$ is a conjugacy. Indeed, the subsets $\core(h)$ and $\overline{h(\mathbb{R})}$ are closed subsets which are respectively $\varphi(G)$- and $\psi(G)$-invariant.
\end{rem}
The following notion corresponds to the \emph{minimalization} considered in \cite[Definition 2.3]{KKMj}.
\begin{dfn} An action $\varphi:G\to\homeo_0(\mathbb{R})$ is a \emph{canonical model} if either it is a minimal action or it is a cyclic action.
\end{dfn}
We have the following consequence of Theorem \ref{prop.basica}.
\begin{cor}\label{cor.basica} Let $\varphi:G\to\homeo_0(\mathbb{R})$ be an action without fixed points that admits a non-empty minimal invariant set (this is automatic if $G$ is finitely generated). Then, $\varphi$ is semi-conjugate to a canonical model. Moreover, the canonical model is unique up to conjugacy.
\end{cor}
A similar discussion holds for actions on the circle. In this case we say that an action $\varphi\colon G\to \homeo_0(\mathbb S^1)$ is in canonical model if it is either minimal or conjugate to an action whose image is a finite cyclic group of rotations (in which case we say that it is \emph{cyclic}). However a crucial simplifying difference is that in this case, by compactness, \emph{every} group action on the circle admits a non-empty minimal invariant set (regardless on whether $G$ is finitely generated or not). This yields the following well-known fact (see e.g.\ \cite{Ghys}).
\begin{prop} \label{p-circle-canonical}
Let $G$ be a group, then every action $\varphi\colon G\to \homeo_0(\mathbb S^1)$ without fixed points is semi-conjugate to a canonical model.
\end{prop}
After Corollary \ref{cor.basica}, semi-conjugacy classes of finitely generated group actions can be divided into two very different families: cyclic actions and minimal actions. However, for practical purposes, it is preferable to split further the case of minimal action. In this work the following classical notion will be important.
\begin{dfn} For $M\in \{\mathbb{R},\mathbb S^1\}$, we say that a minimal action $\varphi:G\to\homeo_0(M)$ is \emph{proximal} if for every non-empty open intervals $I,J\subsetneq M$ with $I$ bounded, there exists an element $g\in G$ such that $g.I\subset J$.
\end{dfn}
\begin{rem}\label{r-centralizer}
Note that if a subgroup $G\subseteq \homeo_0(M)$ commutes with a non-trivial element $h\in \homeo_0(M)$, then the $G$-action cannot be proximal. In fact this is a well-known observation for group actions on arbitrary compact spaces.
\end{rem}
A crucial feature of minimal actions on the circle and the line is that a converse also holds. In the case of the circle this goes back to Antonov \cite{Antonov} and was rediscovered by Margulis \cite{Margulis}, and an analogous result for the real line is provided by Malyutin \cite{Malyutin}. Given a subgroup $G\subseteq\homeo_0(M)$ with $M\in \{\mathbb{R},\mathbb S^1\}$, its \emph{centralizer} (in $\homeo_0(M)$) is the subgroup
\[C(G):=\left \{h\in\homeo_0(M):gh=hg\text{ for every }g\in G\right \}.\]
We will write $C^\varphi:=C({\varphi(G)})$ when $\varphi:G\to\homeo_0(M)$ is an action.
Then we have the following two results.
\begin{thm}[Antonov \cite{Antonov}, see \cite{Ghys}]\label{t-Margulis}
Let $\varphi: G\to \homeo_{0}(\mathbb S^1)$ be a minimal action. Then we have the following alternative:
\begin{itemize}
\item either $C^\varphi$ is trivial, in which case $\varphi$ is proximal, or
\item $C^\varphi$ is a finite cyclic subgroup without fixed points, and the action on the topological circle $\mathbb S^1/C^\varphi$ is proximal, or
\item $C^\varphi$ is conjugate to the group of rotations $\mathbb{R}/\mathbb Z$ and $G$ is conjugate to a subgroup of it.
\end{itemize}
\end{thm}
\begin{thm}[Malyutin \cite{Malyutin}, see also \cite{GOD}]\label{t-centralizer}
Let $\varphi:G\to\homeo_0(\mathbb{R})$ be a minimal action. Then we have the following alternative:\begin{itemize}
\item either $C^\varphi$ is trivial, in which case $\varphi$ is proximal, or
\item $C^\varphi$ is a cyclic subgroup without fixed points, and the action on the topological circle $\mathbb{R}/C^\varphi$ is proximal, or
\item $C^\varphi$ is conjugate to the group of translations $(\mathbb{R}, +)$ and $G$ is conjugate to a subgroup of it.
\end{itemize}
\end{thm}
\subsection{Harmonic actions and Deroin spaces}\label{sec.harmder}\label{ssubsec.Deroin}
In this subsection we recall the definition of the {\em Deroin space}. For that, we fix a finitely generated group $G$, a probability measure $\mu$ on $G$ whose support is finite, generates $G$, and satisfies the symmetry property
\[
\mu(g)=\mu(g^{-1})\quad\text{for every }g\in G.
\]
We will denote by $S\subset G$ the support of $\mu$.
\begin{dfn}
Given a nontrivial action $\varphi:G\to \homeo_{0}(\mathbb{R})$, we say that a Radon measure $\nu$ on $\mathbb{R}$ is \emph{stationary} for the action $\varphi$ if for every Borel subset $A\subset \mathbb{R}$ one has
\[
\nu(A)=\sum_{g\in S}\nu(g^{-1}.A)\,\mu(g).
\]
When the Lebesgue measure $\Leb$ on $\mathbb{R}$ is stationary, we say that the action $\varphi$ is \emph{$\mu$-harnomic}.
\end{dfn}
Properties of $\mu$-harmonic actions are discussed in \cite[\S 4.4]{GOD}, and strongly rely on results of Deroin, Kleptsyn, Navas, and Parwani \cite{DKNP} on symmetric random walks in $\homeo_0(\mathbb{R})$. As remarked by Deroin, the space of $\mu$-harmonic actions of $G$ has several remarkable properties. To describe this, we need first to discuss two fundamental results.
\begin{prop}[\cite{DKNP}]\label{prop.minimalisharmonic}
With notation as above, the following properties hold.
\begin{enumerate}
\item Every non-trivial $\mu$-harmonic action $\varphi:G\to \homeo_{0}(\mathbb{R})$ is a canonical model.
\item Conversely, every canonical model $\varphi:G\to \homeo_{0}(\mathbb{R})$ is conjugate to a $\mu$-harmonic action $\psi:G\to \homeo_{0}(\mathbb{R})$.
\item\label{i:DKNP_affine} Moreover, any two $\mu$-harmonic actions $\psi_1$ and $\psi_2$ of a group $G$ are conjugate by an affine homeomorphism.
\end{enumerate}
\end{prop}
After a careful reading of the proof of \cite[Proposition 8.1]{DKNP}, there is a natural way to reduce the symmetries in part \eqref{i:DKNP_affine} of the statement above, to the group of translations. For this, given a homeomorphism $h\in \homeo_{0}(\mathbb{R})$ consider the following area function:
\[
\mathsf A^h(\xi)=\left\{\begin{array}{lr}
\int_{h^{-1}(\xi)}^{\xi}(h(\eta)-\xi)\,d\eta&\text{if }h(\xi)\ge \xi,\\[.5em]
\int_{h(\xi)}^{\xi}(h^{-1}(\eta)-\xi)\,d\eta&\text{if }h(\xi)\le \xi.\\
\end{array} \right.
\]
Note that $\mathsf A^h(\xi)\ge 0$ for every $\xi \in \mathbb{R}$, and $\mathsf A^h(\xi)=0$ if and only if $\xi\in \operatorname{\mathsf{Fix}}(h)$.
In the case $h(\xi)\ge \xi$, the quantity $\mathsf A^h(\xi)$ represents indeed the area of the bounded planar region delimited by the segments $[h^{-1}(\xi),\xi]\times \{\xi\}$, $\{\xi\}\times [\xi,h(\xi)]$ and the graph of $h$ (when $h(\xi)\le \xi$, one has to switch the endpoints of the segments, and the same interpretation is valid). Being a two-dimensional area, it should be clear that when considering the map $\widetilde h=ah a^{-1}$, obtained by conjugating $h$ by an affine map $a(\xi)=\lambda x+b$, one has the following relation:
\begin{equation}\label{eq.change_area}
\lambda^2 \mathsf A^h(\xi)= \mathsf A^{\widetilde h}(a(\xi))\quad\text{for every }\xi\in \mathbb{R}.
\end{equation}
\begin{lem}[\cite{DKNP}]\label{l.constant_area}
With notation as above, let $\varphi:G\to \homeo_{0}(\mathbb{R})$ be a $\mu$-harmonic action. Then the expected area function $\mathsf{A}^\varphi:\mathbb{R}\to \mathbb{R}$ defined by
\[
\mathsf A^\varphi(\xi)=\sum_{g\in S} \mathsf{A}^{\varphi(g)}(\xi)\,\mu(g),
\]
is constant and positive.
\end{lem}
\begin{dfn}\label{dfn.Deroin}
With notation as above, we introduce the \emph{Deroin space} $\Der_\mu(G)$ of normalized $\mu$-harmonic actions as the subset of $\mu$-harmonic actions $\varphi\in \Hom(G,\homeo_{0}(\mathbb{R}))$ such that $\mathsf A^\varphi=1$, endowed with the induced compact-open topology.
\end{dfn}
Note that as the group $G$ is finitely generated by the subset $S$, the compact-open topology of $\Hom(G,\homeo_{0}(\mathbb{R}))$ is simply given by the product topology of $\homeo_0(\mathbb{R})^S$, where $\homeo_0(\mathbb{R})$ is considered with the topology of uniform convergence on compact subsets.
\begin{thm}[\cite{DKNP, GOD}]\label{t.def_Deroin}
With notation as above, the following properties hold.
\begin{enumerate}
\item\label{i.def_Deroin} The Deroin space $\Der_\mu(G)$ is compact.
\item\label{ii.def_Deroin} Every action $\varphi:G\to \homeo_{0}(G)$ without fixed points is positively semi-conjugate to a normalized $\mu$-harmonic action $\psi\in \Der_\mu(G)$, which is unique up to conjugacy by a translation.
\item\label{iii.def_Deroin} Conjugacy by translations defines a topological flow $\Phi:\mathbb{R}\times \Der_\mu(G)\to \Der_\mu(G)$, called the \emph{translation flow}. Explicitly, this is given by
\[
\Phi^t(\varphi)(g)=T_t\varphi(g)T_{-t},
\]
where $T_t:\xi\to \xi+t$ denotes the translation by $t\in \mathbb{R}$.
\end{enumerate}
\end{thm}
\begin{proof}[Sketch of proof]
After the identity \eqref{eq.change_area} and Lemma \ref{l.constant_area}, if $\varphi$ is a normalized $\mu$-harmonic action, then also $\Phi^t(\varphi)$ is normalized for every $t\in \mathbb{R}$. So the translation flow is well-defined.
After Corollary \ref{cor.basica} and Proposition \ref{prop.minimalisharmonic}, we know that every action $\varphi:G\to \homeo_{0}(\mathbb{R})$ without fixed points, is semi-conjugate to a $\mu$-harmonic action, which is unique up to affine rescaling. Moreover, the identity \eqref{eq.change_area} and Lemma \ref{l.constant_area} give that after an affine rescaling, we can assume that the $\mu$-harmonic action is normalized. This gives \eqref{ii.def_Deroin}.
Details for the fact that $\Der_\mu(G)$ is compact can be found in \cite[Proof of Theorem 5.4]{deroin2020non}.
\end{proof}
Although this will not be used, note that the group $G$ acts on the Deroin space $\Der_\mu(G)$ by the formula $g.\varphi=\Phi^{\varphi(g)(0)}(\varphi)$ (see \cite[(4.5)]{GOD}). In other words, the action on the parameterized $\Phi$-orbit of $\varphi$ is basically the action $\varphi$.
Let us point out the following consequences of Theorem \ref{t.def_Deroin}.
\begin{cor}\label{rem.fixes0}
With notation as above, let $\varphi_1$ and $\varphi_2$ be two normalized $\mu$-harmonic actions in $\Der_\mu(G)$, which are positively conjugate by a homeomorphism $h:\mathbb{R}\to\mathbb{R}$ that
fixes $0$. Then $\varphi_1=\varphi_2$.
\end{cor}
\begin{cor}\label{rem.conjclass}
With notation as above, let $\varphi_1$ and $\varphi_2$ be two normalized $\mu$-harmonic actions in $\Der_\mu(G)$.
Then $\varphi_1$ and $\varphi_2$ are positively conjugate if and only if they belong to the same $\Phi$-orbit.
\end{cor}
\subsection{Preorders and group actions}
\subsubsection{Preordered sets}
In this work, by a \emph{preorder} on a set $X$ we mean a transitive binary relation $\leq$ which is \emph{total}, in the sense that for every $x,y\in X$ we have $x\leq y$ or $y\leq x$ (possibly both relations can hold).
We write $x\lneq y$ whenever $x\leq y$ but it does not hold that $y\leq x$. On the other hand, when $x\leq y$ and $y\leq x$ we say that $x$ and $y$ are \emph{equivalent} and denote by $[x]_\leq$ the equivalence class of $x$ (we will simply write $[x]$ when there is no risk of confusion).
\begin{rem}\label{r.equivalence_po}
A preorder $\leq$ on $X$ induces a \emph{total} order on the set of equivalence classes $\{[x]_\leq:x\in X\}$, by declaring $[x]<[y]$ whenever $x\lneq y$.
\end{rem}
\begin{dfn}
We say that a map between preordered sets $f:(X_1,\leq_1)\to (X_2,\leq_2)$ is \emph{(pre)order-preserving} if $x\leq_1 y$ implies $f(x)\leq_2f(y)$.
On the other hand, given a map $f:X_1\to(X_2,\leq)$ we define the \emph{pull-back} of $\leq$ by $f$ as the preorder $f^\ast(\leq)$ on $X_1$, denoted by $\preceq$ here, so that $x_1\preceq x_2$ if and only if $f(x_1)\leq f(x_2)$.
\end{dfn}
\begin{dfn}
Let $(X,\le)$ be a preordered set. An \emph{automorphism} of $(X,\le)$ is an order-preserving bijection $f:(X,\le)\to (X,\le)$, whose inverse is also order-preserving. We denote by $\Aut(X,\le)$ the group of all automorphisms of $(X,\le)$.
An \emph{order-preserving action} of a group $G$ on a preordered set $(X,\le)$ is a homomorphism $\psi:G\to \Aut(X,\le)$.
\end{dfn}
\begin{rem}\label{r.pullback_po}
Let $G$ be a group with actions $\psi_1:G\to \Bij(X_1)$ and $\psi_2:G\to \Aut(X_2,\le)$. Let $f:X_1\to (X_2,\le)$ be an equivariant map. Then the preorder $f^*(\le)$ is preserved by $\psi_1(G)$.
\end{rem}
\begin{dfn}
Let $(X,\leq)$ be a preordered set and let $A\subset X$ be a subset. We say that $A$ is \emph{$\leq$-convex} if whenever $x\leq y\leq z$ and $x,z\in A$ it holds that $y\in A$.
\end{dfn}
\begin{rem}It is a direct consequence of the definitions that when $f:(X_1,\le_1)\to (X_2,\le_2)$ is order-preserving and $A\subset (X_2,\le_2)$ is $\le_2$-convex, then the preimage $f^{-1}(A)$ is $\le_1$-convex. This fact will be used several times without direct reference.
\end{rem}
\begin{dfn}
Given a partition $\mathfrak{P}$ of $X$, denote by $\mathfrak P(x)$ the atom of $\mathfrak{P}$ containing the point $x\in X$.
We say that a partition $\mathfrak{P}$ of a preordered set $(X,\le)$ is \emph{$\leq$-convex} is every atom of $\mathfrak{P}$ is a $\leq$-convex subset of $(X,\le)$. Compare this with Remark \ref{r.equivalence_po}.
\end{dfn}
\begin{rem}\label{r.convex_partition}
When $\mathfrak{P}$ is a $\le$-convex partition of a preordered set $(X,\le)$, there exists a total order $<_\mathfrak{P}$ on $\mathfrak{P}$ defined by the condition that $\mathfrak P(x)<_\mathfrak{P}\mathfrak P(y)$ if and only if $\mathfrak P(x)\neq \mathfrak P(y)$ and $x\lneq y$ (it is immediate to verify that this does not depend on the choice of points $x$ and $y$).
\end{rem}
\subsubsection{Preorders on groups} A preorder on a group $G$ is \emph{left-invariant} if $h\leq k$ implies $gh\leq gk$ for all $g,h,k\in G$. In other words, the left-multiplication gives an action by automorphisms $G\to \Aut(G,\le)$.
Recall also that a preorder on $G$ is \emph{bi-invariant} if it preserved by left and right multiplications.
By invariance and Remark \ref{r.equivalence_po}, given a left-invariant preorder $\leq$ on $G$, the equivalence class $[1]_\leq$ is a subgroup of $G$ and that $\leq$ is a left-invariant order on $G$ if and only if $[1]_\leq=\{1\}$.
The subgroup $[1]_{\leq}$ is called the \emph{residue}. We say that a left-invariant preorder $\leq$ on $G$ is \emph{trivial} whenever $[1]_\leq=G$ and \emph{non-trivial} otherwise.
We denote by $\LPO(G)$ the set of all non-trivial left-invariant preorders on $G$.
From now on, by a preorder on a group, we \emph{always} mean a non-trivial left-invariant preorder. We endow $\LPO(G)$ with the product topology induced by the realization of preorders as subsets of $\{\leq,\gneq\}^{G\times G}$. It turns out that with this topology, $\LPO(G)$ is a metrizable and totally disconnected topological space, which is moreover compact whenever $G$ is finitely generated (see Antolín and the third author \cite{AntolinRivas}, where preorders are called relative orders). For a modern treatment of left-orders and left-preorders see \cite{GOD} and \cite{AntolinDicksSunic,decaup2019preordered}.
\begin{dfn} Let $G$ be a group and let $\leq\in\LPO(G)$ be a preorder.
The \emph{positive cone} of $\le$ is the subset $P_\leq=\{g\in G:g\gneq 1\}$. Similarly, the subset $N_{\leq}=\{g\in G:g\lneq 1\}$ defines the \emph{negative cone} of $\le$.
\end{dfn}
\begin{rem}\label{r.cones}A preorder $\leq\in\LPO(G)$ induces a partition $G=P_{\leq}\sqcup[1]_\leq\sqcup N_{\leq}$. Also note that $P_\leq$ and $N_\leq$ are semigroups and satisfy $P_\leq^{-1}=N_\leq$ where $P_\leq^{-1}:=\{g^{-1}:g\in P_\leq\}$. Reciprocally, given a partition $G=P\sqcup H\sqcup N$ such that
\begin{enumerate}
\item $P$ is a semigroup,
\item $N=P^{-1}$,
\item $H$ is a proper (possibly trivial) subgroup and
\item $HPH\subseteq P$,
\end{enumerate}
there exists a preorder $\leq\in\LPO(G)$ such that $P=P_\leq$, $H=[1]_\leq$ and $N=N_\leq$. See \cite{decaup2019preordered} for details.
\end{rem}
When $H$ is a $\leq$-convex subgroup, the left coset space $G/H=\{gH:g\in G\}$ defines a $\leq$-convex partition of $G$. The total order $<_{G/H}$ induced on $G/H$ by $\leq$ (see Remark \ref{r.convex_partition}) is preserved by left-multiplication of $G$ on $G/H$. Given a $\leq$-convex subgroup $H\subseteq G$, we define the \emph{quotient preorder} of $\leq$ under $H$ as the preorder $\leq_H\in\LPO(G)$ given by the pull-back $\leq_H:=p^\ast(<_{G/H})$, where $p:G\to G/H$ is the coset projection.
Equivalently we can define $\leq_H$ by setting $P_{\leq_H}=P_\leq\setminus H$.
\begin{rem}\label{r.domination_po}
After Remark \ref{r.pullback_po}, the identity map $\mathsf{id}:(G,\leq)\to(G,\leq_H)$ is order-preserving and equivariant with respect to the actions of $G$ by left-multiplication.
\end{rem}
\subsubsection{Dynamical realizations of actions on totally ordered sets}\label{sec.dynreal}
One general principle that we often use is that for building actions of a group $G$ on the line by homeomorphisms, one may start by finding a totally ordered space $(X,<)$ and an action by order-preserving bijections $\psi:G\to\Aut(X,<)$. Then if the order topology on $X$ is nice enough, for instance when $X$ is countable, then the action $\psi$ can be ``completed'' to an action $\varphi:G\to \homeo_0(\mathbb{R})$ so that, there exists an order-preserving map $i:X \to \mathbb{R}$ satisfying that $\varphi(g)(i(x))=i(\psi(g)(x))$ for all $g\in G.$ Under some mild extra conditions, we call such $\varphi$ a \emph{dynamical realization} of $\psi$. See Definition \ref{d-dynamical-realisation}.
\begin{rem} It is a classical fact that a countable group is left-orderable if and only if it \emph{embeds} into $\homeo_0(\mathbb{R})$ (a fact that we trace back to Conrad \cite{Conrad} in the abelian setting, see \cite{GOD,Ghys} for a proof of the general case). In fact, one side of the proof consist in building a dynamical realization of the left-regular representation of a countable left-ordered group on itself. Analogously, one can show that a countable group admits a left-invariant preorder if and only if it admits a (possibly not faithful) non-trivial action by order-preserving homeomorphisms of the real line \cite{AntolinRivas}.
\end{rem}
We now proceed to formally define what we mean by dynamical realization.
\begin{dfn}\label{dfn.goodbehaved} Let $(X,<)$ be a countable totally ordered set. We say that an injective order-preserving map $i:X\to\mathbb{R}$ is a \emph{good embedding} if its image is unbounded in both directions and every connected component $I$ of the complement of $\overline{i(X)}$ satisfies $\partial I\subset i(X)$.
\end{dfn}
\begin{rem}\label{rem.goodexistence} Following the classical construction of the dynamical realization of a countable left-ordered group (see \cite{GOD, Ghys}), it follows that any countable and totally ordered set $(X,<)$ has a good embedding and that, given two different good embeddings $i_1,i_2:X\to\mathbb{R}$, there exists $h\in\homeo_0(\mathbb{R})$ so that $i_2=h\circ i_1$. \end{rem}
\begin{dfn}\label{d-dynamical-realisation}
Assume that $\psi\colon G\to \Aut(X, <)$ is a group action. An action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is said to be a \emph{dynamical realization} of $\psi$ if there exists a good embedding $\iota\colon X\to \mathbb{R}$ such that the following hold:
\begin{enumerate}
\item $i$ is equivariant for $\varphi$ and $\psi$;
\item \label{i-stab-trivial} for every connected component $I$ of the complement of $\overline{i(X)}$ the (setwise) stabilizer $\stab^\varphi_G(I)$ of $I$ in $G$ acts trivially on $I$.
\end{enumerate}
\end{dfn}
\begin{lem}\label{lem.dynreal} Consider an action $\psi:G\to\Aut(X,<)$ where $(X,<)$ is a countable totally ordered set. Then a dynamical realization of $\psi$ exists and is unique up to positive conjugacy.
\end{lem}
\begin{proof}[Sketch of proof] Consider a good embedding $i:X\to\mathbb{R}$. Then, by transporting the action $\psi$ through $i$ on $i(X)$ and extending it by continuity to the closure we obtain an action $\varphi_0:G\to\homeo_0(\overline{i(X)})$. For every $g\in G$ denote by $\varphi(g)$ the extension of $\varphi_0(g)$ which is affine on every connected component $I$ of $\overline{i(X)}$. It is direct to check that $g\mapsto \varphi(g)$ is a dynamical realization of $\psi$.
Now, consider two actions $\varphi_1,\varphi_2:G\to\homeo_0(\mathbb{R})$ which are dynamical realizations of $\psi$ with associated good embeddings $i_1,i_2:X\to\mathbb{R}$ respectively. Note that by Remark \ref{rem.goodexistence} there exists $h\in\homeo_0(\mathbb{R})$ with $i_2=h\circ i_1$ and after conjugating $\varphi_1$ by $h$ we can suppose that $i_1=i_1=:i$. By equivariance $\varphi_1$ and $\varphi_2$ must coincide on $\overline{i(X)}$. Let $\Omega$ be the set of connected components of $\mathbb{R}\setminus \overline{i(X)}$ and note that the $G$-actions on $\Omega$ induced by $\varphi_1$ and $\varphi_2$ coincide, so that the set of orbits $\Omega/G$ does not depend on the action. For $J\in \Omega$ we denote by $\alpha(J)\in \Omega/G$ its $G$-orbit. Pick a system of representatives $\{I_\alpha\}_{\alpha\in {\Omega/G}}$ of orbits. For $J\in \Omega$ choose $g_J\in G$ such that $g_J(J)=I_{\alpha(J)}$, and for $i\in\{1, 2\}$ set $f_{i, J}=\varphi_i(g_J)\restriction_J$. Note that each $f_{i,J}$ is a homeomorphism from $J$ to $I_{\alpha(J)}$ which does not depend on the choice of $g_J$ by the assumption (\ref{i-stab-trivial}) in the definition of dynamical realization. Thus $f_{2, J}^{-1}f_{1, J}$ is a self-homeomorphism of $J$. Define a map $q\colon \mathbb{R}\to \mathbb{R}$ which is the identity on $\overline{i(X)}$ and satisfies $q\restriction_J=f_{2, J}^{-1}f_{1, J}$ for every $J\in \Omega$. Then one readily checks that $q$ conjugates $\varphi_2$ to $\varphi_1$. \qedhere
\end{proof}
We proceed to give a sufficient condition for minimality of dynamical realizations of actions on totally ordered sets. For this, we need the following definition.
\begin{dfn}\label{dfn.homotype}Let $(X,<)$ be a totally ordered set. We say that a subgroup $H\subseteq\Aut(X,<)$ is of \emph{homothetic type} if the following conditions are satisfied.
\begin{enumerate}
\item There exists $x_0\in \mathsf{Fix}(H)$.
\item For every $x\in X\setminus \{x_0\}$, there exists a sequence of elements $(h_n)\subset H$ such that $h_n(x)\to+\infty$ if $x>x_0$, and $h_n(x)\to-\infty$ if $x<x_0$.
\end{enumerate} When the cyclic subgroup $\langle g \rangle$ is of homothetic type we say that $g$ is a \emph{homothety}. \end{dfn}
\begin{lem}[Minimality criterion]\label{lem.minimalitycriteria} Let $(X,<)$ be a countable totally ordered set and consider an action $\psi:G\to\Aut(X,<)$. Assume that for every four points $x_1,x_2,y_1,y_2$ of $X$ with
$$x_1< y_1< y_2< x_2,$$ there exists $g\in G$ such that $$g.y_1< x_1< x_2< g.y_2.$$ Then the dynamical realization of $\psi$ is minimal.
\end{lem}
\begin{proof} Let $\varphi:G\to\homeo_0(\mathbb{R})$ be a dynamical realization of $\psi$ with its associated good embedding $i:X\to\mathbb{R}$ (Lemma \ref{lem.dynreal}). We first claim that $i$ has dense image. Suppose by contradiction that it is not the case and take a connected component $(\eta_1,\eta_2)$ of the complement of $\overline{i(X)}$. Since $i$ is a good embedding we have that $\{\eta_1,\eta_2\}\subset i(X)$. Choose two points $\xi_1$ and $\xi_2$ in $i(X)$ such that $(\eta_1,\eta_2)\subsetneq (\xi_1,\xi_2)$;
by our assumption, we can find an element $g\in G$ such that $(\eta_1,\eta_2)\supsetneq g.(\xi_1,\xi_2)$,
contradicting the $\varphi(g)$-invariance of $i(X)$. This shows that $i$ has dense image.
Suppose again by contradiction that there exists a proper closed $\varphi(G)$-invariant subset $\Lambda\subset\mathbb{R}$. Take a connected component $(\eta_1,\eta_2)$ of $\mathbb{R}\setminus \Lambda$. By density of $i(\mathbb{R})$, we can find four points $\xi_1,\xi_2,\zeta_1,\zeta_2$ in $i(X)$ such that
$$\zeta_1<\eta_1<\xi_1<\xi_2<\eta_2<\zeta_2.$$
After our assumption, we can find an element $g\in G$ such that $(\eta_1,\eta_2)\supsetneq g.(\zeta_1,\zeta_2)$. This contradicts the $\varphi(g)$-invariance of $\Lambda$ showing that $\varphi$ is minimal, as desired.
\end{proof}
The condition in Lemma \ref{lem.minimalitycriteria} is met in the following situation.
\begin{prop}\label{p.minimalitycriteria}
Let $(X,<)$ be a countable totally ordered set and consider an action $\psi:G\to\Aut(X,<)$. If for every $x\in X$ there exists a subgroup $H\subseteq G$ such that $\psi(H)$ is a subgroup of homothetic type fixing $x$, the dynamical realization of $\psi$ is minimal.
\end{prop}
\begin{proof}
Consider four points $x_1,x_2,y_1,y_2$ of $X$ with $x_1< y_1< y_2< x_2$. By assumption, we can find an element $h_1\in H_{y_1}$ such that $y_1<h_1.y_2<y_2$. Consider the point $x_\ast=h_1.y_2$, so that there exists $h_\ast\in H_{x_\ast}$ such that $h_\ast.y_1<x_1<x_2<h_\ast.y_2$. Thus Lemma \ref{lem.minimalitycriteria} applies.
\end{proof}
\subsection{Bieri--Strebel groups}\label{sc.BieriStrebel} To illustrate our results we will often consider examples of locally moving groups arising as groups of piecewise linear (PL) homeomorphisms of the line. Let us briefly fix the terminology and recall a large family of such groups, following Bieri and Strebel \cite{BieriStrebel}.
We say that a homeomorphism $f\colon X\to X$ of an interval $X\subset \mathbb{R}$ (bounded or unbounded) is \emph{piecewise linear} (PL, for short) if there exists a discrete subset $\Sigma\subset X$ such that in restriction to $X\setminus \Sigma$, the map $f$ is locally affine, that is, of the form $x\mapsto \lambda x+a$, with $\lambda>0$ and $a\in \mathbb{R}$. We denote by $\BP(f)$ the minimal subset $\Sigma$ satisfying such condition, and
points of $\BP(f)$ will be the \emph{breakpoints} of $f$. When $\BP(f)$ is finite, we say that $f$ is \emph{finitary} PL.
Note that with this definition, a PL homeomorphism always preserves the orientation.
\begin{dfn}
\label{d.BieriStrebel}
Given a non-trivial multiplicative subgroup $\Lambda\subseteq \mathbb{R}_+^*$, and a non-trivial $\mathbb Z[\Lambda]$-submodule $A\subset \mathbb{R}$, the \emph{Bieri--Strebel group $G(X;A,\Lambda)$} is the group of finitary PL homeomorphisms $f:X\to X$ with the following properties:
\begin{enumerate}
\item breakpoints of $f$ are in $A$,
\item in restriction to $X\setminus \BP(f)$, the map $f$ is locally of the form $\lambda x+a$, with $\lambda \in \Lambda$ and $a\in A$.
\end{enumerate}
\end{dfn}
For example, Thompson's $F$ is $G([0,1];\mathbb Z[1/2], \langle 2\rangle_* )$. Other interesting examples are provided by the so-called Thompson--Brown--Stein groups.
\begin{dfn}\label{d-Thompson-Stein}
Let $1 < n_1 < \dots < n_k$ be natural numbers such that the multiplicative subgroup $\Lambda=\langle n_i\rangle \subseteq \mathbb{R}_+^*$ is an abelian group of rank $k$. Denote by $A$ the ring $\mathbb Z\left [1/m\right ]$, where $m$ is
the least common multiple of the $n_i$. The group
\[
F_{n_1,\ldots,n_k}:=G([0,1];A,\Lambda)
\]
is the corresponding \emph{Thompson--Brown--Stein group}.
\end{dfn}
The group $F_2$ is simply Thompson's group $F$. For every $n\ge 2$, the group $F_n$ is isomorphic to a subgroup of $F$, and
these groups were first considered by Brown in \cite{Brown}, inspired by the so-called Higman--Thompson groups. Later Stein \cite{Stein} started the investigation of the more general class of groups $F_{n_1,\ldots,n_k}$.
She proved that every Thompson--Brown--Stein group is finitely generated and even finitely presented. Moreover, given any $m$-adic interval $I\subset [0,1]$ (that is, an interval with endpoints in $\mathbb Z[1/m]$), the subgroup $\left (F_{n_1,\ldots,n_k}\right )_I$ is isomorphic to $F_{n_1,\ldots,n_k}$.
We refer to the classical monograph by Bieri and Strebel \cite{BieriStrebel} for an extensive investigation of the groups $G(X;A,\Lambda)$.
\begin{rem}
It would be interesting to see how the results of this text apply to other groups of piecewise projective homeomorphisms, such as Monod's groups $H(A)$ \cite{Monod} (we will not pursue this task).
\end{rem}
\section{An illustrative example: Thompson's group $F$}
\label{s-F}
\subsection{Main results in the case of $F$ and outline of the section} Perhaps the most basic example of finitely generated locally moving group is Thompson's group $F$ (cf.\ Proposition \ref{p-chain}), which also belongs to the class $\mathcal{F}_0${}. It turns out that actions of the group $F$ on $\mathbb{R}$ display a rich combination of rigidity and flexibility phenomena. The goal of this section is to summarize our results in this special case and to go more into detail.
On the one hand, Corollary \ref{c-lm-C1-interval} implies the following for its actions by $C^1$ diffeomorphisms.
\begin{thm}[Rigidity of actions by $C^1$ diffeomorphisms]\label{c-F-C1}
Thompson's group $F$ satisfies the following.
\begin{itemize}
\item For every faithful action $\varphi\colon F\to \Diff^1([0,1])$ without fixed points in $[0,1]$, the $\varphi$-action of $F$ on $(0,1)$ is semi-conjugate to the standard action of $F$ on $(0, 1)$.
\item Every faithful action $\varphi\colon F\to \Diff^1(\mathbb{R})$ without closed discrete orbit is semi-conjugate to the standard action on $(0, 1)$.
\end{itemize}
\end{thm}
\begin{rem}\label{r-F-C1}
Note that the conclusion is optimal: there exist $C^1$ actions (and even $C^{\infty}$ actions) of $F$ on closed intervals which are semi-conjugate, but not conjugate to its standard action: the existence of such actions was shown by Ghys and Sergiescu \cite{GhysSergiescu}, or alternatively can be shown using the ``2-chain lemma'' of Kim, Koberda, and Lodha \cite{KKL} (see Proposition \ref{p-chain}).
\end{rem}
On the other hand this rigidity does not hold for continuous actions: there exists minimal faithful actions of $F$ on $\mathbb{R}$ which are not semi-conjugate to the standard action (see for instance the constructions in \S\S \ref{ssec.cyclicgerm}--\ref{ssec.germtype}). Nevertheless, Theorem \ref{t-C-trichotomy} implies that the dynamics of all exotic actions of $F$ on $\mathbb{R}$ is strongly reminiscent of the standard action on $(0, 1)$, although not via a semi-conjugacy but via the notion of \emph{horograding} of an $\mathbb{R}$-focal action (see \S \ref{ss.horograding}). Indeed for the group $F$, Theorem \ref{t-C-trichotomy} reads as follows (recall that the commutator subgroup $[F, F]$ is simple and coincides with the group $F_c$ of compactly supported elements, so that the largest quotient $F/[F_c, F_c]$ coincides with the abelianization $F^{ab}\cong \mathbb Z^2$).
\begin{thm}[Structure theorem for actions by homeomorphisms]\label{t-F-trichotomy}
Every action $\varphi\colon F\to \homeo_0(\mathbb{R})$ without fixed points is semi-conjugate to one of the following.
\begin{enumerate}[label=(\roman*)]
\item \emph{(Non-faithful)} An action by translations of $F^{ab}\cong \mathbb Z^2$.
\item \emph{(Standard)} The standard piecewise linear action of $F$ on $(0,1)$.
\item \emph{(Exotic)} A minimal $\mathbb{R}$-focal action which can be horograded by the standard action of $F$ on $(0, 1)$.
\end{enumerate}
\end{thm}
Theorem \ref{t-F-trichotomy} implies serious constraints on the structure of actions of $F$ on $\mathbb{R}$. For example, it implies that for all exotic actions, the individual elements of $F$ satisfy a dynamical classification, and the type of each element can be read from the standard action on $(0, 1)$ (see Proposition \ref{prop.dynamic-class-horo}). For ease of reference let us restate the dynamical classification in this special case. Given $g\in F$ and $x\in [0,1]$ we denote by $D^-g(x)\in\{2^n, n\in \mathbb Z\}$ the left derivative of $g$ at $x$. Note that germ of $g\in F$ inside $\Germ(F, 1)$ is uniquely determined by $D^-g(1)$, and $1$ is an attracting fixed point for $g$ if and only if $D^-g(1)<1$.
\begin{prop}[Dynamical classification of elements] \label{p-F-dynclasselements}
Let $\varphi\colon F\to \homeo_0(\mathbb{R})$ be a minimal $\mathbb{R}$-focal action, increasingly horograded by the standard action on $(0, 1)$. Then the following hold.
\begin{enumerate}[label=(\roman*)]
\item For every $x\in (0, 1)$, the image $\varphi(F_{(0, x)})$ is totally bounded. In particular the $\varphi$-image of every element $g\in F$ with $D^-g(1)=1$ is totally bounded.
\item For every $g\in F$ such that $D^-g(1)\neq 1$ the $\varphi$-image of $g$ is a pseudohomothety, which is expanding if $D^-g(1)<1$ and contracting otherwise. If moreover $g\in F$ has no fixed points in $(0, 1)$ then its image is a homothety.
\end{enumerate}
\end{prop}
As in \S\ref{s-lr}, Theorem \ref{t-F-trichotomy} can also be used to analyze the structure of the Deroin space $\Der_\mu(F)$ of normalized harmonic $F$-actions, a representation of which is given in Figure \ref{fig-F-Der}; a detailed description of this picture is provided in \S \ref{s-F-Deroin} below. This description implies in particular that a representative of the standard action in $\Der_\mu(F)$ cannot be accumulated by exotic actions, which in turns implies the following (which is a particular instance of Theorem \ref{t-local-rigidity}).
\begin{thm}[Local rigidity of the standard action] \label{t-F-lr}
The standard piecewise linear action of $F$ on $(0,1)$ is locally rigid.
\end{thm}
\begin{figure}[ht]
\includegraphics[scale=1]{Deroin_F.pdf}
\caption{The Deroin space of Thompson's group $F$. See \S \ref{s-F-Deroin} for a detailed explanation. }\label{fig-F-Der}
\end{figure}
It also allows to say that the Borel complexity structure of the semi-conjugacy relation on $\Homirr(F,\homeo_{0}(\mathbb{R}))$ is the simplest possible (see the discussion in \S\ref{ssc.complexity}).
In other words, there exists a Borel invariant for actions of $F$ on the real line without fixed points, which completely determine the semi-conjugacy class.
\begin{thm}[Smoothness of the semi-conjugacy relation]\label{t-F-classification}
The semi-conjugacy relation on the space $\Homirr(F,\homeo_{0}(\mathbb{R}))$ of actions of $F$ without fixed points is smooth. In particular, the conjugacy relation on the space of minimal actions of $F$ on $\mathbb{R}$ is smooth.
\end{thm}
\begin{proof}
Direct consequence of Corollary \ref{cor.cross-section} and the description of the Deroin space of $F$ in \S\ref{s-F-Deroin}.
\end{proof}
Leaving temporary aside the setting of orientation-preserving actions, we may consider \emph{Thompson's group with flips} $F^\pm$, which is defined as the subgroup of $\homeo((0, 1))$ defined analogously to $F$ but by allowing negative slopes (equivalently, the group generated by $F$ and by the reflection along $1/2$). Then Theorem \ref{t-F-trichotomy} implies the following (which is a special case of Corollary \ref{c-C-flip}).
\begin{thm}[Rigidity in the non-orientation-preserving case] \label{t-F-flip}
Let $F^\pm$ denote Thompson's group with flips. Then every minimal action $\varphi\colon F^\pm \to \homeo(\mathbb{R})$ is conjugate to its standard action on $(0,1)$.
\end{thm}
\begin{rem}
In Theorem \ref{t-F-flip}, the assumption that the action be faithful is actually redundant. In fact by analyzing the proper quotients of $F^\pm$, one can check that the only ones that act non-trivially on the line by homeomorphisms are isomorphic to $\mathbb Z$ or to the dihedral group $D_\infty$, so that none of them can act minimally.
\end{rem}
While the above results can all be seen as rigidity properties for actions of $F$ on the line, a large part of this section will be devoted to illustrate their considerable flexibility. Indeed within the constraints imposed by Theorem \ref{t-F-trichotomy}, it turns out the group $F$ admits a wild zoo of exotic (hence $\mathbb{R}$-focal) actions. Some first examples of $\mathbb{R}$-focal actions of $F$ can be obtained by the constructions in \S \S\ref{ssec.cyclicgerm}--\ref{ssec.germtype}, which already show that $F$ admits an uncountable family of non-conjugate such actions. In \S\S \ref{s-F-plethora1}--\ref{s-F-hyperexotic} we illustrate the abundance of $\mathbb{R}$-focal actions of the group $F$ by providing various other constructions, and describing some subtle differences in their dynamical behavior. While we will focus more on some significant examples, the reader will notice that these constructions admit various variants involving several choices, which depend on the data and are not always compatible between them. Trying to take them all into account simultaneously would result in an obscure treatment.
Along the way we will observe that many examples of minimal $\mathbb{R}$-focal actions of $F$ share an interesting property: the image of the commutator subgroup $[F, F]$ does not act minimally on the real line (equivalently it does not admit any closed minimal invariant subset, see Lemma \ref{l-normal-minimal}). In \S \ref{s-F-simplicial} we show that this condition is particularly relevant and has many equivalent characterizations (see Proposition \ref{prop.hyperequiv}). In particular it characterizes the $\mathbb{R}$-focal actions of $F$ which can be encoded by an action on a \emph{simplicial} planar directed tree (cf.\ Proposition \ref{p-focal-simplicial}). Nevertheless it turns out that not \emph{all} $\mathbb{R}$-focal actions of $F$ have this property: in \S \ref{s-F-hyperexotic} we will construct a family of minimal $\mathbb{R}$-focal actions of $F$ such that $[F, F]$ still acts minimally.
To conclude this discussion, we would like to emphasize that the abundance of exotic actions relies on some specific features of the group $F$ . For instance in the next section (Section \ref{s-few-actions}) we will see that a class of finitely generated groups of piecewise linear homeomorphisms of the whole \emph{real line} (with finitely many breakpoints) defined analogously to $F$ admit exactly \emph{two} minimal faithful exotic actions up to conjugacy, and we will also construct a finitely generated locally moving group which does not admit any exotic action at all.
\subsection{An analysis of the Deroin space of $F$}\label{s-F-Deroin} The goal of this subsection is to analyze the Deroin space of $F$ and to explain and justify Figure \ref{fig-F-Der}. From now on, we fix a symmetric probability measure $\mu$ on $F$ whose support is finite and generates $F$, and consider the associated Deroin space $\Der_\mu(F)$ (see \S \ref{ssubsec.Deroin}), with its translation flow $\Phi$. We will follow closely the discussion in \S \ref{s-lr}, and refine it when possible to obtain more precise information in this special case.
\begin{rem}As mentioned in the introduction, the Deroin space is a continous analogue of the space of \emph{left-invariant} orders for a finitely generated group. We point out that the space of \emph{bi-invariant} orders on $F$ was described in \cite{NavasRivas}.
\end{rem}
We take notation analogous to Assumption \ref{a-lr}. Namely we fix any element $f\in F$ which in the standard action satisfies $f(x)>x$ for every $x\in (0, 1)$. For definiteness we choose $f$ to be the element of the standard generating pair of $F$ given by
\begin{equation} \label{e-F-big-generator}f(x)=\left\{\begin{array}{lr}2x & x\in [0, \frac{1}{4}],\\[.5em] x+ \frac{1}{4} & x\in [\frac{1}{4}, \frac{1}{2}],\\[.5em] \frac{1}{2}x & x\in [\frac{1}{2}, 1].
\end{array}\right.\end{equation}
By Theorem \ref{t-F-trichotomy}, we have a decomposition of $\Der_\mu(F)$ into $\Phi$-invariant subspaces
\[\Der_\mu(F)=\mathcal{N}\sqcup \mathcal{I} \sqcup \widehat{\mathcal{I}} \sqcup \mathcal{P},\]
defined as follows:
\begin{itemize}
\item we let $\mathcal{N}\subset \Der_\mu(F)$ be the subset consisting of actions by translations of $F^{ab}$;
\item we fix a representative $\iota\in \Der_\mu(F)$ of the standard action on $(0, 1)$, and let $\mathcal{I}=\{\Phi^t(\iota)\colon t\in \mathbb{R}\}$ denote its $\Phi$-orbit, whilst $\widehat{\mathcal{I}}$ denotes the $\Phi$-orbit of the reversed action $\widehat{\iota}$;
\item we let $\mathcal{P}=\mathcal{P_+}\sqcup \mathcal{P}_-$ be the subset of $\Der_\mu(F)$ of $\mathbb{R}$-focal actions, where $\mathcal{P}_+$ (respectively, $\mathcal{P}_-$) is the subset of $\mathbb{R}$-focal actions which are increasingly (respectively, decreasingly) horograded by the standard action on $(0, 1)$. For $\varphi\in \mathcal{P}$ we let $\xi_\varphi$ be the unique fixed point of $\varphi(f)$. We say that $\varphi\in \mathcal{P}$ is $f$-centered if $\xi_\varphi=0$, and let $\mathcal{P}^0\subset \mathcal{P}$ be the subset of $f$-centered $\mathbb{R}$-focal actions. Finally we set $\mathcal{P}^0_\pm=\mathcal{P}^0\cap \mathcal{P}_\pm$.
\end{itemize}
Let us now give a closer description of the topology of these subsets and of the dynamics of the flow $\Phi$ on them.
First of all, from the elementary structure of the maximal quotient $F/[F_c,F_c]\cong \mathbb Z^2$ we obtain an explicit description of the subset $\mathcal{N}$ in this case. Indeed $\mathcal{N}$ is homeomorphic to $\Der_{\bar{\mu}}(\mathbb Z^2)$, where $\bar{\mu}$ is the projection of $\mu$ to $F^{ab}\cong \mathbb Z^2$. Every element of $\Der_{\bar{\mu}}(\mathbb Z^2)$ corresponds to a $\mathbb Z^2$-action by translations given by a non-trivial homomorphism $\varphi\colon \mathbb Z^2\to (\mathbb{R}, +)$ up to rescaling by a positive real, so that $\Der_{\bar{\mu}}(\mathbb Z^2)$ is homeomorphic to the circle $\mathbb S^1$. We deduce that $\mathcal{N}$ is a closed subset of $\Der_\mu(F)$ homeomorphic to a circle, and it consists of points which are fixed by the translation flow $\Phi$.
Let now $\tau_0, \tau_1\colon F\to \mathbb Z$ be the two natural homomorphisms obtained by identifying the groups of germs $\Germ(F,0)$ and $\Germ(F, 1)$ with $\mathbb Z$, with the convention that $\tau_x(g)>0$ if and only if the corresponding endpoint $x\in\{0, 1\}$ is an attracting fixed point of $g$. Explicitly,
\begin{equation} \label{e-F-germs}\tau_0(g)=-\log_2 D^+g(0) \quad \text{and}\quad \tau_1(g)=-\log_2 D^-g(1).\end{equation}
We identify $\tau_0$ and $\tau_1$ with the elements of $\mathcal{N}$ given by their corresponding cyclic action. One readily shows that the actions $\iota$ and $\widehat{\iota}$ satisfy the following (compare with Lemma \ref{l-lr-standard-accumulation}):
\[\ \lim_{t\to+\infty} \Phi^t(\iota)=\tau_1, \lim_{t\to -\infty} \Phi^t(\iota)=\widehat{\tau_0}, \quad \quad \lim_{t\to +\infty} \Phi^{t}(\widehat{\iota})=\tau_0, \lim_{t\to -\infty} \Phi^{t}(\widehat{\iota})=\widehat{\tau_1}.\]
Thus $\mathcal{I}$ is a copy of $\mathbb{R}$ inside $\Der_\mu(F)$ which connects the points $\widehat{\tau_0}$ and $\tau_1$, while $\widehat{\mathcal{I}}$ connects $\tau_0$ to $\tau_1$ as shown in Figure \ref{fig-F-Der}.
Recall that by Lemma \ref{l-lr-section} each subset $\mathcal{P}_\pm$ is open, and $\mathcal{P}^0_\pm$ is a cross section for $\Phi$ inside $\mathcal{P}_\pm$. Moreover, in this setting we have the following more precise version of Proposition \ref{p-lr-attractor}.
\begin{prop}\label{p-F-limits}
The subsets $\mathcal{P}_-^0$ and $\mathcal{P}_+^0$ are closed in $\Der_\mu(F)$. Moreover, for every $\varphi\in \mathcal{P}^0_\pm$ the limits $\lim_{t\to \pm \infty} \Phi^t(\varphi)$ exist and the following holds:
\begin{enumerate}
\item if $\varphi\in \mathcal{P}_+^0$ then
\[\lim_{t\to +\infty} \Phi^t(\varphi)=\tau_1\quad\text{and}\quad\lim_{t\to -\infty }\Phi^t(\varphi)=\widehat{\tau}_1,\]
where the convergence is uniform on $\varphi\in \mathcal{P}^0_+$;
\item if $\varphi\in \mathcal{P}_-^0$ then
\[\lim_{t\to +\infty} \Phi^t(\varphi)=\tau_0\quad\text{and}\quad\lim_{t\to -\infty} \Phi^t(\varphi)=\widehat{\tau}_0,\]
where the convergence is uniform on $\varphi\in \mathcal{P}^0_-$.
\end{enumerate}
\end{prop}
\begin{proof}
We already know by Lemma \ref{l-lr-section} that $\overline{\mathcal{P}^0}_\pm \subset \mathcal{P}^0_\pm \cup \mathcal{N}$. Let us
show that in this case the sets $\mathcal{P}_\pm^0$ are actually closed. Assume by contradiction that $(\varphi_n)_{n\in\mathbb N}\subset \mathcal{P}^0_+$ is a sequence converging to $\psi\in \mathcal{N}$. Since $\varphi_n(f)$ fixes $0$ for every $n\in\mathbb N$, so does $\psi(f)$. Since $\psi$ is an action by translations, this is possible only if $\psi(f)=\mathsf{id}$. Now fix $x\in (0, 1)$ and choose any $h\in F_{(0, x)}$ with non-trivial germ at $0$. Recall the notation $\Iphi(x, \xi)$ from Definition \ref{d-I-phi}. After the claim in Lemma \ref{l-lr-bound}, there exists $D=D(x, \mu)>0$ such that $|\IphiN(x, 0)| \le D$ for every $n\in \mathbb N$. Thus, upon extracting a subsequence we can suppose that $\eta_n:=\sup \IphiN(x, 0)$ converges to a limit $\eta\in \mathbb{R}$ as $n\to \infty$. The choice of $h$ implies that $\varphi_n(h)$ fixes $\eta_n$ for every $n$, so that $\psi(h)$ fixes $\eta$, and we deduce again that $\psi(h)=\mathsf{id}$. However note that in the standard action the element $f$ has non-trivial derivatives at both endpoints, while $h$ has non-trivial germ only at $0$. Thus $h$ and $ f$ project to two linearly independent elements of $F^{ab}\cong \mathbb Z^2$. Since $\psi$ is an action by translations and it vanishes on both, we have that $\psi$ is trivial, which is a contradiction. This shows that $\mathcal{P}^0_+$ is closed, and the proof for $\mathcal{P}^0_-$ is closed is similar.
Let us now show that $\lim_{t\to +\infty} \Phi^t(\varphi)=\tau_1$ uniformly on $\varphi\in \mathcal{P}^0_+$. Assume that $(\varphi_n)_{n\in \mathbb N}\subset \mathcal{P}^0_+$ and $(t_n)_{n\in \mathbb N}\subset \mathbb{R}$ are sequences such that $t_n$ increases to $+\infty$, and let $\psi$ be a cluster point of $\psi_n:=\Phi^{t_n}(\varphi_n)$.
Repeating the proof of Proposition \ref{p-lr-attractor} in this case, one sees that $\operatorname{\mathsf{Fix}}^\psi(F_+)\neq\varnothing$, hence $\psi(F_+)=\{\operatorname{\mathsf{id}}\}$. This is only possible when $\psi\in \{\tau_1,\widehat{\tau}_1\}$. However note that $\psi_n(f)$ is an expanding homothety with fixed point $-t_n$ for every $n\in \mathbb N$, so that $\psi_n(f)(0)> 0$ for sufficiently large $n$. Passing to the limit we obtain that $\psi(f)(0)\ge 0$. Thus necessarily $\psi=\tau_1$. \qedhere
\end{proof}
An immediate consequence of the uniform convergence in Proposition \ref{p-F-limits} is the following. Given a topological space $K$, we define the \emph{double cone} over $K$ to be the quotient $K\times [-1, 1]/\sim$ where $(x, -1)\sim (y, -1)$ and $(x, 1)\sim (y, 1)$ for every $x, y \in K$.
\begin{cor}
The closures of the subsets $\mathcal{P}_+$ and $\mathcal{P}_-$ in $\Der_\mu(F)$ are given by $\overline{\mathcal{P}}_+=\mathcal{P}_+\cup \{\tau_1, \widehat{\tau}_1\}$ and $\overline{\mathcal{P}}_-=\mathcal{P}_- \cup \{\tau_0, \widehat{\tau}_0\}$, and both are homeomorphic to the double cone over the corresponding cross section $\mathcal{P}^0_\pm$.
\end{cor}
With this discussion in mind, illustration of $\Der_\mu(F)$ is provided by Figure \ref{fig-F-Der}. Note that in particular it is visible from this description that $\overline{\mathcal{P}}\cap \mathcal{I}=\varnothing$, so that $\iota$ is transversely isolated. In particular the standard action of $F$ is locally rigid (Theorem \ref{t-F-lr} above).
To conclude this discussion, we observe that it is a tantalizing problem to obtain further results on the topology of the compact cross sections $\mathcal{P}^0_+$ and $\mathcal{P}^0_-$, which is at the moment quite mysterious. Note that these do not depend up to homeomorphism on the choice of the generator $f\in F$ made above, and by symmetry they are homeomorphic one to the other. The constructions of $\mathbb{R}$-focal actions which will appear in the rest of the section show that the spaces $\mathcal{P}^0_+$ and $\mathcal{P}^0_-$ are uncountable, and contain homeomorphic copies of a Cantor set. However we were not able to construct a non-trivial connected subset of $\mathcal{P}^0_+$, and we do not know whether they are totally disconnected. We also do not know the answer to the following question.
\begin{ques}
Do the cross sections $\mathcal{P}^0_+$ and $\mathcal{P}^0_-$ admit isolated points?
\end{ques}
By Corollary \ref{prop.rigidityderoin} this is equivalent to the question whether $F$ admits minimal $\mathbb{R}$-focal actions which are locally rigid.
\subsection{Simplicial $\mathbb{R}$-focal actions of $F$ and minimality of $[F, F]$} \label{s-F-simplicial}
Before discussing examples of minimal $\mathbb{R}$-focal actions of $F$, we would like to single out an important question to address about any such action: whether the commutator subgroup $[F, F]$ acts minimally or not. By Lemma \ref{l-normal-minimal}, the second possibility is equivalent to the fact that it does not admit any non-empty closed minimal invariant subset of $\mathbb{R}$. The $\mathbb{R}$-focal actions of $F$ with this property turn out to be very special due to the following result, which elaborates on Proposition \ref{p-focal-simplicial}.
\begin{prop}[Simplicial $\mathbb{R}$-focal actions of $F$]\label{prop.hyperequiv} Let $\varphi:F\to\homeo_{+}(\mathbb{R})$ be a minimal $\mathbb{R}$-focal action which can be increasingly horograded by the standard action on $(0, 1)$. Then, the following are equivalent:
\begin{enumerate}[label=(\roman*)]
\item \label{i-F+-minimal} The image $\varphi(F_+)$ admits no non-empty closed minimal invariant set.
\item \label{i-F'-minimal} The image $\varphi([F, F])$ admits no non-empty closed minimal invariant set.
\item \label{i-simplicial} $\varphi$ is represented by a focal action on a planar directed tree $(\mathbb T, \triangleleft, \prec)$, such that $\mathbb T$ is a simplicial tree of countable degree, and the action of $F$ on $\mathbb T$ is by simplicial automorphisms.
\item \label{i-isometric} $\varphi$ is represented by a focal action on a planar
directed tree $(\mathbb T, \triangleleft, \prec)$ such that $F$ acts by isometries with respect to a compatible $\mathbb{R}$-tree metric on $\mathbb T$.
\item \label{i-homothety} Every pseudohomothety in the image of $\varphi$ is a homothety.
\item \label{i-not-type-III} There exist bounded open intervals $I,J\subset \mathbb{R}$ such that for every $g\in F_+$ the image $g.I$ does not contain $J$.
\end{enumerate}
\end{prop}
\begin{rem}
The fact that some minimal $\mathbb{R}$-focal actions of $F$ can be encoded by a simplicial action on a simplicial tree can seem in contradiction with the fact that they can also be horograded by the standard action of $F$, which is highly not isometric. However the tree provided by the proof Proposition \ref{prop.hyperequiv} is not the same directed tree as the one provided by the proof of Theorem \ref{t-C-trichotomy} (it arises from a different CF-cover). This is an illustration of the fact that a planar directed tree encoding an $\mathbb{R}$-focal action is not unique, and identifying a tree with good properties may be important for some purposes. We note however that the focal germ representations associated to both trees are semi-conjugate to the germ homomorphism $F\to \Germ(F, 1)\simeq \mathbb Z$ associated to the standard action (recall Lemma \ref{l-focal-germ-well-defined}).
\end{rem}
\begin{proof}[Proof of Proposition \ref{prop.hyperequiv}] Let us first show that \ref{i-F+-minimal} and \ref{i-F'-minimal} are equivalent. If $[F, F]$ admits a closed minimal invariant subset, then by Lemma \ref{l-normal-minimal} it acts minimally on $\mathbb{R}$, and thus so does $F_+$. Conversely, assume by contradiction that $[F, F]$ does not act minimally on $\mathbb{R}$, but $F_+$ does. Then we can apply Proposition \ref{p-focal-simplicial} to $G=F_+$ and $N=[F, F]$ and we deduce that $\varphi\restriction_{F_+}$ is $\mathbb{R}$-focal and is the dynamical realization of an action of $F_+$ on an planar directed simplicial tree $(\mathbb T, \triangleleft, \prec)$. Moreover, the argument in its proof shows that the focal germ representation of the action on $(\mathbb T, \triangleleft,\prec)$ coincides with the projection to $F_+/[F, F]\cong \mathbb Z$. Since the quotient $F_+/[F, F]$ is simply the group of germs $\Germ(F_+, 0)$ we deduce that the $\varphi$-image of every $g\in F_+$ with a non-trivial germ at $0$ must be a pseudohomothety. However we were assuming that $\varphi$ is an $\mathbb{R}$-focal action of $F$ increasingly horograded by the standard action on $(0, 1)$, so that the image of every element $g\in F_+$ must be totally bounded. This is a contradiction.
The implication \ref{i-F+-minimal}$\Rightarrow$\ref{i-simplicial} follows from Proposition \ref{p-focal-simplicial}, and \ref{i-simplicial}$\Rightarrow$\ref{i-isometric} is obvious.
Let us prove that \ref{i-isometric} implies \ref{i-homothety}. Note that the image of an element $g\in F$ is a pseudohomothety if and only if $g$ has a non-trivial germ at $1$. As in the proof of Proposition \ref{p-focal-simplicial} we can consider a horofunction $\pi\colon \mathbb T\to \mathbb{R}$ given by $\pi(v)=d_\mathbb T(v\wedge v_0, v_0)-d_\mathbb T(v\wedge v_0, v)$,
where $v_0$ is some basepoint, and we see that this provides a horograding by an action by translations on $\mathbb{R}$ coming from the germ homomorphism $F\to \Germ(F, 1)\cong \mathbb Z\to (\mathbb{R}, +)$. Then from Proposition \ref{prop.dynclasselements} we obtain that every element with a non-trivial germ at $1$ must act as a homothety under $\varphi$.
To show that \ref{i-homothety} implies \ref{i-not-type-III}, we show that \ref{i-homothety} actually implies the following more explicit condition, which clearly implies \ref{i-not-type-III}.
\begin{itemize}
\item[\emph{(vi')}] \emph{For every element $h\in F$ which in the standard action satisfies $h(x)>x$ for every $x\in (0, 1)$ (so that $\varphi(h)$ is an expanding homothety) there exists an open bounded interval $I\subset \mathbb{R} $ such that $I\subset h.I$ and such that for every $g\in F_+$, the interval $h.I$ is not contained in $g.I$. }
\end{itemize}
Indeed assume that $h$ is such that $h(x)>x$ for every $x\in X$ and let $\xi\in \mathbb{R}$ be the unique fixed point of $\varphi(h)$. Fix $x\in X$ and let $I:=\Iphi(x,\xi)$ (note that this connected component is defined for every $x\in X$).
As $\varphi(h)$ is an expanding homothety, we have $I\subset h.I$. Suppose by contradiction that there exists $g\in F_+$ such that $h.I\subset g.I$.
Note that we have the equalities \[g.I=g.\Iphi(x,\xi)=\Iphi(g(x),g.\xi)=\Iphi(g(x),\xi)\]
(the last equality follows from the assumption that $\xi\in g.I$). Moreover we have that $h.I=\Iphi(h(x),\xi)$ and therefore we conclude that $\Iphi(h(x),\xi)\subset \Iphi(g(x),\xi)$ implying that $g(x)\ge h(x)$. Since $g\in F_+$ this implies that for some $y\ge x$ it holds that $g(y)=h(y)$. Therefore,
\[g.\Iphi(y,\xi)=\Iphi(g(y),\xi)=\Iphi(h(y),\xi)=h.\Iphi(y,\xi)\]
(the first equality follows from the fact that $\Iphi(y,\xi)\supset I$).
Finally, this implies that $\varphi(g^{-1}h)$ preserves $\Iphi(y,\xi)$. Since $g^{-1}h$ has the same germ as $h$ at $1$, it acts as a pseudohomothety, so that by \ref{i-homothety} it is a homothety, and this is a contradiction with the fact that it preserves $\Iphi(y, \xi)$.
Finally to show that \ref{i-not-type-III} implies \ref{i-F+-minimal}, assume by contradiction (using Lemma \ref{l-normal-minimal}) that the action of $F_+$ is minimal. As we are assuming \ref{i-not-type-III}, the action cannot be proximal, so by Theorem \ref{t-centralizer} the centralizer of $\varphi(F_+)$ in $\homeo_0(\mathbb{R})$ must be infinite cyclic generated by an element $T$ without fixed point. Since $F_+$ is normal, we deduce that the whole group $\varphi(F)$ must normalize $\langle T\rangle$ and thus centralize it, contradicting that a minimal $\mathbb{R}$-focal action is always proximal (Proposition \ref{prop.minimalimpliesfocal}). \qedhere
\end{proof}
\begin{dfn}\label{d-F-simplicial}
We will say that a minimal $\mathbb{R}$-focal action $\varphi\colon F\to \homeo_0(\mathbb{R})$ is simplicial if it satisifies one of the equivalent conditions in Proposition \ref{prop.hyperequiv}.
\end{dfn}
For many constructions of $\mathbb{R}$-focal actions of $F$ discussed below one can easily check conditions \ref{i-F+-minimal} or \ref{i-F'-minimal}, and thus they turn out to be simplicial (although a simplicial tree does not always appear naturally in the construction, and it might also be not obvious to check directly condition \ref{i-homothety}). It is more delicate to construct $\mathbb{R}$-focal actions of $F$ which are \emph{not} simplicial: this will be done in \S \ref{s-F-hyperexotic}.
\subsection{A plethora of $\mathbb{R}$-focal actions I: restriction preorders} \label{s-F-plethora1}
Starting from now we will present various constructions of $\mathbb{R}$-focal actions of the group $F$ and study some of their properties. We will focus on some examples, and indicate how they can be modified to obtain more.
\begin{rem}
The attentive reader will notice that a common feature of all our constructions of $\mathbb{R}$-focal actions of $F$ is the choice of a closed subset $K\subset (0,1)$, which is invariant under the generator $f$ given by \eqref{e-F-big-generator}. These sets appear quite naturally with the point of view of focal actions. To understand this, take a minimal $\mathbb{R}$-focal action $\varphi\colon F\to \homeo_0(\mathbb{R})$. By Theorem \ref{t-F-trichotomy} we know that it can be horograded by the standard action on $(0, 1)$, meaning that one can find a focal action $\Phi:G\to\Aut(\mathbb T, \triangleleft, \prec)$ on a directed planar tree which is horograded by the standard action, and whose dynamical realization is (conjugate to) $\varphi$.
By Proposition \ref{p-F-dynclasselements}, the element $f$ fixes a unique end $\xi_0\in \partial^*\mathbb T$, so that it preserves the axis $]\xi_0, \omega[\subset \mathbb T$, which is naturally identified with the interval $(0,1)$ via the horograding map $\pi:\mathbb T\to (0,1)$. In particular, the $\pi$-image of the closure $\overline{]\xi_0, \omega[\cap \Br(\mathbb T)}$ of the subset of branching points on this axis defines an $\varphi(f)$-invariant closed subset $K\subset (0,1)$. Although for some choices of the action $\Phi$ (as the one in the proof of Theorem \ref{t-C-trichotomy}) the subset $K$ is the whole interval $(0,1)$, in most examples it is not the case with correct choice of $\Phi$.
\end{rem}
\subsubsection{A reinterpretation in terms of preorders}
We start with a simple observation which is useful to understand $\mathbb{R}$-focal actions of $F$. Recall that we write $\tau_1\colon F\to \mathbb Z \cong \Germ (F, 1)$ for the germ homomorphism given by \eqref{e-F-germs}, and $f\in F$ for the element given by \eqref{e-F-big-generator}. We will also write $1_F$ for the trivial element of $F$ (and we will simply denote it by $1$ when there is no risk of confusion).
Since $\tau_1(f)$ is a generator of $\Germ(F, 1)$, we have a splitting
\[F=F_+\rtimes\langle f \rangle,\]
where as usual we write $F_+=\ker \tau_1$. Then we can make $F$ act on $F_+$ ``affinely'' by letting $F_+$ act on itself by left translations and $f$ act on $F_+$ by conjugation. In formula, for $g=hf^n\in F$, with $h\in F_+$ and $n\in \mathbb Z$ and for $r\in F_+$, this action is given by $g\cdot r=hf^nrf^{-n}$.
Assume that $\preceq$ is a left preorder on $F_+$ which is invariant under conjugation by $f$. In particular its residual subgroup $H=[1]_\preceq$ is normalized by $f$, so that the action of $F$ on $F_+$ descends to an order-preserving action on $(F_+/H, \prec)$, where $\prec$ is the total order induced by $\preceq$. Then, we can consider the dynamical realization $\varphi\colon F\to \homeo_0(\mathbb{R})$ of this action. We have the following equivalence.
\begin{prop} \label{p-F-focal-plo}
Let $\varphi\colon F\to \homeo_+(\mathbb{R})$ be an action. The following are equivalent.
\begin{enumerate}[label=(\roman*)]
\item \label{i-F-plo-focal} $\varphi$ is a minimal $\mathbb{R}$-focal action increasingly horograded by the standard action on $(0, 1)$.
\item \label{i-F-plo-preorder} There exists a preorder $\preceq$ on $F_+$ invariant under conjugation by $f$, such that, writing $H=[1]_\preceq$, the map $f$ acts as a homothety on $(F_+/H,\prec)$, and $\varphi$ is conjugate to the dynamical realization of the action of $F$ on $(F_+/H, \prec)$.
\end{enumerate}
Moreover two distinct preorders as in \ref{i-F-plo-preorder} give rise to non-conjugate minimal $\mathbb{R}$-focal actions.
\end{prop}
\begin{proof}
Let us prove that \ref{i-F-plo-preorder} implies \ref{i-F-plo-focal}. Assume that $\preceq$ verifies the conditions, and let $\varphi$ be the dynamical realization of the action of $F$ on $(F_+/H, \prec)$. Since $f$ is a homothety on $(F_+/H, \prec)$, Proposition \ref{p.minimalitycriteria} implies that $\varphi$ is minimal, and $\varphi(f)$ is a homothety. Since moreover $\varphi$ must fall into one of the cases of Theorem \ref{t-F-trichotomy}, the only possibility is that $\varphi$ is $\mathbb{R}$-focal, increasingly horograded by the standard action on $(0, 1)$.
For the converse, let $\varphi$ be is as in \ref{i-F-plo-focal}. Then $\varphi(f)$ is an expanding homothety (see Proposition \ref{p-F-dynclasselements}); let $\xi\in \mathbb{R}$ be its unique fixed point, and consider the preorder $\preceq$ on $F_+$ associated with this point: $g\precneq h$ if and only if $g.\xi< h.\xi$. Using that $\xi$ is fixed by $f$, we see that $\preceq$ is invariant under conjugation by $f$, and that the natural action of $F$ on $(F_+/[1]_\preceq, \prec)$ can be identified with the action of $F$ on the orbit of $\xi$, showing the claim.
Finally note that these two constructions are inverse to each other, and since $\xi$ is the unique fixed point of $f$, the preorder $\prec$ is uniquely determined by the conjugacy class of the action.
\qedhere
\end{proof}
\subsubsection{Restriction preorders on $F_+$}\label{s-restriction-preorder}
We now explain a concrete construction of preorders on $F_+$ satisfying \ref{i-F-plo-preorder} in Proposition \ref{p-F-focal-plo}. This yields a family of $\mathbb{R}$-focal actions of $F$ which contains as special cases the constructions in \S\S \ref{ssec.cyclicgerm}--\ref{ssec.germtype}.
Let $K\subseteq (0, 1)$ be a closed $f$-invariant subset. To the subset $K$ we associate a preorder $\preceq^{K}$ on $F_+$ which is obtained by looking at the restriction of elements of $F_+$ to $K$, as follows.
We first consider the subgroup $H=\{g\in F_+\colon g(x)=x\text{ for every }x\in K\}$ of elements which fix $K$ pointwise, and for $g\in F_+$ define
\[
x_g=\left\{
\begin{array}{lr}
0&\text{if }g\in H,\\
\sup \{x\in K\colon g(x)\neq x\}&\text{if }g\in F_+\setminus H.
\end{array}
\right.
\]
We immediately observe that $x_g=x_{g^{-1}}$ for every $g\in F_+$. Moreover, we have the following behavior when considering compositions.
\begin{lem}\label{l.xg-restriction} Let $K\subseteq (0,1)$ be a non-empty closed subset, and take $g,h\in F_+$. Then we have the inequality $x_{gh}\le \max\{x_g,x_h\}$, and when $x_h\neq x_g$ the equality $x_{gh}= \max\{x_g,x_h\}$ holds.
\end{lem}
\begin{proof}
Note that if $x\in K$ is such that $x>\max\{x_g,x_h\}$, then $gh(x)=g(x)=x$. This gives the inequality $x_{gh}\le \max\{x_g,x_h\}$.
Assume now $x_h<x_g$ and assume first we are in the case $g(x_g)\neq x_g$. Then $gh(x_g)=g(x_g)\neq x_g$, proving that $x_g\le x_{gh}$, hence $x_{gh}=x_g$ (using the previous inequality). When
$g(x_g)=x_g$, then $x_g$ is accumulated from the left by points of $K$ which are moved by $g$; in particular for every such point $x$ with $x_h<x<x_g$, we have $gh(x)=g(x)\neq x$, giving $x\le x_{gh}$. Taking the supremum we obtain the desired equality $x_g=x_{gh}$.
Note also that the same assumption $x_h<x_g$ (which is equivalent to $x_{h^{-1}}<x_{g^{-1}}$) gives $x_{g^{-1}h^{-1}}=x_{g^{-1}}=x_{g}$.
As $x_{hg}=x_{g^{-1}h^{-1}}$, we deduce from the previous case that $x_{hg}=x_g$. This concludes the proof.
\end{proof}
We next introduce the subset
\begin{equation}\label{eq.cone_restriction}
P_K=\left \{g\in F_+\setminus H\colon \text{either }g(x_g)>x_g,\text{ or }g(x_g)=x_g\text{ and }D^-g(x_g)>1\right \}
\end{equation}
and observe the following.
\begin{lem}\label{l.restriction_cone}
For any non-empty closed subset $K\subseteq (0,1)$, the subset $P_K$ defines a positive cone in $F_+$.
\end{lem}
\begin{proof}
We have to verify the conditions in Remark \ref{r.cones}. Let us first prove that $F_+=P_K\sqcup H\sqcup P_K^{-1}$. For this notice that, since $x_g=x_{g^{-1}}$, we have $$P_K^{-1}=\{g\in F_+\setminus H\colon \text{either }g(x_g)<x_g,\text{ or }g(x_g)=x_g\text{ and }D^-g(x_g)<1\}.$$ Thus, we automatically get that $H\cap \big(P_K\cup P_K^{-1}\big)=\emptyset$ and $P_K^{-1}\cap P_K=\emptyset$. It only remains to shows that $F_+\subseteq P_K\sqcup H\sqcup P_K^{-1}$. For this, take $g\in F_+\setminus H$, so that $x_g>0$. If $x_g\neq g(x_g)$ we are done. In the complementary case, $x_g$ must be accumulated from the left by points that are moved by $g$. Since $g$ is piecewise linear we must have $D^-g(x_g)\neq 1$ showing that $g\in P_K\sqcup P_K^{-1}$.
Next, let us check that $P_K$ is a semigroup and $HP_KH\subseteq P_K$.
Take $g,h\in P_K$, and assume first $x_h< x_g$. Then Lemma \ref{l.xg-restriction} gives $x_{gh}=x_g$ and
$gh(x_{gh})= g(x_g)$. If $g(x_g)>x_g$, we deduce immediately $gh\in P_K$; otherwise $x_g$ is accumulated from the left by points of $K$, which must be fixed by $h$, so that $D^-h(x_g)=1$. Then $D^-(gh)(x_{gh})=D^-g(x_g)\,D^-h(x_g)>1$, and we conclude that $gh\in P_K$.
Assume now that $x_g<x_h$, so that $x_{gh}=x_h$ by Lemma \ref{l.xg-restriction}.
Consider first the case $h(x_h)=x_h$.
Then $gh(x_{gh}) =g(x_h)=x_h=x_{gh}$, and as in the previous case
we see that $D^-g(x_h)=1$, so that
$D^-(gh)(x_{gh})=D^-h(x_h)>1$. When $h(x_h)>x_h$, then
$gh(x_{gh})>g(x_h)=x_h$. In both cases we have $gh\in P_K$.
Note that the previous argument works also when one of the two elements is in the residue $H$, proving that $HP_KH\subseteq P_K$.
Finally, consider the case $x_g=x_h$. As $h(x_h)\ge x_h$ and $g(x_g)\ge x_g$, then if any of the two inequalities is strict we deduce $gh(x_{g})>x_{g}$, and thus $x_{gh}=x_g$ (by the inequality of Lemma \ref{l.xg-restriction}) and $gh\in P_K$. Otherwise, assume that both $g$ and $h$ fix $x_g=x_h$. Then we have the relation $D^-(gh)(x_{gh})=D^-g(x_g)\,D^-h(x_h)>1$, showing that $x_{gh}=x_g$ (again by Lemma \ref{l.xg-restriction}) and $gh\in P_K$ also in this case.
\end{proof}
The previous lemma leads to the following definition.
\begin{dfn}
Given a closed subset $K\subseteq(0, 1)$, the preorder $\preceq^K$ on $F_+$ defined by the positive cone $P_K$ in \eqref{eq.cone_restriction} will be called the \emph{restriction preorder} associated with $K$. We will always write $H=[1]_{\preceq^K}$ for its residue.
\end{dfn}
Let us describe some elementary properties related to the preorder $\preceq^K$ that will be useful in the sequel.
\begin{lem}\label{lem.Lx-convex}
Let $K\subseteq (0,1)$ be a non-empty closed subset, and let $\preceq^K$ be the corresponding restriction preorder on $F_+$. Then the following hold.
\begin{enumerate}[label=(\roman*)]
\item \label{ii.Lx-convex} For $g,h\in F_+$ with $1\preceq^K g\preceq^K h$, we have $x_g\le x_h$.
\item\label{iii.Lx-convex} For $x\in (0,1)$, the subset $L_x:=\{g\in F_+\colon x_g\le x\}$ is a $\preceq^K$-convex subgroup.
\end{enumerate}
\end{lem}
\begin{proof}
We first prove \ref{ii.Lx-convex}. We can assume $g\in P_K$ otherwise $x_g=0$ and the result follows.
Assume for contradiction that $x_g>x_h$. Then from Lemma \ref{l.xg-restriction} we have $x_{g^{-1}h}=x_g$. Consider first the case $g(x_g)>x_g$, then $g^{-1}h(x_{g})=g^{-1}(x_g)<x_g$, so that $g^{-1}h\precneq^K 1_F$, contradicting the assumption $g\preceq^K h$. Consider next the case $g(x_g)=x_g$, so that $D^-g(x_g)>1$ and $D^-h(x_g)=1$ (as in this case $x_g$ is accumulated from the left by points of $K$). Then $g^{-1}h(x_g)=x_g$ and $D^-(g^{-1}h)(x_g)=D^-g(x_g)^{-1}<1$, giving again the contradiction $g^{-1}h\precneq^K 1_F$.
The inequality $x_{gh}\le \max\{x_g,x_h\}$ from Lemma \ref{l.xg-restriction} shows that the subset $L_x$ in \ref{iii.Lx-convex} is a subgroup, whilst \ref{ii.Lx-convex} proves that $L_x$ is $\preceq^K$-convex.
\end{proof}
Note that the coset space $F_+/H$ can be identified with the set of restrictions
$\{g\restriction_K \colon g\in F_+\}$,
so that two elements $g, h\in F_+$ are equivalent for $\preceq^K$ if and only if their restrictions to $K$ coincide.
\begin{lem}
For every $f$-invariant closed set $K\subseteq (0, 1)$, the restriction preorder $\preceq^K$ on $F_+$ is invariant under conjugation by $f$, and the conjugacy induces a homothety on $(F_+/H, \prec^K)$ fixing $H$.
\end{lem}
\begin{proof}
The verification that $\preceq^K$ is invariant under conjugation follows easily from $f$-invariance of $K$.
Let us check that $f$ acts a homothety on $(F_+/H,\prec^K)$. It is clear that it fixes the point corresponding to $H$.
We next verify that conjugation by $f$ preserves the positive cone $P_K$. Take $h\in P_K$, write $x_*=x_h$ and note that $f(x_h)=x_{fhf^{-1}}$; when $h(x_h)>x_h$, we have $fhf^{-1}(x_{fhf^{-1}})=fh(x_h)>x_h$, otherwise we have $h(x_{h})=x_{h}$ and \[D^-(fhf^{-1})(x_{fhf^{-1}})=D^-h(x_h)>1.\] Hence $fhf^{-1}\in P_K$, as wanted.
More generally, for $n\in \mathbb N$, consider $h_n=f^nhf^{-n}$ and observe that the point $x_{h_n}=f^n(x_h)$ tends to $1$ as $n\to \infty$.
Take $r\in P_K$ and let $y\in (0, 1)$ be such that $r$ acts trivially on $(y, 1)$. If $n$ is large enough so that $h_n(x_{h_n})=f^nh(x_h)$ and $x_{h_n}$ are greater than $y$, we have that $x_{r^{-1}h_n}=x_{h_n}$ and $r^{-1}h_n$ coincides with $h_n$ on a neighborhood of $x_{h_n}$. Since $h_n\in P_K$, and this depends only on the behavior of $h_n$ on some neighborhood of $x_{h_n}$, we must have $r^{-1}h_n\in P_K$ for $n$ large enough, and thus $h_n\succneq^K r$. Since $h$ and $r$ were arbitrary $\preceq^K$-positive elements and we can repeat the same reasoning for arbitrary $h,r\in P_K^{-1}$, this shows that the conjugation by $f$ is a homothety.
\end{proof}
For every $f$-invariant subset $K\subseteq (0, 1)$ let us denote by $\psi_K\colon F\to \homeo_0(\mathbb{R})$ the dynamical realization of the action of $F$ on $(F_+/H, \prec^K)$ defined above. By Proposition \ref{p-F-focal-plo} this action is $\mathbb{R}$-focal and increasingly horograded by the standard action on $(0, 1)$. Note also that since the residue $H$ is the fixator of $K$, and two distinct closed subsets of $(0, 1)$ have different fixators, then $\preceq^K$ determines $K$ completely. In particular, by the last part of Proposition \ref{p-F-focal-plo}, if $K_1\neq K_2$ their associated $\mathbb{R}$-focal actions $\psi_{K_1}$ and $\psi_{K_2}$ are not conjugate.
\subsubsection{Some properties of the actions arising from restriction preorders}
Given a non-empty $f$-invariant closed subset $K\subseteq (0, 1)$, we keep denoting by $\psi_K:F\to \homeo_{0}(\mathbb{R})$ the action constructed above. We want to point out some dynamical properties of this family of actions.
Recall that a minimal action of a group $G$ on a locally compact space $Y$ is \emph{topologically free} if the set of fixed points $\operatorname{\mathsf{Fix}}(g)$ has empty interior for every $g\in G$. By Baire's theorem this is equivalent to the requirement that a dense $G_\delta$-set of points in $Y$ have a trivial stabilizer in $G$.
\begin{prop}[Freeness and non-freeness] \label{p-F-restriction-topfree}
Let $K\subseteq (0, 1)$ be a non-empty $f$-invariant closed subset. Then the $\mathbb{R}$-focal action $\psi_K\colon F\to \homeo_0(\mathbb{R})$ defined above is topologically free if and only if $K=(0, 1)$. In particular, $F$ admits both topologically free and non-topologically free minimal $\mathbb{R}$-focal actions.
\end{prop}
\begin{proof}
Assume $K=(0, 1)$. We claim that the action $\psi:=\psi_{K}$ is topologically free. Indeed in this case the preorder $\preceq^{K}$ is actually a total order on $F_+$. Thus there is a dense subset of points in $\mathbb{R}$ with a trivial stabilizer for $\psi(F_+)$, which implies that the action of $F_+$ is topologically free. Assume by contradiction that $g\in F$ is such that $\operatorname{\mathsf{Fix}}^\psi(g)$ has non-empty interior, and let $I$ be a connected component of its interior. Note that $g\notin F_+$ so that by Propositions \ref{p-F-focal-plo} and \ref{p-F-dynclasselements}, the image $\psi(g)$ must be a pseudohomothety; in particular $I$ is bounded. As the action $\psi$ is proximal (see for instance Lemma \ref{l.exotic_proximal}), there exists $h\in F$ such that $\psi(h)(I)\Subset I$. Then it is not difficult to see that the commutator $[g, h]=ghg^{-1}h^{-1}$ is non-trivial, belongs to $F_+$ and fixes $\psi(h)(I)$ pointwise. This is a contradiction since we have already shown that the action of $F_+$ is topologically free.
Now consider the case $K\neq (0, 1)$. We can take a connected component $U=(y,z)$ of the complement $(0,1)\setminus K$, and consider a non-trivial element $h\in F_+$ whose support is contained in $U$. Fix $x<y$ and consider the $\preceq^K$-convex subgroup $L_x$ from Lemma \ref{lem.Lx-convex}. Take an element $g\in L_x$, and let us prove that the conjugate $g^{-1}hg$ belongs to $H$. For this, note that the condition $x_g<x$ implies $g^{-1}(U)=U$, so that the restriction of $g^{-1}hg$ to the complement $(0,1)\setminus U$ is trivial. This immediately implies that $g^{-1}hg$ fixes every point of $K$, so that $g^{-1}hg$ belongs to the residue $H$. This proves that $hgH=gH$ for any element $g\in L_x$, so that the element $h$ fixes the $\prec^K$-convex subset $L_x/H$ pointwise. We deduce that $\psi_K(h)$, which is non-trivial as the action $\psi_K$ is faithful, fixes a non-trivial interval.
\end{proof}
\begin{rem}
Proposition \ref{p-F-restriction-topfree} should be compared with the fact that many groups arising via a micro-supported action by homeomorphisms satisfy rigidity results for their non-topologically free actions on \emph{compact} spaces, as shown in \cite{LBMB-sub, bon2018rigidity, boudec2020commutator} using results on uniformly recurrent subgroups (URS) and confined subgroups.
As an example tightly related to this setting consider Thompson's group $F$ and its sibling $T$ acting on the circle. Then every minimal action of $T$ on any compact space is either topologically free or factors onto its standard action on the circle, while every faithful minimal action of $F$ on a compact space is topologically free \cite{LBMB-sub}.
Proposition \ref{p-F-restriction-topfree} shows that actions on the line behave very differently from this perspective, and the notion of topological freeness is much less relevant.
\end{rem}
Another feature of this family of actions is that they are all simplicial in the sense of \S \ref{s-F-simplicial}.
\begin{prop}[Simpliciality] \label{p-F-restriction-not-minimal}Let $K\subset (0,1)$ be a non-empty $f$-invariant closed subset and consider the corresponding action $\psi_K\colon F\to \homeo_0(\mathbb{R})$, as constructed above. Then the image of $F_+$ does not act minimally on $\mathbb{R}$. In particular every action $\psi_K$ is simplicial.
\end{prop}
\begin{proof}
Fix $x\in (0,1)$ and consider the $\preceq^K$-convex subgroup $L_x=\{g\in F_+\colon x_g\le x\}$ (Lemma \ref{lem.Lx-convex}). In the dynamical realization $\psi_K$ of the action $F\to\Aut\left (F_+/H, \prec^K\right )$, the cosets of $L_x$ span an $F_+$-invariant family of disjoint open intervals, showing that the $\psi_K$-action of $F_+$ is not minimal. In particular neither is the action of $[F, F]\subseteq F_+$. \end{proof}
One way to analyze finer properties of $\mathbb{R}$-focal actions of the group $F$ is to apply Theorem \ref{t-F-trichotomy} iteratively, by exploiting the self-similarity of $F$. Namely assume that $\varphi\colon F\to \homeo_+(\mathbb{R})$ is a minimal $\mathbb{R}$-focal action increasingly horograded by the standard action on $(0, 1)$. Recall that for every dyadic $x\in (0,1)$ the group $F_{(0,x)}$ is isomorphic to $F$, and its image under $\varphi$ is totally bounded, that is $\operatorname{\mathsf{Fix}}^\varphi(F_{(0, x)})$ accumulates on both $\pm \infty$. Thus we can apply Theorem \ref{t-F-trichotomy} to the action of $F_{(0, x)}$ on every connected component $J$ of $\suppphi(F_{(0, x)})$. It follows that this action still falls into one of the three cases up to semi-conjugacy: action by translations, the standard action, and $\mathbb{R}$-focal actions. In the third case, this analysis can of course be iterated. We will speak of ``sublevels'' of the action $\varphi$ to refer to the actions of the subgroups $F_{(0, x)}$ obtained in this way. From this point of view the actions $\psi_{K}$ arising from restriction preorder are very special: indeed they are not exotic on any sublevel (in contrast with other $\mathbb{R}$-focal actions of $F$; see Proposition \ref{p-F-CB-sublevels} below).
\begin{prop}[Absence of exotic sublevels] \label{p-restriction-sublevels}
Let $K\subset (0,1)$ be a non-empty $f$-invariant closed subset and consider the corresponding action $\psi_K\colon F\to \homeo_0(\mathbb{R})$, as constructed above. Let $x\in X$ be a dyadic point, and let $J$ be a connected component of $\supp^{\psi}(F_{(0, x)})$. Then the $\psi$-action of $F_{(0, x)}$ on $J$ is semi-conjugate either to its standard action on $(0, x)$, or to a cyclic action by translations induced from group of germs $\Germ(F_{(0, x)}, x)\cong \mathbb Z$.
\end{prop}
\begin{proof}
Let $\xi_0$ be the unique fixed point of $\psi(f)$. Let us first show the claim for the action of $F_{(0, x)}$ on $J=\Ipsi(x, \xi_0)$ (the connected component of $\supp^{\psi}(F_{(0,x)})$ containing $\xi_0$). The semi-conjugacy type of this action is determined by the preorder $\preceq_{\xi_0}\in\LPO(F_{(0,x)})$ induced by the point $\xi_0$ on $F_{(0, x)}$, which coincides with the restriction of $\preceq^{K}$ to $F_{(0, x)}$. Now we distinguish two cases.
First assume that $K\cap (0, x)$ does not accumulate on $x$. Write $y=\sup \{K\cap (0, x)\} < x$ and let $\preceq_y\in\LPO(F_{(0,x)})$ be its induced preorder on $F_{(0, x)}$. By definition of $\preceq^{K}$, it follows that $\preceq_{\xi_0}$ is dominated (in the sense of Definition \ref{d-preorder-dominates}) by $\preceq_y$. Since the dynamical realization of $\preceq_y$ is the standard action of $F_{(0,x)}$, the conclusion in this case follows from Lemma \ref{lem.domsemicon}.
Assume now that $\sup \{K\cap (0, x)\}= x$. In this case, by definition of $\preceq^K$ we get that $\preceq_{\xi_0}$ is dominated by a preorder obtained as the pull-back of one of the two non-trivial preorders on $\Germ(F_{(0, x)}, x)\cong \mathbb Z$. This shows the conclusion for $\xi=\xi_0$.
If now $\xi\in \supp^\psi(F_{(0, x)})$ is arbitrary, then by minimality we can choose $h\in F$ such that $\psi(h)(\xi_0)\in \Ipsi(x, \xi)$. Then the conclusion follows from the previous case applied to the action of $F_{(0, h^{-1}(x))}=h^{-1} F_{(0, x)}h$ on $\Ipsi(h^{-1}(x), \xi_0)$. \qedhere
\end{proof}
\subsubsection{Some variations on the restriction preorder construction}
The restriction preorder construction can be modified in multiple ways to produce new families of minimal $\mathbb{R}$-focal actions, which are not conjugate to the actions $\psi_K$ defined above. We indicate some of them, without detailed exploration nor attempting to include them all in a unified family.
\begin{enumerate}[leftmargin=*]
\item \emph{Twisting with sign choices.} In addition to the $f$-invariant set $K\subseteq (0, 1)$ consider an $f$-invariant choice of signs $u\colon K\to \{+1, -1\}$. We proceed to define a preorder $\preceq^{(K, u)}$ in $F_+$. For this, given $g\in F_+$ we say that $g\succneq^{(K,u)}1_F$ if either $u(x_g)=1$ and $g\succneq^K1_F$ or $u(x_g)=-1$ and $g\precneq^K1_F$. It is direct to check (following the proof of Lemma \ref{l.restriction_cone}) that $\preceq^{(K,u)}$ is an invariant preorder on $F_+$ and that the $f$-invariance of $u$ makes $\preceq^{(K,u)}$ invariant under conjugation by $f$. Of course, when $u\equiv 1$ the preorders $\preceq^{(K,u)}$ and $\preceq^K$ coincide.
There are some straightforward variations to this twist. For instance one may consider two different $f$-invariant functions $u, v\colon K\to \{\pm 1\}$ to determine the sign in the two different cases $g(x_g)\neq x_g$ and $g(x_g)=x_g$.
\item \emph{Twisting with derivative morphisms.} In this case, in addition to the $f$-invariant set $K\subseteq (0, 1)$ consider a total left order $<_0$ on the abelian group \[A=\{(2^n, 2^m)\colon n, m\in \mathbb Z\}\cong \mathbb Z^2\]
(note that $A$ can be though as the set of derivatives that an element of $F$ can take at a dyadic point). As before, we will define a preorder on $F_+$ which is invariant under conjugacy by $f$. For this, consider a different definition of $x_g$, namely define
\[x'_g:=\sup\left \{x\in K:g(x)\neq x, \text{ or } g(x)=x \text{ and }(D^-g(x),D^+g(x))\neq (1,1)\right \}.\] Then, set $\preceq^{K}_0\in\LPO(F_+)$ so that $g\succneq^{K}_0 1_F$ if either $g(x'_g)>x'_g$, or $g(x'_g)=x'_g$ and $(D^-g(x'_g),D^+g(x'_g))>_0(1,1)$. Again, it is straightforward to check (following the proof of Lemma \ref{l.restriction_cone}) that the preorder $\preceq^K_0$ is left invariant and also invariant under conjugation by $f$.
To compare these preorders with the preorders of the form $\preceq^{(K,u)}$, consider $p\in(0,1)\cap\mathbb Z[1/2]$ and the closed subset $K_p=\{f^n(p):n\in\mathbb Z\}$. In this case all the twists $\preceq^{(K_p,u)}$ given by sign choices coincide with $\preceq^{K_p}$, while the preorder $\preceq^{K_p}_0$ just defined does not. The interested reader can also check that in this case, the dynamical realization of $\preceq^{K_p}_0$ has sublevels (in the sense of Proposition \ref{p-restriction-sublevels}) semi-conjugate to $\varphi\circ\pi^{ab}$ where $\pi^{ab}:F_{(0,x)}\to F_{(0,x)}^{ab}\cong\mathbb Z^2$ is the abelianization of $F_{(0,x)}$ (i.e.\ $x$ is dyadic) and $\varphi$ is the dynamical realization of $<_0$.
\item \emph{Twisting with new orderings of $(0,1)$.} In the construction of the preorder $\preceq^K$ one can modify the definition of the point $x_g$ by taking the supremum with respect to an order $\prec_0$ on $K$ which is different from the order induced from the embedding $K\subseteq (0, 1)$. The whole construction will still be well-defined provided $\prec_0$ is $f$-invariant and satisfies suitable assumptions, which are not difficult to figure out but are rather technical to state. Instead of discussing this in general, let us give an example.
Take $0<x_0<p_1<p_2<f(x_0)<1$ and define $K$ as the union of the orbits of $p_1$ and $p_2$. Then, we define the total order $\prec_0$ on $K$ so that $f^m(p_i)\prec_0 f^n(p_j)$ if either $m+i<n+j$, or $m+i=n+j$, $i=1$ and $j=2$. More explicitly we have $$\cdots \prec_0 f^{-2}(p_2)\prec_0 p_1\prec_0 f^{-1}(p_2)\prec_0 f(p_1)\prec_0\cdots.$$ It is clear that $\prec_0$ is $f$-invariant. We can then define a preorder $\preceq^{K, \prec_0}$ in the same way as the restriction preorder $\preceq^K$, except that we replace the point $x_g$ by the point $x''_g$ consisting on the $\prec_0$-greatest element of the subset $\{x\in K:g(x)\neq x \}$. It is straightforward to check that $\preceq^{K,\prec_0}$ is left invariant and also invariant under conjugation by $f$, inducing an order-preserving action $F\to \Aut \left (F_+/[1]_{\preceq^{K,\prec_0}},\prec^{K,\prec_0}\right )$ as above. Denote by $\Psi_0=\Psi_{K,\prec_0}$ the dynamical realization of this action and assume that its associated good embedding satisfies $\iota([1]_{\preceq^{K,\prec_0}})=0$. It can be shown that different choices of $p_1$ and $p_2$ produce non-conjugate actions. On the other hand, the interested reader can check that the semi-conjugacy classes of the sublevels $F_{(0,x)}\curvearrowright \mathrm I^{\Psi_0}(x,0)$ only depend on the choice of $p_2$ but not of $p_1$. This shows that exotic actions cannot be reconstructed with the information of the semi-conjugacy classes of its sub-levels as defined in Proposition \ref{p-restriction-sublevels}.
Again there are some obvious variations of this, such as considering preorders on $A$ instead of orders, and modifying the definition of the point $x'_g$ accordingly.
\end{enumerate}
Of course one can consider appropriate combinations of the variants defined above. However whether such combinations make sense or not depends on the choice of the parameters, and a unified treatment would be obscure and pointless. All constructions obtained using these methods yield simplicial actions.
\subsection{A plethora of $\mathbb{R}$-focal actions II: ordering the orbit of a closed subset of $(0, 1)$} \label{s-F-orbit-construction}
We now describe another method to construct $\mathbb{R}$-focal actions of $F$. The starting ingredient of this method is again a non-empty closed subset $K\subset (0, 1)$ which is invariant under the generator $f$ given by \eqref{e-F-big-generator}. We assume now $K\neq (0, 1)$, and consider the $F$-orbit of $K$ among closed subsets of $(0, 1)$, and denote it by \[\mathcal{O}_K:=\{g(K) \colon g\in F\}.\]
As $K\subset (0,1)$ is a proper subset, we clearly have that the orbit $\mathcal{O}_K$ is infinite.
It is then natural to try to define an $F$-invariant order $\prec$ on $\mathcal{O}_K$, and then consider its dynamical realization. While this may seem similar to the construction just discussed in \S \ref{s-restriction-preorder}, it turns out to be quite different and to produce actions with more exotic dynamical properties. Note that we are not aware of any general receipt to build $F$-invariant orders on $\mathcal{O}_K$ which works \emph{for all} $K$: the way such orders arise depend subtly on the properties of the subset $K$. However, there is a general strategy which is conveniently described in the language of directed trees. We will first describe this strategy in general, and then illustrate it in practice with a concrete choice of subset $K$. More examples of actions obtained using this method will appear later in \S \ref{s-F-hyperexotic}.
\subsubsection{A strategy to order $\mathcal{O}_K$} \label{s-F-orbit-strategy}Assume that $K\subset (0, 1)$ is an $f$-invariant closed subset.
Since the germ of $f$ at $1$ generates the group of germs $\Germ(F, 1)\cong \mathbb Z$ and $K$ is $f$-invariant, it follows that every $K_1=g(K)\in \mathcal{O}_K$ must coincide with $K$ on an interval of the form $(1-\varepsilon, 1)$, with $\varepsilon>0$. Thus, it follows that for every pair $K_1, K_2\in \mathcal{O}_K$, the subsets $K_1$ and $K_2$ coincide on some interval of the form $(1-\varepsilon, 1)$, so that we can define
\begin{equation} \label{e-alpha-K1-K2}\alpha(K_1, K_2)=\inf\{x\in (0,1)\colon K_1\cap[x, 1)=K_2\cap[x, 1)\}.\end{equation}
As $K$ is closed, we have $\alpha(K_1,K_2)\in K_1\cap K_2$, unless $K_1=K_2$ (in which case $\alpha(K_1,K_2)=0$).
Moreover, in light of the previous discussion, we get that $\alpha(K_1,K_2)<1$ for every $K_1,K_2\in\mathcal{O}_K$.
It is clear from the definition that for every $K_1,K_2,K_3\in \mathcal{O}_K$ with $\alpha(K_1,K_3)\le\alpha(K_2,K_3)$, we have $\alpha(K_1,K_2)\le\alpha(K_2,K_3)$ (indeed, when $K_2\neq K_3$, the three intersections $K_i\cap [\alpha(K_2,K_3),1)$ for $i\in\{1,2,3\}$ coincide). This gives the ultrametric inequality
\[
\alpha(K_1,K_2)\le\max\{\alpha(K_2,K_3),\alpha(K_1,K_3)\}.
\]
Note also that for given $g\in F$ and $K_1,K_2\in \mathcal{O}_K$, we have
\begin{align*}
\alpha(g(K_1),g(K_2))&=\inf\{x\in (0,1)\colon K_1\cap[g^{-1}(x), 1)=K_2\cap[g^{-1}(x), 1)\}\\
&=\inf\{g(y)\in (0,1)\colon K_1\cap[y, 1)=K_2\cap[y, 1)\}=g\left (\alpha(K_1,K_2)\right ).
\end{align*}
In the terminology of \S \ref{subsec.ultra}, we have just verified that the map $\alpha:\mathcal{O}_K\times\mathcal{O}_K\to[0,1)$ is an $F$-equivariant ultrametric kernel with respect to the standard $F$-actions on $\mathcal{O}_K$ and $(0,1)$ respectively. Moreover, we have the following.
\begin{lem}\label{l-tree-K-focal}
For every closed $f$-invariant subset $K\subsetneq (0, 1)$, the action of $F$ on $\mathcal{O}_K$ expands $\alpha$-balls.\end{lem}
\begin{proof}
As the action of $F$ on $\mathcal{O}_K$ is transitive, it is enough to check that there exists a sequence of elements $(g_n)\subset F$ such that the sequence of balls $g_n.B_\alpha(K,x)$ defines an increasing exhaustion of $\mathcal{O}_K$. For this, note that by $f$-invariance we have $f^n.B_\alpha(K,x)=B_\alpha(K,f^n(x))$ and thus $\mathcal{O}_K=\bigcup_{n \ge 0}f^n.B_\alpha(K,x)$, as desired.
\end{proof}
As a consequence of the discussion above and Corollary \ref{cor.ultra}, for every $\alpha$-convex order $<$ on $\mathcal{O}_K$ we get a minimal $\mathbb{R}$-focal action increasingly horograded by the standard $F$-action. However, it is not clear \textit{a priori} that for a given subset $K$ such an $\alpha$-convex order exists, and this is why what we have just described is simply a \emph{strategy}. We will see in \S \ref{subsub.concrete} and \S \ref{s-F-hyperexotic}, that for some choices of $K$, $\alpha$-convex orders actually exist, although this is false in general (see Example \ref{rem.nonplanar} below).
\begin{rem}In practice, $\alpha$-convex orders on $\mathcal{O}_K$ are such that the order relation between $K_1$ and $K_2$ only depends on how $K_1,K_2$ behave ``right before'' the point $\alpha(K_1, K_2)$, in an $F$-invariant manner. For a formal presentation of this correspondence see Proposition \ref{prop.deltaconvex}.
\end{rem}
\begin{ex}[Non planarly orderable actions]\label{rem.nonplanar}
Recall from Definition \ref{dfn.ultratree}, that there is a natural construction of directed tree associated with the ultrametric kernel $\alpha$ on $\mathcal{O}_K$. More precisely, we obtain an action $$\Phi:F\to\Aut(\mathbb{T}_K,\triangleleft)$$ together with a $F$-equivariant injective map $i:\mathcal{O}_K\to\partial^\ast\mathbb{T}_K$.
Roughly speaking, the directed tree $(\mathbb T_K,\triangleleft)$ is obtained by taking a copy of $(0,1)$ for each $K_1\in \mathcal{O}_K$, and by gluing the two copies corresponding to $K_1$ and $K_2$ along the interval $[\alpha(K_1, K_2), 1)$. We denote by $p:\mathcal{O}_K\times (0,1)\to\mathbb{T}_K$ the quotient projection and $[K_1,x]:=p(K_1,x)$. Then, two points $v,w\in \mathbb T_K$ satisfy $v\triangleleft w$ (that is, $v$ lies below $w$) if and only there exists $K_1\in\mathcal{O}_K$ and $x,y\in(0,1)$ so that $v=[K_1,x]$, $w=[K_1, y]$, and $x<y$. The diagonal action of $F$ on $\mathcal{O}_K \times (0, 1)$ descends to an action on $(\mathbb T_K,\triangleleft)$ and the projection to the second coordinate descends to an increasing $F$-equivariant horograding $\pi\colon \mathbb T_K \to (0, 1)$. Finally, the embedding $i:\mathcal{O}_K\to\partial^\ast\mathbb{T}_K$ is defined so that each $K_1\in \mathcal{O}_K$ is sent to the infimum of the $\triangleleft$-chain $\{[K_1,x]:x\in (0,1)\}$, which naturally belongs to $\partial^\ast\mathbb{T}_K$.
Although non-strictly necessary for the sequel, this point of view is well-suited for understanding in a more conceptual way whether for given $K$ the action $\Phi:F\to\Aut(\mathbb{T}_K,\triangleleft)$ admits a $\Phi$-invariant planar order, and this is the same (after Proposition \ref{prop.deltaconvex}) to the condition that $\mathcal{O}_K$ admit an $\alpha$-convex order.
This turns out to depend on the local geometry of $K$ relatively to dilations by 2. We explain this with an example, but first we introduce some terminology to discuss the local geometry.
We say that two closed sets $K_1,K_2\subseteq (0,1)$ have equivalent left-germs at $x$ if for some $\varepsilon>0$ it holds $K_1\cap(x-\varepsilon,x]=K_2\cap(x-\varepsilon,x]$. We denote by $K^-_x$ the left-germ class of the subset $K$ at $x$. Notice that the group $\Germ_-(x)$ of left-germs of homeomorphisms fixing $x$ naturally acts on the set of left-germs of closed sets at $x$. We denote by $h_x\in\Germ_-(x)$ the germ of the homothety that fixes $x$ and has derivative $2$.
Recall from Remark \ref{rem.planarexistence} that the existence of such invariant planar ordering boils down to the existence, for each branching point $v\in \mathbb T_K$, of an ordering of the set of connected components $E^-_v$ below $v$, which is invariant under the action of the stabilizer $\stab^\Phi(v)$. An obstruction for this is clearly given by finite orbits.
With this in mind, consider an $f$-invariant closed subset $K\subseteq (0,1)$ containing a dyadic point $x\in K\cap\mathbb Z[\frac{1}{2}]$ such that $h_x(K^-_x)\neq K^-_x$ but $h_x^n(K^-_x)= K^-_x$ for some $n>1$. Write $v=[K,x]$ and let $e_v(K)$ be the component of $E_v^-$ corresponding to the ray $\{[K,x]:x\in(0,1)\}$. Since $\Germ_-(F,x)=\langle h_x\rangle$, it holds that the component $e_v(K)$ has a finite orbit which is not a fixed point, so that there exists no $\mathsf{Stab}^\Phi(v)$-invariant total order on $E_v^-$.
\end{ex}
\subsubsection{A concrete example}\label{subsub.concrete}
We now illustrate the flexibility of the method described in \S\ref{s-F-orbit-strategy}, with an explicit example of subset $K$. More precisely we will construct a subset $K\subset (0, 1)$ with the following property: \emph{there is an explicit (continuous) injective map from the set $\mathsf O(\mathbb N)$ of orders on the natural numbers $\mathbb N$ to the set of $F$-invariant orders on the orbit $\mathcal{O}_K$}. This will provide a family of $\mathbb{R}$-focal actions of $F$ which are naturally indexed by orders on $\mathbb N$.
We start by choosing an irrational point $x_0\in (0, 1)$, and consider the interval $I=(f^{-1}x_0, x_0]$, which is a fundamental domain for $f$. Next we choose a sequence of open intervals $(J_n)_{n\ge 1}$ with dyadic endpoints such that $ J_n\Subset J_{n+1}\subset I$ for every $n\ge 1$, and such that $\bigcup_{n\ge 1} J_n=(f^{-1}x_0, x_0)$. For every $n\ge 1$, write $y_n=\sup J_n$ and choose an element $h_n\in F_{J_n}$ with the following properties:
\begin{itemize}
\item $h_n(x)>x$ for every $x\in J_n$,
\item $h_n(J_{n-1})\cap J_{n-1}=\varnothing$ for $n\ge 2$.
\item $D^-h_n(y_n)=1/2$ (in other words, the germ of $h_n$ at $y_n$ generates the group of germs $\Germ(F_{(0, y_n)}, y_n)$).
\end{itemize}
Choose now a dyadic point $z_0\in J_0$, and let $\Sigma_0=\{h_1^n(z_0) \colon n \in\mathbb N\}$ be its forward orbit under $h_1$. By construction we have the inclusion $\Sigma_0\subset J_0$ and equality $\overline{\Sigma_0}=\Sigma_0\cup \{y_1\}$. Set $\Sigma_1=\bigcup_{n\ge 0} h_2^n(\overline{\Sigma_0})$, so that $\overline{\Sigma_1}=\Sigma_1\cup\{y_2\}$. Continue in this way by defining for every $i\ge 1$ a subset $\Sigma_i=\bigcup_{n\ge 0} h_{i+1}^n(\overline{\Sigma_{i-1}})$. Set $\Sigma_\omega=\bigcup_{i\in \mathbb N} \Sigma_i$, and note that $\overline{\Sigma_\omega}=\Sigma_\omega\cup \{x_0\}$. Note also that $\overline{\Sigma_\omega}$ is contained in the fundamental domain $I$ on $f$. Thus we obtain an $f$-invariant closed subset $K$ as
\begin{equation}\label{ex.K}K=\bigcup_{n\in \mathbb Z} f^n(\overline{\Sigma_\omega}).\end{equation}
By construction the subset $\overline{\Sigma_\omega}$ is invariant under the semigroup $S:=\langle h_n\colon n\ge 1\rangle_+$, in the sense that $s(\overline{\Sigma_\omega})\subset \overline{\Sigma_\omega}$ for every $s\in S$. See Figure \ref{fig.exotic_CB}
\begin{figure}[ht]
\includegraphics[width=\textwidth]{Exotic_F_CB-1.pdf}
\caption{Construction of the compact set $K$.}\label{fig.exotic_CB}
\end{figure}
The subset $\overline{\Sigma_\omega}$ is countable and compact, and its points can be classified according to their \emph{Cantor--Bendixson rank} (see \cite[\S6]{Kechris}), as follows. Points of rank 0 are the isolated points: these are exactly points in the orbit of $z_0$ under the semigroup $S$. Points of rank 1 are those that are not isolated, but become isolated after removing the isolated points: these are exactly points in the $S$-orbit of $y_1$. Continuing in this way, points of rank $n$ are precisely points in the $S$-orbit of $y_{n}$. Finally there is a unique point whose rank is the first countable ordinal $\omega$, namely the point $x_0$. This discussion can be directly extended to the subset $K$. We write $\rk_K(x)$ for the Cantor--Bendixson rank of a point $x\in K$. Note that for every $g\in F$ and $x\in K$, we have the relation $\rk_{g(K)}(g(x))=\rk_K(x)$.
We next consider the $F$-equivariant ultrametric kernel
$\alpha:\mathcal{O}_K\times\mathcal{O}_K\to [0,1)$ defined as in \eqref{e-alpha-K1-K2} and the key observation is that the particular choice of the subset $K$ allows to directly relate $\alpha$ with the Cantor--Bendixson rank.
\begin{lem} \label{l-F-CB}
Let $K\subset (0,1)$ be the subset defined at \eqref{ex.K}. For every $K_1, K_2\in \mathcal{O}_K$, the point $x=\alpha(K_1, K_2)$ is such that $\rk_{K_1}(x)$ and $\rk_{K_2}(x)$ are both finite, and moreover $\rk_{K_1}(x)\neq \rk_{K_2}(x)$ unless $\rk_{K_1}(x)=\rk_{K_2}(x)=0$.
Conversely, for every distinct $n, m\in \mathbb N$ there exist $K_1, K_2\in \mathcal{O}_K$ such that the point $x=\alpha(K_1, K_2)$ satisfies $\rk_{K_1}(x)=n$ and $\rk_{K_2}(x)=m$.
\end{lem}
\begin{proof}
We first need some observations.
\setcounter{claimnum}{0}
\begin{claimnum}\label{cl.CB1}
For every $x\in K$ and every $g\in F$ such that $g(x)=x$, there exists $\varepsilon>0$ such that $g(K)\cap (x-\varepsilon, x]=K\cap (x-\varepsilon, x]$.
\end{claimnum}
\begin{proof}[Proof of claim]
Up to replace $g$ by its inverse, we can assume $D^-g(x)\le 1$.
Also, upon conjugating by powers of $f$, we can assume $x\in \overline{\Sigma_\omega}$. If $x=x_0$ then this follows from the fact that we chose $x_0$ to be irrational, so that every element of $F$ that fixes $x_0$ must actually fix a neighborhood of it. If $x$ is isolated in $K$ the conclusion is obvious. Finally assume that $n:=\rk_K(x)\notin\{0, \omega\}$. Then $x$ is in the $S$-orbit of the point $y_{n}$, so that it is fixed by a conjugate $h$ of $h_n$, which has therefore the property that $D^-h(x)=1/2$. Hence the restriction of $g$ to a left-neighborhood of $x$ must coincide with the restriction of some non-negative power of $h$, so that we can conclude from the fact that $K$ is forward invariant under $h$.
\end{proof}
\begin{claimnum}\label{cl.CB2}
For every pair of points $x, y\in K$ with $\rk_k(x)=\rk_k(y)$, there exist an element $h\in F$ and $\varepsilon>0$ such that $h(x)=y$ and $h(K)\cap (y-\varepsilon, y]=K\cap (y-\varepsilon, y]$.
\end{claimnum}
\begin{proof}[Proof of claim]Upon replacing $x,y$ with $f^m(x), f^n(y)$ for suitable $n, m$ we can assume that $x, y\in \overline{\Sigma_\omega}$. Then $x$ and $y$ are in the same $S$-orbit, and so it is enough to observe that elements of $S$ and their inverses have this property.
\end{proof}
With this in mind, let us prove the lemma. We can assume without loss of generality that $K_1=K$. Take $g\in F$ such that $K_2=g(K)$ and set $x=\alpha(K, K_2)$ and $y=g^{-1}(x)\in K$, so that $\rk_{K_2}(x)=\rk_K(y)$. Assume by contradiction that $\rk_K(x)=\rk_K(y)\ge 1$. After Claim \ref{cl.CB2}, we can choose $h\in F$ such that $h(x)=y$ and $\varepsilon>0$ such that $h(K)\cap (y-\varepsilon, y]=K\cap (y-\varepsilon, y]$. Then the element $g'=hg$ is such that $g(y)=y$, so that upon taking a smaller $\varepsilon$, by Claim \ref{cl.CB1} we also have $g'(K)\cap (y-\varepsilon, y]=K\cap(y-\varepsilon, y]$. Applying $h^{-1}$ we deduce that there is $\varepsilon'>0$ such that $g(K)\cap (x-\varepsilon', x]=K\cap (x-\varepsilon', x]$, and the latter intersection is not reduced to $\{x\}$ since we assume that $\rk_K(x)\ge 1$. This contradicts the definition of $x=\alpha(g(K),K)$.
Thus $\rk_K(x)\neq \rk_{g(K)}(x)$ unless both ranks are 0. Finally this also implies that we cannot have $\rk_K(x)=\omega$, indeed since points of rank $\omega$ are the only non-dyadic points in $K$ this would imply that $\rk_{g(K)}(x)=\omega$ as well, contradicting the previous reasoning. \qedhere
\end{proof}
Now let $\mathsf{O}(\mathbb N)$ be the set of total orders on the natural numbers.
To every order $\prec$ in $\mathsf{O}(\mathbb N)$ we associate an $F$-invariant order $\prec^*$ on $\mathcal{O}_K$, as follows. Given distinct $K_1, K_2\in \mathcal{O}_K$, set $n_1=\rk_{K_1}(\alpha(K_1, K_2))$ and $n_2= \rk_{K_2}(\alpha(K_1. K_2))$. If $n_1\neq n_2$, then we declare $K_1\prec^* K_2$ if and only if $n_1\prec n_2$. Else, by Lemma \ref{l-F-CB} we have $n_1=n_2=0$, namely the point $\alpha(K_1, K_2)$ is isolated in both $K_1$ and $K_2$. In this case set
\begin{equation}\label{eq:Kisolated}
x_i=\max\{x\in K_i\colon x<\alpha(K_1, K_2)\}\quad\text{for }i\in \{1, 2\}.
\end{equation}
Then for $i\in \{1,2\}$ we must have $x_i<\alpha(K_1, K_2)$ and $x_1\neq x_2$ by definition of $\alpha(K_1, K_2)$. In this case we declare $K_1 \prec^* K_2$ if and only if $x_1<x_2$. It is routine to verify that this defines indeed a total order relation, and it is clear from the construction, and $F$-equivariance of the ultrametric kernel $\alpha$ and the Cantor--Bendixson rank, that this order is $F$-invariant.
Denote by $\varphi_{\prec}:F\to \homeo_{0}(\mathbb{R})$ the dynamical realization of the action of $F$ on $(\mathcal{O}_K,\prec^\ast)$. We want to prove that $\varphi_{\prec}$ is a minimal $\mathbb{R}$-focal action, increasingly horograded by the standard action of $F$. After the discussion in \S\ref{s-F-orbit-strategy}, this is equivalent to the property that the $\alpha$-balls are $\prec^*$-convex.
This is what we verify next.
\begin{lem}\label{lem.Kisconvex} With notation as above, the $\alpha$-ball \[B_\alpha(L,x)=\left \{L'\in\mathcal{O}_K:\alpha(L,L')\leq x\right \}\] is $\prec^\ast$-convex for every $L\in\mathcal{O}_K$ and $x\in (0,1)$.
\end{lem}
\begin{proof} First notice that the $\prec^\ast$-order relation between $K_1,K_2\in\mathcal{O}_K$ is determined by the intersections $K_1\cap[x,1)$ and $K_2\cap[x,1)$ for any $x\in(0,1)$ such that these intersections do not coincide.
Now take elements $K_1,K_2\in B_\alpha(L,x)$ for some $L\in\mathcal{O}_K$ and $x\in (0,1)$. This is equivalent to the condition that
\begin{equation}\label{eq.ball_intersection}
K_1\cap [x,1)=K_2\cap [x,1)=L\cap[x,1).
\end{equation}
Consider next an element $K_3$ between $K_1$ and $K_2$ (with respect to $\prec^*$) and assume by contradiction that $K_3\notin B_\alpha(L,x)$. This means that $K_3\cap [x,1)\neq L\cap [x,1)$ and therefore, considering the equalities \eqref{eq.ball_intersection}, the $\prec^\ast$-order relation between $K_i$ and $K_3$ is determined by the intersections $K_3\cap[x,1)$ and $L\cap[x,1)$ for every $i\in \{1,2\}$. Hence we conclude that the $\prec^\ast$-order relation between $K_1$ and $K_3$ coincides with that of $K_2$ and $K_3$. As we are assuming that $K_3$ lies between $K_1$ and $K_2$, we necessarily have $K_1=K_2=K_3$, but this contradicts the assumption $K_3\notin B_\alpha(L,x)$.
\end{proof}
As a conclusion of our discussion, we have the following.
\begin{prop} \label{p-F-CB}
With notation as above, for any $\prec\in \mathsf O(\mathbb N)$, the dynamical realization $\varphi_{\prec}:F\to \homeo_{0}(\mathbb{R})$ of the action of $F$ on $(\mathcal{O}_K,\prec^*)$ is a minimal $\mathbb{R}$-focal action increasingly horograded by the standard action of $F$.
Moreover, if $\prec_1$ and $\prec_2$ are distinct orders on $\mathbb N$, then the actions $\varphi_{\prec_1}$ and $\varphi_{\prec_2}$ are not conjugate.
\end{prop}
\begin{proof}
After Lemma \ref{lem.Kisconvex}, the $\alpha$-balls are $\prec^*$-convex, so the first statement is a consequence of Corollary \ref{cor.ultra}.
For a given order $\prec\in \mathsf O(\mathbb N)$, observe that by definition of dynamical realization, the $F$-action on $(\mathcal{O}_K, \prec^*)$ can be identified with the $\varphi_\prec$-action on the orbit of the unique fixed point $\xi$ of $\varphi_\prec(f)$ with the order induced by $\mathbb{R}$, so that the order $\prec^*$ can be reconstructed from $\varphi_{\prec}$. Finally the order $\prec$ on $\mathbb N$ can be reconstructed from $\prec^*$ by the last statement in Lemma \ref{l-F-CB}.
\end{proof}
We now point out a qualitative difference which distinguishes the family of $\mathbb{R}$-focal actions constructed here from the one obtained via the restriction preorder construction as in \S \ref{s-restriction-preorder}. Indeed, in this case the actions of the subgroups $F_{(0, x)}\cong F$ on the components of their support can remain exotic (compare this with Proposition \ref{p-restriction-sublevels}).
\begin{prop}[Presence of exotic sublevels]\label{p-F-CB-sublevels}
Fix an order $\prec$ on $\mathbb N$ and let $\varphi:=\varphi_\prec\colon F\to \homeo_0(\mathbb{R})$ be the $\mathbb{R}$-focal action constructed above. Then there exist a dyadic point $x\in (0,1)$ and a connected component $J$ of $\suppphi(F_{(0,x)})$ such that the action of $F_{(0, x)}$ on $J$ is semi-conjugate to a faithful $\mathbb{R}$-focal action.
\end{prop}
\begin{proof}
Note first that for every $x\in (0,1)$ and $g\in F$ such that
$g(K\cap [x,1))=K\cap [g(x), 1)$ then $g.B_\alpha(K,x)=B_\alpha(K,g(x))$. In particular,
the $\alpha$-ball $B_\alpha(K,x)$ is preserved by the subgroup $F_{(0,x)}$.
Let $\iota\colon (\mathcal{O}_K, \prec^*)\to \mathbb{R}$ be an equivariant good embedding associated with $\varphi$ (in the terminology of Definition \ref{dfn.goodbehaved}) and let $I_x$ be the open interval spanned by $\iota(B_\alpha(K,x))$, namely $I_x$ is the interior of the closure of $\iota(B_\alpha(K,x))$ (using minimality of the action and that the $\alpha$-balls are $\prec^*$-convex after Lemma \ref{lem.Kisconvex}).
Consider the element $h_1$ from the construction of $K$, and consider the points $z_0\in J_1$ and $y_1=\sup J_1$ as in the construction; for $n\ge 1$ set $z_n=h_1^n(z_0)$, which by construction is an increasing sequence converging to $y_1$. For every $n\ge 0$ we have $h_1^n(K\cap [z_0, 1))=K\cap [z_n, 1)$, so that $h_1^n.B_\alpha(K,z_0)=B_\alpha(K,z_n)$. The corresponding intervals $I_{z_n}$ satisfy $I_{z_n}\Subset I_{z_{n+1}}$ and $h_1.I_{z_n}=I_{z_{n+1}}$. Set $B:=\bigcup_{n \ge 0} B_\alpha(K,h_1^n(z_0))$, and let $J=\bigcup_{n \ge 0} I_{z_n}$ be the interval spanned by $\iota(B)$. Then $B$ is preserved by $F_{(0, y_1)}$ and the action of $F_{(0, y_1)}$ on $B$ is cofinal (with respect to the order $\prec^*$ restricted to $B$). As a consequence, $F_{(0, y_1)}$ preserves $J$ and acts on it without fixed points, so that $J$ is a connected component of $\suppphi(F_{(0, y_1)})$. Since moreover
$h_1.I_{z_n}=I_{z_{n+1}}$, we deduce that $\varphi(h_1)$ acts on $J$ as a pseudohomothety.
This cannot happen if the action of $F_{(0, y_1)}$ on $\Iphi(y_1, \xi)$ is semi-conjugate to an action by translations, nor if it is semi-conjugate to the standard action on $(0, y_1)$. Thus by Theorem \ref{t-F-trichotomy} the action of $F_{(0, y_1)}$ must be semi-conjugate to an $\mathbb{R}$-focal action.
\end{proof}
Nonetheless, this family of examples
still turns out to produce simplicial $\mathbb{R}$-focal actions in the sense of \S \ref{s-F-simplicial}.
\begin{prop}[Simpliciality] \label{p-F-orbit-not-minimal}
For an order $\prec$ on $\mathbb N$, let $\varphi_\prec\colon F\to \homeo_0(\mathbb{R})$ be the $\mathbb{R}$-focal action constructed above. Then $\varphi_\prec(F_+)$ does not act minimally on $\mathbb{R}$. In particular each action $\varphi_\prec$ is simplicial
\end{prop}
\begin{proof}
We keep the same notation as in the proof of Proposition \ref{p-F-CB-sublevels}. Let $x_0\in K$ be the point as in the construction of $K$. We claim that the $\alpha$-ball $B_\alpha(K,x_0)\subset \mathcal{O}_K$ has the property that for every $g\in F_+$ we have either $g(B_\alpha(K,x_0))=B_\alpha(K,x_0)$ or $g(B_\alpha(K,x_0))\cap B_\alpha(K,x_0)=\varnothing$. It then follows that the interval $\Iphi(x_0, \xi)$ has the same property for $\varphi_\prec(F_+)$, so that the union of its translates defines a proper invariant open subset, contradicting minimality. Indeed, suppose that $g\in F_+$ and $K_1\in B_\alpha(K,x_0)$ are such that $g(K_1)\in B_\alpha(K,x_0)$, namely we assume
\[K_1\cap [x_0,1)=g(K_1)\cap[x_0,1)=K \cap [x_0, 1).\]
The key observation is that this implies that $g$ must actually fix $K\cap [x_0, 1)$. First of all observe that $g$ must send points of rank $\omega$ in $K_1$ to points of rank $\omega$ in $g(K_1)$, and the set of such points in both $K_1\cap [x_0,1)$ and $g(K_1)\cap [x_0, 1)$ consists precisely of the sequence $x_n:=f^n(x_0)$ for $n\ge 0$. This is a discrete increasing sequence and $g(x_n)=x_n$ for $n$ large enough, hence we deduce from the condition $g\in F_+$ that $g(x_n)=x_n$ for every $n\ge 0$. As a consequence the cyclic subgroup $\langle g\rangle$ must preserve every interval $[x_n, x_{n+1}]$, with $n\ge 0$, and thus every intersection $K\cap [x_n, x_{n+1}]$ for $n \ge 0$.
Assume by contradiction that there exist $n\ge 0$ and a point $t\in K\cap [x_n,x_{n+1}]$ which is not fixed by $g$, and consider the orbit $\Omega=\{g^m(t)\colon m\in \mathbb Z\}$, which is a subset of $K$. As $K\cap [x_n,x_{n+1}]$ is compact, the point $\inf \Omega$ is in $K\cap [x_n,x_{n+1}]$, and it is accumulated by points of $\Omega$ (and hence of $K$) from the right. This is in contradiction with the choice of $K$, as by construction every point of $K$ is isolated from the right hand side. Hence $g$ fixes $K\cap [x_0, 1)$, which implies that $g.B_\alpha(K,x_0)=B_\alpha(K,x_0)$. \qedhere
\end{proof}
\subsection{A plethora of $\mathbb{R}$-focal actions III: existence of non-simplicial $\mathbb{R}$-focal actions} \label{s-F-hyperexotic}
All the example of $\mathbb{R}$-focal actions of $F$ discussed so far are simplicial in the sense of Definition \ref{d-F-simplicial}, so that it is tempting to try to prove that all $\mathbb{R}$-focal actions of $F$ must be simplicial. However, we build an exotic action of Thompson's group $F$ which fails to have this property. With Proposition \ref{prop.hyperequiv} in mind, we will prove the following.
\begin{thm} \label{t-F-hyperexotic}
There exist faithful $\mathbb{R}$-focal actions $\varphi \colon F\to \homeo_0(\mathbb{R})$ such that $\varphi([F, F])$ acts minimally (and thus are not simplicial). More precisely, there exist uncountably many such actions which are pairwise non-conjugate, and whose restriction to $[F, F]$ yield pairwise non-conjugate actions of $[F, F]$.
\end{thm}
Note the following consequence of independent interest.
\begin{cor}
The group $[F, F]$ admits uncountably many, pairwise non-conjugate minimal actions $\varphi\colon [F, F]\to \homeo_0(\mathbb{R})$.
\end{cor}
This should be compared with the general constructions of exotic actions of groups of compactly supported homeomorphisms described in \S \ref{ss.exoticactions}, which provide actions without any closed minimal invariant subset. The construction given here relies on the classical symbolic coding of the the standard action of $F$ by binary sequences, which is specific to Thompson's groups.
For the proof it will be convenient to see $F$ as a group of homeomorphisms of $X=\mathbb{R}$ rather than of the interval $(0,1)$. Namely we realize $F\subseteq \homeo_0(\mathbb{R})$ as the group of piecewise linear maps of the line, with dyadic breakpoints, slopes in the group $\langle 2^n:n\in\mathbb Z\rangle$ and which coincide with integer translations near $\pm\infty$. It is well-known that this action is conjugate to the natural action of $F$ on $(0,1)$ (see e.g.\ \cite[Lemma E18.4]{BieriStrebel}).
\emph{From now and until the end of this subsection, the term \emph{standard action} will refer to the action of $F$ on $\mathbb{R}$ described above. We will denote by $f\in F$ the translation $f(x)=x+1$. } (Note that the element $f$ corresponds to the element given by \eqref{e-F-big-generator} in the action on $(0, 1)$).
The proof of Theorem \ref{t-F-hyperexotic} employs the strategy described in \S \ref{s-F-orbit-strategy}, namely we will start with a closed $f$-invariant subset $K\subset \mathbb{R}$, and define an invariant order on its orbit $\mathcal{O}_K:=\{g(K)\colon g\in F\}$. The main difficulty is that we need to construct a subset $K$ satisfying a somewhat delicate combination of properties. We begin with a definition.
\begin{dfn} We say that a subset $K\subset \mathbb{R}$ has property $(O)$ if it is proper, non-empty, closed, $f$-invariant, and moreover $g(K)\cap K$ is open in $K$ for every $g\in F$.
\end{dfn}
\begin{rem}
Note that the last condition for property $(O)$ is actually equivalent to that $K_1\cap K_2$ be open in $K_1$ and $K_2$ for every $K_1, K_2\in\mathcal{O}_K$.
\end{rem}
\begin{ex}
Property $(O)$ is clearly satisfied when $K$ is a non-empty $f$-invariant discrete subset, as for example the $f$-orbit of a point. However, this is not a good example for the construction described in this subsection, as the stabilizer of such $K$ in $[F,F]$ is trivial (cf.\ Proposition \ref{p-F-hyperexotic}).
\end{ex}
Assume that $K$ is a subset with property $(O)$ (many examples are exhibited by Lemma \ref{lem.propertyO}).
We consider the $F$-equivariant ultrametric kernel $\alpha:\mathcal{O}_K\times \mathcal{O}_K\to \mathbb{R}\cup \{-\infty\}$ defined as in \S \ref{s-F-orbit-strategy}, namely \[\alpha(K_1,K_2):=\inf\left \{x\in\mathbb{R}:[x,+\infty)\cap K_1=[x,+\infty)\cap K_2\right \}.\]
Reasoning as in the example of \S \ref{subsub.concrete}, we proceed to construct an $F$-invariant order on $\mathcal{O}_K$ and then prove that the dynamical realization of the action of $F$ on $(\mathcal{O}_K,\prec)$ is minimal and $\mathbb{R}$-focal.
This is the content of the next result.
\begin{prop}\label{prop O implica orden} If a subset $K\subset \mathbb{R}$ has property $(O)$, the relation $\prec$ on $\mathcal{O}_K$ defined by $K_1\prec K_2$ if and only if
\[
\max\left \{x\in K_1\colon x<\alpha(K_1,K_2)\right \}<\max\left \{x\in K_2\colon x<\alpha(K_1,K_2)\right \},\]
is an $F$-invariant total order on $\mathcal{O}_K$. Moreover, the dynamical realization $\varphi_K:F\to \homeo_{0}(\mathbb{R})$ of the action of $F$ on $(\mathcal{O},\prec)$ is a minimal $\mathbb{R}$-focal action.
\end{prop}
\begin{proof} Recall that $\alpha(K_1,K_2)\in K_1\cap K_2$ whenever $K_1$ and $ K_2$ are different elements of $ \mathcal{O}_K$. As $K$ satisfies property $(O)$, we have that $K_1\cap K_2$ is open inside both $K_1$ and $K_2$ and hence $K_1\cap K_2$ is an open neighborhood of $\alpha(K_1,K_2)$ inside $K_1$ and $K_2$. Therefore $\alpha(K_1,K_2)$ is isolated from the left hand side in both $K_1$ and $K_2$.
As for \eqref{eq:Kisolated}, we deduce that the points
\[
x_i:=\max\{x\in K_i\colon x<\alpha(K_1,K_2)\}\quad\text{for }i\in \{1,2\}
\]
are distinct, so we can declare
$K_1\prec K_2$ if and only if $x_1<x_2$.
As for the order $\prec^*$ from \S\ref{subsub.concrete}, it is routine to check that this defines indeed an $F$-invariant total order on $\mathcal{O}_K$.
Similarly one proceeds as in \S \ref{subsub.concrete} to check that $\varphi_K$ is minimal and $\mathbb{R}$-focal. Namely, one verifies that the order $\prec$ is $\alpha$-convex, and the proof of Lemma \ref{lem.Kisconvex} can be adapted \textit{verbatim} to this case (just replacing $\prec^\ast$ with $\prec$ and $(0,1)$ with $\mathbb{R}$). Then Corollary \ref{cor.ultra} gives the desired conclusion.
\end{proof}
The main difference from the construction in \S \ref{subsub.concrete} is the way that the commutator subgroup $[F,F]$ acts in the actions $\varphi_K$.
\begin{prop} \label{p-F-hyperexotic} Given a subset $K\subset (0,1)$ with property $(O)$, let $\varphi_K:F\to \homeo_{0}(\mathbb{R})$ be the corresponding minimal $\mathbb{R}$-focal action from Proposition \ref{prop O implica orden}. Then the following hold.
\begin{enumerate}[label=(\roman*)]
\item \label{i-F-O-minimal} If $K$ has property $(O)$, then $\varphi_K([F, F])$ acts minimally provided that the stabilizer of $K$ in $[F, F]$ (with respect to the standard action) acts on $K$ without fixed points. Moreover in this case the induced action of $[F,F]$ is minimal and $\mathbb{R}$-focal.
\item \label{i-F-O-distinct} If two distinct subsets $K,K'\subset \mathbb{R}$ have property $(O)$, then the restrictions of $\varphi_K$ and $\varphi_{K'}$ to $[F, F]$ are not conjugate actions of $[F, F]$. In particular $\varphi_K$ and $\varphi_{K'}$ are not conjugate.
\end{enumerate}
\end{prop}
\begin{proof}
To prove \ref{i-F-O-minimal}, assume that the stabilizer of $K$ in $[F, F]$ acts without fixed points on $K$.
Fix $x\in (0,1)$ and choose a sequence of elements of $[F, F]$ which preserve $K$ and such that $g_n(x)$ tends to $+\infty$. Then $g_n.B_\alpha(K,x)=B_\alpha(K,g_n(x))$, so that $\mathcal{O}_K=\bigcup_{n \ge 0}g_n.B_\alpha(K,x)$. Then by Proposition \ref{p-focal-semiconj}, the subgroup $[F,F]$ admits a unique minimal invariant set $\Lambda\subset \mathbb{R}$, which is preserved by $F$ because $[F,F]$ is a normal subgroup. We deduce that the action of $[F,F]$ is also minimal.
To prove \ref{i-F-O-distinct} take $K\neq K'$ with property $(O)$ and assume without loss of generality that $K'\not\subset K$.
Write $\alpha:\mathcal{O}_K\times \mathcal{O}_K\to \mathbb{R}\cup\{-\infty\}$ and $\beta:\mathcal{O}_{K'}\times \mathcal{O}_{K'}\to \mathbb{R}\cup\{-\infty\}$
for the corresponding ultrametric kernels. Fix $x\in K$ and let $D$ be the subgroup of $[F,F]_{(x,+\infty)}$ which fixes $K$ pointwise. Then for every $g\in D$ we have $g.B_\alpha(K,x)=B_\alpha(K,x)$ and actually $g$ fixes $B_\alpha(K,x)$ pointwise, so that the dynamical realization $\varphi_K$ fixes a non-empty open interval pointwise. On the contrary, for any $y\ge x$ with $y\in K'\setminus K$, we can consider an element $h\in D$ such that $h(y)\notin K'$ and $h(y)>y$.
Let us show that for such choices we have $h.B_\beta(K',y)\cap B_\beta(K',y)=\varnothing$. Indeed, assume there exists $L\in h.B_\beta(K',y)\cap B_\beta(K',y)$; then, as $h.B_\beta(K',y)=B_\beta(h(K'),h(y))$, we have
\[L\cap [h(y),+\infty)=h(K')\cap [h(y),+\infty)\]
and in particular $h(y)\in L$. However, if $L\in B(K',y)$, then $L\cap [y,+\infty)=K'\cap [y,+\infty)$
and thus $h(y)\notin L$, which is an absurd.
By $f$-invariance of $K'\setminus K$, we can find arbitrarily large points $y$, and thus elements $h\in D$, satisfying such properties. As $\mathcal{O}_{K'}=\bigcup_{y}B_\beta(K',y)$, this implies that $D$ acts without fixed points, so that the actions $\varphi_K$ and $\varphi_{K'}$ cannot be conjugate.
\end{proof}
After the previous proposition, in order to prove Theorem \ref{t-F-hyperexotic} we need to show the existence of subsets $K\subset \mathbb{R}$ with property $(O)$ and with the additional property that the stabilizer of $K$ in $[F, F]$ does not have fixed points. For this, we are going to use the symbolic description of real numbers by binary expansions.
To each infinite sequence $(a_n)_{n\geq 1}\in \{\mathtt{0},\mathtt{1}\}^{\mathbb{N}}$ we associate the real number $\ev((a_n)):=\sum_{n\geq 1}a_n2^{-n}\in[0,1]$. Note that this association is continuous if we endow $\{\mathtt{0},\mathtt{1}\}^\mathbb N$ with the product topology. If $z, w_1,w_2$ are finite binary sequences (\emph{binary words} for short), we consider the \emph{cylinder} over $z$, defined by \[C_z=\{w\in\{\mathtt{0},\mathtt{1}\}^{\mathbb{N}}:z\text{ is a prefix of }w\},\]
and denote by $\tilde{K}_0(w_1,w_2)\subseteq\{\mathtt{0},\mathtt{1}\}^{\mathbb{N}}$ the subset of all infinite concatenations of $w_1$'s and $w_2$'s. Clearly, both images $\ev(C_z)$ and $K_0(w_1, w_2):=\ev(\tilde{K}_0(w_1,w_2))$ are closed subsets of $[0,1]$, and the former is a closed interval with dyadic endpoints (a \emph{dyadic interval} for short). Note that, conversely, any closed dyadic interval is the union of (the real numbers represented by) finitely many cylinders.
With this in mind, if $z_1$ and $z_2$ are binary words, the \emph{substitution map} $S(z_1,z_2):C_{z_1}\to C_{z_2}$ defined by $S(z_1,z_2)(z_1w)=z_2w$ represents an affine map $\overline{S(z_1,z_2)} $ between
dyadic intervals of $[0,1]$. Therefore, in the action of $F$ on $\mathbb{R}$, every element of $F$ locally coincides (except at breakpoints, which are finitely many dyadic rationals) with transformations of the form $f^n\circ \overline{S(z_1,z_2)}\circ f^m$, for some powers $n,m\in\mathbb Z$ and some finite sequences $z_1,z_2$.
We say that a pair of binary words $w_1,w_2$ has the \emph{cancellation property} if whenever $zw=w'$ for $w,w'\in \tilde{K}_0(w_1,w_2)$, it holds that $z$ is a finite concatenation of $w_1$'s and $w_2$'s. As a concrete example of a pair of words with the cancellation property, we may take $w_1=\mathtt{0}$ and $w_2=\mathtt{1}$ but these are \emph{constant} binary words (i.e.\ made of a single repeated bit). As a concrete example of non-constant binary words with the cancellation property we can take $w_1=\mathtt{10001}$ and $w_2=\mathtt{01110}$.
\begin{lem}\label{lem.propertyO} Let $w_1$ and $w_2$ be non-constant binary words satisfying the cancellation property and write $K_0:=K_0(w_1,w_2)$. Then, the subset $K:=\bigcup_{n\in\mathbb Z}f^n(K_0) $ has property $(O)$.
\end{lem}
\begin{proof} Since $K_0$ is a closed subset of $[0,1]$, the subset $K$ is a closed and $f$-invariant subset of $\mathbb{R}$. Also, since $w_1$ and $w_2$ are non-constant, the set $\tilde{K}_0(w_1,w_2)$ has no eventually constant sequences, and so the subset $K_0$ contains no dyadic points. It follows that $\ev\colon \tilde{K}_0(w_1, w_2)\to K_0$ is a homeomorphism onto its image, and that the set of intersections of the form $(p/2^n,(p+1)/2^n)\cap K$ with $p\in \mathbb Z$ and $n\in \mathbb N$, forms a basis of its topology. The restriction of every element of $F$ to $K$ is locally given by maps of the form $f^n\circ\overline{ S(z_1,z_2)}\circ f^m$, and since $K$ is $f$-invariant, in order to check property $(O)$ it is enough to check that $\overline{S(z_1,z_2)}(K_0\cap \ev(C_{z_1}))$ is open in $K_0$ for every pair of finite binary words $z_1,z_2$.
For this, consider two binary finite words $z_1,z_2$ and also $w\in \tilde{K}_0(w_1,w_2)\cap C_{z_1}$ so that $S(z_1,z_2)(w)\in \tilde{K}_0(w_1,w_2)$. We need to check that $S(z_1,z_2)(\tilde{K}_0(w_1,w_2)\cap C_{z_1})$ contains a neighborhood of $S(z_1,z_2)(w)$. Since the pair $w_1,w_2$ has the cancellation property, we can write $w=z_1'w'$ with $z_1'=z_1z_1''$ and $w'\in \tilde{K}_0(w_1,w_2)$. Since $S(z_1,z_2)(w)$ equals $z_2z_1''w'$ and belongs to $\tilde{K}_0(w_1,w_2)$, again by the cancellation property we conclude that $z_2z_1''$ is a finite concatenation of $w_1$'s and $w_2$'s. Therefore
\[S(z_1,z_2)\left (C_{z_1'}\cap \tilde{K}_0(w_1,w_2)\right )=C_{z_2z_1''}\cap \tilde{K}_0(w_1,w_2),\] showing that $S(z_1,z_2)(\tilde{K}_0(w_1,w_2))$ contains a neighborhood of $S(z_1,z_2)(w)$ inside $\tilde{K}_0(w_1,w_2)$. This concludes the proof.
\end{proof}
\vspace{0,3cm}
In order to ensure that the stabilizer of $K$ in $[F, F]$ has no fixed points, we need to to impose one last extra condition on $K$.
Say that a map $h:I\to J$ is a \emph{dyadic affine map} between intervals if $I$ is a dyadic interval and $h$ is of the form $x\mapsto ax+b$, where $ a\in \{2^n\colon n\in {\mathbb{Z}}\}$, and $b\in\mathbb Z[1/2]$. Consider now a compact subset $K_0\subseteq (0,1)$. We say that $K_0$ admits a \emph{self similar decomposition} if there exists a pair of dyadic affine maps $h_1,h_2:I\to (0,1)$ such that
\begin{itemize}
\item $h_1(I)\cap h_2(I)=\emptyset$,
\item $h_1(K_0)\cup h_2(K_0)=K_0$.
\end{itemize}
For example, the subset $K_0(w_1, w_2)$ admits a self similar decomposition provided the words $w_1$ and $w_2$ are such that $\ev(C_{w_1})$ and $\ev(C_{w_2})$ are disjoint. Indeed in this case we have that for $i\in \{1,2\}$, the symbolic maps $w\mapsto w_iw$ correspond to dyadic affine maps $h_i:[0,1]\to [0,1]$ with disjoint images and such that $K_0=h_1(K_0)\sqcup h_2(K_0)$.
\begin{lem}\label{lem.horseshoe}
Let $K_0\subset (0,1)$ be a closed subset admitting a self similar decomposition, and let $K=\bigcup_{n\in\mathbb Z}f^n(K_0)$. Then, the action of $H=\{g\in [F, F] \colon g(K)=K\}$ on $K$ has no fixed points.
\end{lem}
To show Lemma \ref{lem.horseshoe} we use its self-similarity to build elements in $H$ moving points of the real line arbitrarily far away. But before giving the formal proof, let us see how this ends the proof of Theorem \ref{t-F-hyperexotic}.
\begin{proof}[Proof of Theorem \ref{t-F-hyperexotic} given Lemma \ref{lem.horseshoe}]
Let $K\subsetneq \mathbb{R}$ be a subset satisfying property $(O)$ and such that $K\cap (0,1)$ admits a self similar decomposition. As concrete example we may take $K=\bigcup_{n\in \mathbb Z} f^n(K_0) $ with $K_0=K_0(w_1, w_2)$ for $w_1=\mathtt{10001}$ and $w_2=\mathtt{01110}$. By Lemma \ref{lem.propertyO} we may consider the $\mathbb{R}$-focal action $\varphi_K$, and by Lemma \ref{lem.horseshoe} and Proposition \ref{p-F-hyperexotic} we have that $\varphi_K([F, F])$ acts minimally. To finish the proof, we show that from the existence of one subset $K$ with these properties, we can deduce the existence of uncountably many.
Let $K$ be one such subset. Clearly $K$ is locally a Cantor set, so $\mathbb{R}\setminus K$ is a countable union of open intervals, that we call the \emph{gaps} of $K$. Pick $\beta\in (0,1)$. For each gap $I$ of $K$, consider the point $p_{I}(\beta)$ where $p_I:(0,1)\to I$ is the unique order-preserving affine map. We let $K^\beta$ be the subset resulting from adding to $K$ all the points of the form $ p_I(\beta)$ where $I$ runs over gaps of $K$. Clearly $K^\beta$ is still closed and $f$-invariant. Moreover, $K^\beta$ still admits a self similar decomposition since the maps involved in the definition of self similar decomposition are affine maps sending gaps of $K$ to gaps of $K$, so in particular they preserve the proportion of the subdivision we have introduced in the gaps. We claim, that for uncountably many $\beta\in (0,1)$, the subset $K^\beta$ also satisfies (the last condition of) property $(O)$, that is $g.K^\beta \cap K^\beta$ is open in $K^\beta$ for every $g\in F$.
Fix $g\in F$. The only problem that may arise is that a point of the form $p_I(\beta)$ (which is an isolated point) might land inside $K$ under the action of $g$. But if we fix a gap $I$ of $K$, the set of parameters $\beta$ such that $g(p_I(\beta))$ does not belong to $K$ is open and dense in $(0,1)$ since $K$ has empty interior. In particular, since there are only countably many gaps and $F$ is also countable, with a Baire argument we obtain that $K^\beta$ satisfies property $(O)$ for a generic choice of $\beta\in (0,1)$.
\end{proof}
We conclude this subsection with the proof of Lemma \ref{lem.horseshoe}. For this we need the following elementary interpolation lemma. Its proof follows from the transitivity of the action of $[F, F]$ on unordered $n$-tuples of dyadic numbers (see for instance \cite{BieriStrebel}), and details are left to reader. To simplify the statement, given (possibly unbounded) intervals $I,J$, we write $I<J$ whenever $\sup I<\inf J$.
\begin{lem}\label{sublem.gluing} Consider intervals $I_1<I_2<\cdots<I_k$ and $J_1<J_2<\cdots<J_k$ with dyadic endpoints such that $I_1=J_1=(-\infty,p]$, $I_k=J_k=[q,+\infty)$,
and such that $h_n:I_n\to J_n$ are dyadic affine maps for $n\in \{1,\ldots,k\}$.
Assume moreover that $h_1$ and $h_k$ are restrictions of the identity.
Then, there exists $g\in [F, F]$ such that $g\restriction_{I_n}=h_n$ for $n\in \{1,\ldots,k\}$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem.horseshoe}] Consider the dyadic affine maps $h_1,h_2:I_0\to(0,1)$ given by the self similar decomposition of $K_0$. Since $K_0$ is a closed subset of $(0,1)$, we can assume that $I_0$ is a closed dyadic interval inside $(0,1)$. For $i\in\{1,2\}$, write $K_0^i=h_i(K_0)$ and $I_0^i=h_i(I_0)$. Note that $I_0^1\cap I_0^2=\varnothing$ and $K_0=K_0^1\sqcup K_0^2$.
Now, for $n\in\mathbb Z$ and $i\in \{1, 2\}$, write $I_n=f^n(I_0)$, $K_n^i=f^n(K_0^i)$, and $I_n^i=f^n(I_0^i)$. Then consider the following locally dyadic affine maps.
\begin{itemize}
\item $a:I_0^1\sqcup I_0^2\to I_0\sqcup I_1$ defined by
\[a(x)=\left\{\begin{array}{lc} h_1^{-1}(x) & \text{if } x\in I_0^1, \\ f\circ h_2^{-1}(x) & \text{if } x\in I_0^2, \end{array} \right. \]
\item $b:I_{3}\sqcup I_4\to I_4^{1}\sqcup I_4^2$ defined by \[b(x)=\left\{\begin{array}{lc} f^4\circ h_1\circ f^{-3}(x) & \text{if } x\in I_{3}, \\
f^4\circ h_2\circ f^{-4} & \text{if }x\in I_{4}, \end{array} \right. \]
\item $c:[1,2]\to[2,3]$ defined by $c(x)=x+1$.
\end{itemize}
Then, we can apply Lemma \ref{sublem.gluing} to construct $h\in [F, F]$ which simultaneously extends $a, b, c$ and $\mathsf{id}\restriction_{(-\infty,0]\cup[4,+\infty)}$. By construction $h$ preserves $K$ and has no fixed points in $[1,2]$. Thus the subgroup $H=\{h\in [F, F] \colon h(K)=K\}$ has no fixed points inside $[1, 2]$. Finally, note that $f$ normalizes $H$, so that it preserves its set of fixed points. Since $\bigcup_{n\in \mathbb Z} f^n([1, 2])=\mathbb{R}$, we deduce that $H$ has no fixed points on $\mathbb{R}$, whence on $K$. \qedhere \end{proof}
\section{Uncountable groups}\label{s-uncountable}
Locally moving groups also contain several natural ``large'' groups, such as the group $\homeo_0(\mathbb{R})$ or $\Diff^r_0(\mathbb{R})$ or the subgroups $\homeo_c(\mathbb{R})$ and $\Diff_c^r(\mathbb{R})$ of compactly supported elements. Actions of such groups on the line are well understood thanks to work of Militon \cite{Militon}, and of the recent work of Chen and Mann \cite{ChenMann}. In fact, such results fit in a program started by Ghys \cite{MR1115743}, asking when the group of all diffeomorphisms (or homeomorphisms) of a manifold may act on another manifold; in the recent years, very satisfactory results have been obtained, and we refer to the survey of Mann \cite{MannSurvey} for an overview.
In this section we provide a rigidity criterion for locally moving groups whose standard action has uncountable orbits and satisfies an additional condition. We then explain how this criterion recovers some of the results in \cite{Militon,ChenMann}, and unifies them with the setting of the other results of this paper.
Recall from the introduction (Definition \ref{d-Schreier}), that for a a group $G$ and subgroup $H\subset G$, we say that the pair $(G, H)$ has \emph{relative Schreier property} if every countable subset of $H$ is contained in a finitely generated subgroup of $G$.
We have the following result, which is a more general version of Theorem \ref{t-intro-uncountable} from the introduction.
\begin{thm}\label{t-uncountable}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be locally moving, such that for every open subinterval $I\subset X$, all $G_I$-orbits in $I$ are uncountable. Suppose that for every subinterval $I\Subset X$, the pairs $(G_+, [G_I, G_I])$ and $(G_-, [G_I, G_I])$ have the relative Schreier property.
Then every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ without fixed points is either topologically conjugate to the standard action of $G$ on $X$, or semi-conjugate to an action that factors through $G/[G_c, G_c]$.
\end{thm}
\begin{rem}
Note that if the pair $(G_c, [G_I, G_I])$ has the relative Schreier property, then so do the pairs $(G_+, [G_I, G_I])$ and $(G_-, [G_I, G_I])$, since $G_c$ is contained in $G_+$ and $G_-$.
\end{rem}
For the proof we need the following lemmas.
\begin{lem}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be a subgroup such that every $x\in X$ has an uncountable $G$-orbit. Then every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ which is semi-conjugate to the standard action on $X$ is topologically conjugate to it.
\end{lem}
\begin{proof}
Note that if orbits are uncountable, then the action is minimal. Indeed if by contradiction $\Lambda \subset X$ is a closed invariant subset, then the set $\partial \Lambda$ is countable and $G$-invariant, thus $\partial \Lambda=\varnothing$, hence either $\Lambda=\varnothing$ or $\Lambda=X$.
Assume that $\varphi\colon G\to \homeo_0(\mathbb{R})$ is semi-conjugate to the standard action on $X$, by a monotone equivariant map $q\colon \mathbb{R}\to X$. As the action on $X$ is minimal, the semi-conjugacy $q$ is continuous. If it is not injective, there exists points $x\in X$ for which $q^{-1}(x)$ is a non-trivial interval. But the set of such points is $G$-invariant and at most countable, which is a contradiction. \qedhere \end{proof}
\begin{lem}\label{l-fix-countable}
Let $G$ be a group of homeomorphisms of a second countable Hausdorff space. Then there exists a countable subgroup $G_0\subseteq G$ such that $\operatorname{\mathsf{Fix}}(G)=\operatorname{\mathsf{Fix}}(G_0)$.
\end{lem}
\begin{proof}
The statement is non-empty only when $G$ is uncountable, and we will assume so.
Let $\mathcal{U}$ be a countable basis of open subsets of the space. For every $z\in \supp(G)$, we can find a neighborhood $U\in \mathcal{U}$ of $z$ and $g_U\in G$ such that $g_U(U)\cap U=\varnothing$. Thus we can cover $\supp(G)$ with countably many subsets with this property and the subgroup $G_0$ generated by the corresponding $g_U\in G$ is countable and satisfies the desired condition.
\end{proof}
\begin{proof}[Proof of Theorem \ref{t-uncountable}.]
Assume by contradiction that $\varphi \colon G \to \homeo_0(\mathbb{R})$ is an action which is not topologically conjugate to the action on $X$ and such that $\varphi([G_c, G_c])$ has no fixed point. As $G$ is locally moving, by Theorem \ref{p-lm-trichotomy} we can assume, say, that every finitely generated subgroup $\Gamma$ of $G_+$ is totally bounded.
\begin{claim}
For every subinterval $I\Subset X$, the subgroup $[G_I, G_I]$ is totally bounded.
\end{claim}
\begin{proof}[Proof of claim]
By Lemma \ref{l-fix-countable}, we can find a countable subgroup $H\subseteq [G_I, G_I]$ with $\operatorname{\mathsf{Fix}}^\varphi\left ([G_I, G_I]\right )=\operatorname{\mathsf{Fix}}^\varphi(H)$. By the relative Schreier property of $(G_+,[G_I,G_I])$, the subgroup $H$ is contained in a finitely generated subgroup of $G_+$, and thus is totally bounded. Hence $[G_I, G_I]$ is also totally bounded.
\end{proof}
Fix $I=(c,d)\Subset X$. Let $\mathcal{A}$ be the collection of connected components of $\suppphi\left ([G_I, G_I]\right )$ and fix $L\in \mathcal{A}$. For every $x\in (d, b)$ we have the inclusion $[G_I,G_I]\subseteq [G_{(c,x)},G_{(c,x)}]$, so that $L$ is contained in some connected component $L_x$ of $\suppphi\left ([G_{(c, x)}, G_{(c, x)}]\right )$.
Consider the function
\[
\dfcn{F_L}{(d,b)}{\mathbb{R}}{x}{\sup L_x},
\]
which is well-defined after the claim, monotone increasing, and which tends to $+\infty$ as $x$ tends to $b$. Moreover for every $x\in (d, b)$ the point $F_L(x)$ belongs to $\operatorname{\mathsf{Fix}}^\varphi\left ([G_{(c, x)}, G_{(c, x)}]\right )$, thus to $\operatorname{\mathsf{Fix}}^\varphi([G_I, G_I])$, and thus the function $F_L$ cannot be continuous at every point. The group $G_{(d, b)}$ centralizes $[G_I, G_I]$, so it permutes the intervals in $\mathcal{A}$, hence for every $g\in G_{(d, b)}$, the family of functions $\left \{F_L\colon L\in \mathcal{A}\right \}$ is equivariant in the following sense:
\[ g.F_L(x)= F_{g.L}(g(x)).\]
In particular, a point $x$ is a discontinuity point for $F_L$ if and only if $g(x)$ is a discontinuity points for $F_{g.L}$. But since all functions $F_L$ are monotone, and there are countably many of them, there are at most countably many points $x\in (d, b)$ which are discontinuity points of some $F_L$, for $L\in \mathcal{A}$. Thus the $G_{(d, b)}$-orbit of every such point must be countable, which is in contradiction with our assumption. \qedhere \end{proof}
Let us now give some examples of groups that satisfy the hypotheses of Theorem \ref{t-uncountable}. In several situations the relative Schreier property can be established using an embedding technique for countable groups due Neumann and Neumann \cite{NeumannNeumann}, based on unrestricted permutational wreath product. This method has been exploited by Le Roux and Mann \cite{LeRouxMann} to show that many homeomorphisms and diffeomorphisms groups of manifolds have the Schreier property. In order to run this method, it is enough that the group $G$ be closed under certain infinitary products, in the following sense.
\begin{dfn}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be a subgroup. Let $(I_n)_{n\in \mathbb N}$ be a collection of disjoint open subintervals of $X$. For every $n\in \mathbb N$, take an element $g_n\in G_{I_n}$. We denote by $\prod g_n$ the homeomorphism of $X$ defined by
\[ \prod g_n:x\mapsto \left\{\begin{array}{lr} g_n(x) & \text{if }x\in I_n, \\[.5em] x &\text{if }x\notin \bigcup_n I_n.\end{array}\right.\]
We say that the group $G$ is closed under \emph{monotone infinitary products} if for every monotone sequence $(I_n)_{n\in \mathbb N}$ of disjoint open subintervals (in the sense that the sequence $(\inf I_n)_{n\in \mathbb N}$ is monotone), and every choice of $g_n\in G_{I_n}$, we have $\prod g_n\in G$. We also say that $G$ is closed under \emph{discrete infinitary products} if the latter holds for every sequence of intervals $(I_n)_{n\in \mathbb N}$ such that $(\inf I_n)_{n\in \mathbb N}$ has no accumulation point in the interior of $X$.
\end{dfn}
The following lemma provides a criterion for the relative Schreier property in our setting.
\begin{lem}\label{l-Schreier-homeo}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be locally moving. Suppose that $G$ satisfies one of the following conditions:
\begin{enumerate}[label=\roman*)]
\item \label{i-monotone-infinitary} $G$ is closed under monotone infinitary products;
\item \label{i-discrete-infinitary} $G$ is closed under discrete infinitary products and contains elements acting without fixed points on a neighborhood of $a$ and trivially on a neighborhood of $b$, and viceversa.
\end{enumerate}
Then for every open subinterval $I\Subset X$ the pairs $(G_+, [G_I, G_I])$ and $(G_-, [G_I, G_I])$ have the relative Schreier property.
\end{lem}
\begin{proof}
Fix $I=(x, y)$ and let us show that the pair $(G_+, [G_I, G_I])$ has relative Schreier property (the other case is analogous). It is enough to show that for every sequence $(f_n)_{n\in \mathbb N}\subset G_I$, there exists a finitely generated subgroup $\Gamma \subset G_+$ such that $[f_n, f_m]\in \Gamma$ for all $n\neq m$. So let $(f_n)$ be such a sequence. Choose $t\in G_+$ such that $t(y)<x$, and if we are in case \ref{i-discrete-infinitary} suppose additionally that $t$ has no fixed point in $(a, y)$, so that $\lim_{n\to \infty}t^n(y)=a$. Also choose an increasing sequence of positive integers $(k_n)$ which is \emph{parallelogram-free}, that is, if $k_{n_1}-k_{n_2}=k_{m_1}-k_{m_2}\neq 0$, then $n_1=m_1$ and $n_2=m_2$ (for example the sequence $k_n=2^n$ has this property). Set $g_n=t^{k_n} f_n t^{-k_n}$ and note that $g_n\in G_{I_n}$ with $I_n:=t^{k_n}(I)$. By construction, in either case \ref{i-monotone-infinitary} or \ref{i-discrete-infinitary} the element $h:=\prod g_n$ belongs to $G$. It is not difficult to check that for every $n\in \mathbb N$ the element $t^{-k_n}ht^{k_n}$ coincides with $f_n$ on $I$, and the fact that $(k_n)$ is parallelogram-free implies that for $n \neq m$ the intersection of the supports of $t^{-k_m}ht^{k_m}$ and of $t^{-k_n}ht^{k_n}$ is contained in $I$. Thus we have $[t^{-k_n} h t^{k_n}, t^{-k_m} ht^{k_m}]=[h
_n, h_m]$. It follows that the finitely generated subgroup $\Gamma:=\langle h, t\rangle \subset G_+$ satisfies the desired conclusion.
\end{proof}
Combining this with Theorem \ref{t-uncountable}, one can show that various sufficiently ''large'' locally moving groups do not admit any exotic action at all. For example, we have the following criterion.
\begin{cor}\label{c-uncountable-complete}
For $X=(a,b)$, let $G\subseteq\homeo_c(X)$ be a locally moving, perfect subgroup of compactly supported homeomorphisms. Assume that for every open subinterval $I\subset X$, all $G_I$-orbits in $I$ are uncountable, and that $G$ is closed under monotone infinitary products. Then every action $\varphi\colon G\to \homeo_0(\mathbb{R})$ without fixed points is topologically conjugate to the standard action of $G$ on $X$.
\end{cor}
This criterion is clearly satisfied by the group $G=\homeo_c(\mathbb{R})$ of all compactly supported homeomorphisms of $\mathbb{R}$ (it is well-known that it is perfect, see for instance \cite[Proposition 5.11]{Ghys} for a very short proof). Thus, Corollary \ref{c-uncountable-complete} recovers the following result due to Militon \cite{Militon}.
\begin{cor}[Militon]
Every action $\varphi\colon \homeo_c(\mathbb{R})\to \homeo(\mathbb{R})$, without fixed points is topologically conjugate to the standard action.
\end{cor}
Let us now consider the group $G=\Diff_c^r(\mathbb{R})$ of compactly supported diffeomorphisms of $\mathbb{R}$ of class $C^r$, with $r\in [1, \infty]$. This case is more subtle since the group $G$ is not closed under monotone infinitary products: if $g_n\in G_{I_n}$, the element $\prod g_n$ need not be a diffeomorphism on a neighborhood of any accumulation point of the intervals $I_n$. A way to go around this is provided by the arguments of Le Roux and Mann in \cite[\S 3]{LeRouxMann}, where they show that the group $\Diff^r(M)$ has Schreier property for every closed manifold $M$ whenever $r\neq \dim (M)+1$. Note that this result does not apply to the group $\Diff_c^r(\mathbb{R})$, which in fact does not have Schreier property, since it can be written as countable strictly increasing union of subgroups (see \cite{LeRouxMann}). However, the same argument of their proof can be adapted to show the following.
\begin{prop} \label{p-Schreier-diff}
For every $r\in [1, \infty]\setminus \{2\}$ and interval $I\Subset \mathbb{R}$, the pair $(\Diff^r_c(\mathbb{R}), \Diff^r_c(I))$ has the relative Schreier property.
\end{prop}
The reason for the assumption $r \neq 2$ is that in this case the group $\Diff^r_c(\mathbb{R})$ is known to be simple, by famous results of Thurston \cite{Thurston} (for $r=\infty$) and Mather \cite{Mather} (for $r\neq 2$ finite). Whether this holds for $r=2$ remains an open question.
\begin{proof}[Proof of Proposition \ref{p-Schreier-diff}]
We outline the steps, and refer to \cite[\S 3]{LeRouxMann} for details. First of all, observe that in order to prove the proposition it is enough to show find a generating set $S$ of $\Diff^r_c(I)$ with the property that every sequence $(b_n)$ of elements of $S$ is contained in a finitely generated subgroup of $\Diff^r_c(\mathbb{R})$ (see Lemma 3.6 in \cite{LeRouxMann}).
Set $I=(x, y)$. Following the same strategy as in the proof of Proposition \ref{l-Schreier-homeo}, let $t\in \Diff^r_c(\mathbb{R})$ be such that $t(y)<x$, and choose a parallelogram-free increasing sequence $(k_n)\subset \mathbb Z_+$, so that the intervals $I_n:=t^{k_n}(I)$ are pairwise disjoint. The main difference with the proof of Proposition \ref{l-Schreier-homeo} is that if we choose $g_n\in \Diff^r_c(I_n)$ arbitrarily, the element $\prod g_n$ does not necessarily belong to $\Diff^r_c(\mathbb{R})$. However, if the elements $(g_n)$ are such that their $C^r$ norms satisfy $\left\lVert g_n \right\rVert_r \le 2^{-n}$ then the sequence of truncated products $\prod_{n=1}^m g_n$ is a Cauchy sequence, so that the infinite product $\prod_{n=1}^\infty g_n$ belongs to $\Diff^r_c(\mathbb{R})$. Since conjugation by $t^{k_n}$ is continuous in the $C^r$ topology, this implies that there exists a sequence $(\varepsilon_n)$ such that whenever the elements $f_n\in \Diff^r_c(I)$ are such that $\left\lVert f_n\right\rVert_r\le \varepsilon_n$, then the product $\prod (t^{k_n} f_n t^{-k_n})$ is indeed in $\Diff^r_c(\mathbb{R})$ (compare \cite[Lemma 3.7]{LeRouxMann}).
With these preliminary observations in mind, the key idea of \cite[\S 3]{LeRouxMann} is to consider a well-chosen generating set of $\Diff^r_c(I)$, consisting of elements belonging to suitable copies of the {affine group}. By an affine group inside $\Diff_c^r(I)$ we mean a subgroup generated by two one-parameter subgroups $\left \{a^t\right \}_{t\in \mathbb{R}}$ and $\left \{b^t\right \}_{t\in \mathbb{R}}$ of $\Diff^r_c(\mathbb{R})$, varying continuously in the $C^r$ topology, which satisfy the relations $a^s b^t a^{-s}=b^{e^st}$. Existence of affine subgroups in $\Diff_c^r(I)$ can be obtained by applying the trick of Muller and Tsuboi described in Section \ref{s-differentiable} to the two vector fields generating the affine group (see \cite[Lemma 3.3]{LeRouxMann}). Let now $S \subset \Diff_c^r(I)$ be the set of all time-one maps $b:=b^1$ of a flow $b^t$ belonging to an affine subgroup. Since the set $S$ is non-empty and stable under conjugation in $\Diff_c^r(I)$, it is a generating set by simplicity of $\Diff_c^r(I)$. Now let $(b_n)\subset S$ be a sequence, where each $b_n=b_n^1$ belongs to an affine subgroup $\langle a_n^t, b_n^t \rangle$. The relations in the affine subgroups imply that for every $t, s\in \mathbb{R}$ we have $[a_n^s, b_n^t]=b_n^{(e^s-1)t}$. This equality implies that for $\delta>0$ small enough, the element $b_n^\delta$ can be written as a commutator of elements with arbitrarily small $C^r$ norm (see \cite[Corollary 3.4]{LeRouxMann}). Thus, if for every $n\in \mathbb N$ we choose a sufficiently large positive integer $\ell_n>0$, we have $b_n^{1/\ell_n}=[f_{2n}, f_{2n+1}]$ for some sequence $(f_n)\subset \Diff^r_c(I)$ such that $\left\lVert f_n\right\rVert_r\le \varepsilon_n$. By the choice of the sequence $(\varepsilon_n)$ made above, the element $h:=\prod_{n=1}^\infty t^{k_n} f_n t^{-k_n}$ is in $\Diff^r_c(\mathbb{R})$. The same argument in the proof of Lemma \ref{l-Schreier-homeo} then implies that $[t^{-k_2n} ht^{k_2n}, t^{-k_{2n+1}} ht^{k_{2n+1}}]=[f_{2n}, f_{2n+1}]=b_n^{1/{\ell_n}}$, so that the subgroup $\Gamma=\langle h, t\rangle$ contains the sequence $(b_n)$. By the remark made at the beginning of the proof, this proves the proposition. \qedhere
\end{proof}
Combined with Proposition \ref{p-Schreier-diff}, Theorem \ref{t-uncountable} provides an alternative proof of the following recent result of Chen and Mann \cite{ChenMann}.
\begin{cor}[Chen--Mann]
For $r\in [1,\infty]\setminus \{2\}$, every action $\varphi\colon \Diff^r_c(\mathbb{R}) \to \homeo_0(\mathbb{R})$ without fixed points is topologically conjugate to its standard action.
\end{cor}
\section{Some minimal $\mathbb{R}$-focal actions}
\label{sec.examplesRfocal}
The goal of this section is to present some general constructions of exotic actions on $\mathbb{R}$. The first three examples describe $\mathbb{R}$-focal actions of locally moving groups. As in Section~\ref{sec.locallymgeneral}, given $X=(a, b)$ and a locally moving subgroup $G\subseteq \homeo_0(X)$, we say that an action $\varphi\colon G\to \homeo_0(\mathbb{R})$ is exotic if it is not semi-conjugate to the standard action on $X$, nor to any action of the largest quotient $G/[G_c, G_c]$. Some basic examples of exotic actions have already been provided in \S \ref{ss.exoticactions}; those examples are not satisfactory because they yield actions which are not minimal (and that do not admit any non-empty closed minimal invariant subset), and they essentially rely on the fact that the groups considered there are not finitely generated. On the contrary, here we will construct examples of exotic \emph{minimal} actions for some classes of locally moving groups, including finitely generated ones. Along the way we will observe that these constructions share the property to produce $\mathbb{R}$-focal actions.
Finally, the last example is a generalization of the Brin--Navas group defined in Example~\ref{e-BN}. The new examples, called \emph{generalized Brin--Navas groups}, serve to show that some results in this article concerning locally moving groups do not generalize to the general context of micro-supported groups (notably Corollary \ref{cor.unique} and Theorem \ref{t-lm-C1} fail).
\subsection{Locally moving groups with cyclic group of germs}\label{ssec.cyclicgerm}
Throughout the subsection we let $X=(a, b)$, and we let $G\subseteq\homeo_0(X)$ be a locally moving group such that $\Germ(G,b)$ is infinite cyclic and acts freely near $b$ (that is, for every $g\in G$ whose projection to $\Germ(G,b)$ is non-trivial, there is an interval $(x,b)$ on which $g$ has no fixed points). We will say for short that $G$ has \emph{cyclic germs} at $b$.
\begin{ex}One example of group with cyclic germs is Thompson's group $F\subseteq \homeo_0((0, 1))$, and more generally any Higman--Thompson's group $F_n$ (see $\S$\ref{sc.BieriStrebel}). A much wider class of examples with this property is given by chain groups in the sense of Kim, Koberda, and Lodha \cite{KKL}.\end{ex}
\subsubsection{Actions from escaping orbits} For $G\subseteq \homeo_0(X)$ with cyclic germs, we present a mechanism build a continuum of pairwise non-conjugate minimal exotic actions, which is a modification of the escaping sequence construction from \S \ref{ss.exoticactions}.
For this, we identify $\Germ(G, b)$ with $\mathbb Z$ in such a way that a germ for which $b$ is an attractive fixed point is sent to the positive generator of $\mathbb Z$. We denote by $\tau\colon G\to \mathbb Z$, the homomorphism obtained via this identification.
Then we fix an element $f_0\in G$ such that $\tau(f_0)=1$ (that is, the germ of $f_0$ generates $\Germ(G, b)$ and we have $f_0(x)>x$ near $b$). Choose next a bi-infinite sequence $\s=(s_n)_{n\in\mathbb Z}\subset X$ with
\begin{equation}\label{e-conditions-sequence} s_{n+1}=f_0(s_n)\text{ for }n\in\mathbb Z,\quad \text{and} .\end{equation}
Consider the action of the group $G$ on the set of sequences $(0, 1)^\mathbb Z$, where the action of $g\in G$ on a sequence $\mathsf{t}=(t_n)_{n\in \mathbb Z}$ is given by
\begin{equation}\label{e-action-sequences} g\cdot \mathsf{t}=\left (g(t_{n-\tau(g)} )\right )_{n\in \mathbb Z}.\end{equation}
It is straightforward to check that this defines an action of $G$ on $X^\mathbb Z$, using that $\tau$ is a homomorphism. We let $\mathsf{S}\subset X^\mathbb Z$ be the orbit of $\s$ under this action. Note that $\s$ is fixed by $f_0$.
\begin{lem}\label{l-cyclic-eventual}
With notation as above, for every sequence $\mathsf{t}=(t_n)_{n\in \mathbb Z}\in \mathsf{S}$, there exists $n_0\in \mathbb Z$ such that $t_n=s_n$ for every $n\ge n_0$.
\end{lem}
\begin{proof}
Let $g\in G$ be such that $\mathsf{t}=g\cdot \s$. Then $t_n=g (s_{n-\tau(g)} )$. As we required $\lim_{n\to+\infty} s_n=b$ in \eqref{e-conditions-sequence}, and $g$ coincides with $f_0^{\tau(g)}$ on a neighborhood of $b$, the conclusion follows.
\end{proof}
It follows from the lemma that for every two distinct sequences $\mathsf{t}=(t_n)$ and $\mathsf{t}'=(t'_n)$ in $\mathsf{S}$, the integer
\begin{equation}\label{eq.mts}
m(\mathsf{t}, \mathsf{t}')= \max\left \{n\in \mathbb Z\colon t_n\neq t'_n\right \}\end{equation}
is well-defined and finite.
Thus we can introduce the total order relation $\prec$ on $\mathsf{S}$, given by $\mathsf{t}\prec \mathsf{t}'$ if and only if $t_m<t_m'$, with $m=m(\mathsf{t}, \mathsf{t}')$
\begin{lem}\label{l-construction-cyclic-order}
With notation as above, the total order $\prec$ on $\mathsf S$ is preserved by the action of $G$ on $\mathsf{S}$ defined by \eqref{e-action-sequences}. Moreover, the element $f_0$ acts as a homothety on $(\mathsf{S}, \prec)$ (in the sense of Definition \ref{dfn.homotype}) with fixed point $\s$.
\end{lem}
\begin{proof}
It is routine verification that the order $\prec$ is $G$-invariant. Let us check that $f_0$ is a homothety. We have already noticed that the sequence $\s$ is a fixed point for $f_0$. Fix sequences $\mathsf{t},\mathsf{t}'\in \mathsf S$ such that
\begin{equation}\label{eq:choose_sequences}
\s\prec \mathsf t\prec \mathsf t'.
\end{equation}
We need to show that there exists $n\in \mathbb Z$ such that $f_0^n\cdot \mathsf t\succ \mathsf t'$ (in fact, we will find some $n\ge 1$, showing that $f_0$ acts as an \emph{expanding} homothety).
Write $m_0=m(\mathsf{t}, \s)$ and $m_1=m(\mathsf{t}', \s)$ and note that the condition \eqref{eq:choose_sequences} gives $m_0\le m_1$ and $t_{m_0}>s_{m_0}$. We claim that $n=m_1-m_0+1$ is fine for our purposes. For this, we compute directly:
\begin{align*}
\left (f_0^n\cdot \mathsf t\right )_{m_1+1}&=f_0^n(t_{m_1+1-n})=f_0^n(t_{m_0})\\
&>f_0^n(s_{m_0})=s_{m_0+n}=s_{m_1+1}=t'_{m_1+1},
\end{align*}
while for every $m> m_1+1=n+m_0$ we have
\[
\left (f_0^n\cdot \mathsf t\right )_{m}=f_0^n(t_{m-n})=f_0^n(s_{m-n})=s_m=t'_m.\]
Thus $m(\mathsf{t}'', \mathsf{t}')=m_0+1$ and $f_0^n\cdot \mathsf t\succ \mathsf{t}'$, as desired. Similarly one argues for $\mathsf t'\prec \mathsf t\prec \s$.\qedhere
\end{proof}
Assume now that $G$ is countable, so that the set $\mathsf{S}$ is countable as well. Then we can consider the dynamical realization $\varphi_{\s}\colon G\to \homeo_0(\mathbb{R})$ of the action of $G$ on $(\mathsf{S}, \prec)$.
\begin{prop} For $X=(a,b)$, let $G\subseteq \homeo_0(X)$ be a countable locally moving group with cyclic germs at $b$.
For every sequence $\s=(s_n)_{n\in \mathbb Z}$ as in \eqref{e-conditions-sequence}, the action $\varphi_{\s}\colon G\to \homeo_0(\mathbb{R})$ constructed above is minimal and faithful. Moreover if $\s'$ is another such sequence, whose image is different from that of $\s$ (that is, if they are not the same after a shift of indices), then $\varphi_{\s}$ and $\varphi_{\s'}$ are not conjugate. In particular $G$ has uncountably many minimal faithful non-conjugate actions on the real line. \end{prop}
\begin{proof}
The fact that $\varphi_{\s}$ is minimal follows from Lemma \ref{l-construction-cyclic-order} and Proposition \ref{p.minimalitycriteria} (the action on $\mathsf S$ is transitive, so it is enough to describe what happens at $\s$).
Let $\iota\colon (\mathsf{S}, \prec)\to \mathbb{R}$ be an equivariant good embedding associated with $\varphi_{\s}$. Since $f_0$ is a homothety on $(\mathsf{S}, \prec)$, it follows that its image of $\varphi_{\s}$ is a homothety of $\mathbb{R}$ whose unique fixed point is $\iota(\s)$. In particular the stabilizer of this point inside $G_+=\ker{\tau}$, which after \eqref{e-action-sequences} coincides with the stabilizer of $\s$ for the natural diagonal action of $G_+$ on $X^\mathbb Z$, is a well-defined invariant of the conjugacy class of $\varphi_{\s}$. Now note that if $\s$ and $\s'$ are sequences with distinct images, using that $G$ is locally moving it is not difficult to construct $g\in G_+$ such that $g\cdot \s= \s$ and $g\cdot \s'\neq \s'$, showing that $\varphi_{\s}$ and $\varphi_{\s'}$ are not conjugate. \qedhere
\end{proof}
\subsubsection{A simplicial tree and $\mathbb{R}$-focality}\label{subsubsec.simplicialfocal}
We keep the same standing assumption on $G$ and the same setting as above, with a fixed sequence $\mathsf{s}=(s_n)_{n\in \mathbb Z}$ as in \eqref{e-conditions-sequence}. We now want to observe that the action $\varphi_{\s}$ constructed above is $\mathbb{R}$-focal. What is more, we will interpret the action $\varphi_{\s}$ as the dynamical realization of a $G$-action on a planar directed tree which is \emph{simplicial} (of infinite degree), by simplicial automorphisms.
For $n\in \mathbb Z$ denote by $\mathbb Z_{\ge n}$ the set of integers $j\ge n$. We let $\mathsf{S}_{\ge n}\subset X^{\mathbb Z_{\ge n}}$ be the subset of sequences indexed by $\mathbb Z_{\ge n}$ obtained by restricting sequences in $\mathsf{S}$ to $\mathbb Z_{\ge n}$:
\[\mathsf{S}_{n}=\{(t_j)_{j\ge n}\colon (t_j)_{j\in \mathbb Z}\in \mathsf{S}\}.\]
We will call \emph{truncation} this operation. Given a sequence $(t_j)_{j\ge n} \in \mathsf{S}_n$ we say that $(t_j)_{j\ge n+1}\in \mathsf{S}_{n+1}$ is its \emph{successor}.
The disjoint union $\bigsqcup_{n\in \mathbb Z} \mathsf{S}_n$ is naturally the vertex set of a simplicial tree $\mathbb T$, obtained by connecting each element to its successor. Indeed it is clear to see that the graph obtained in this way has no cycles; moreover it is connected, because of the fact that all elements of $\mathsf{S}$ eventually coincide with the sequence $\s$ (Lemma \ref{l-cyclic-eventual}). Thus we obtain a simplicial tree.\footnote{Note that by slight abuse of notation we do not distinguish between the tree $\mathbb T$ as a graph, which is the pair $(V, E)$ consisting of the set of vertices and of edges $E\subset V^2$, and its simplicial realization, which is the topological space obtained by gluing a copy of the interval $[0, 1]$ for each edge in the obvious way and with the natural quotient topology.}
If we endow all edges of $\mathbb T$ with the orientation from a point to its successor, then all edges point to a common natural end $\omega\in \partial \mathbb T$, we get a directed tree $(\mathbb T,\triangleleft)$ with focus $\omega$.
Note that every $\mathsf{t}=(t_j)_{j\in \mathbb Z}\in \mathsf{S}$ defines a bi-infinite ray of $\mathbb T$, whose vertices are the successive truncations of $\mathsf{t}$.
For every $\mathsf{t}\in \mathsf{S}$, this sequence converges to $\omega$ as $n\to +\infty$. As $n\to -\infty$, it converges to some end $\alpha_{\mathsf{t}} \in \partial^*\mathbb T=\partial \mathbb T\setminus \{\omega\}$. The map $\mathsf{t}\mapsto \alpha_{\mathsf{t}}$ is clearly injective, and thus allows to identify $\mathsf{S}$ with a subset of $\partial^* \mathbb T$.
The group $G$ has a natural action on $(\mathbb T,\triangleleft)$, namely for every vertex $v=(t_j)_{j\ge n}$ of $\mathbb T$ and $g\in G$ we set
\[g\cdot (t_j)_{j\ge n}=\left (g(t_{j-\tau(g)})\right )_{j\ge n+\tau(g)}.\]
Note in particular that if $v\in \mathsf{S}_n$ then $g\cdot v\in \mathsf{S}_{n+\tau(g)}$. This action is by simplicial automorphism and fixes the end $\omega$. Moreover if $g\in G$ is such that $\tau(g)\neq 0$, then $g$ has no fixed point on $\mathbb T$, and thus acts as a hyperbolic isometry. If $\tau(g)=0$, then $g$ preserves each of the sets $\mathsf{S_n}$ and acts as an elliptic isometry (indeed since $g$ acts trivially on some neighborhood of $b$ in $x$, it must fix all vertices $(s_j)_{j\ge n}$ for $n$ large enough).
Let us now define a planar order on $(\mathbb T,\triangleleft)$. In this case, this just means an order $<^v$ for every $v=(t_j)_{j\ge n}\in \mathbb T$ on the set of edges $E^-_v$ which lie below $v$ (i.e.\ opposite to $\omega$). Fix $v=(t_j)_{j\ge n}$, and consider two distinct edges $e_1, e_2\in E_v^-$. Then for $i\in\{1,2\}$ we have $e_i=(w, v)$ for some $w=\left (t^{(i)}_j\right )_{j\ge {n-1}}$ with $t^{(i)}_j=t^{(2)}_j=t_j$ for $j\ge n$, and $t^{(1)}_{n-1}\neq t^{(2)}_j$. Thus we set $e_1<^v e_2$ if and only if $t^{(1)}_j<t^{(2)}_j$. Then the collection $\{<^v\colon v\in \mathbb T\}$ defines a planar order $\prec$ on $(\mathbb T, \omega)$ which is invariant under the action of $G$.
Finally note that the map $\mathsf{t}\mapsto \alpha_{\mathsf{t}}$ is $G$-equivariant and increasing with respect to the order on $\mathsf{S}$ and the order on $\partial^* \mathbb T$ induced from the planar order.
Thus the $G$-action on $(\mathsf{S}, \prec)$ can be identified with an action on an orbit in $\partial^* \mathbb T$. It follows that $\varphi_{\s}$ is conjugate to the dynamical realization of the action on the planar directed tree $(\mathbb T, \prec, \omega)$.
Finally we note that the above discussion also shows that the action $\varphi_{\s}$ can be increasingly horograded by the cyclic action of $G$ on $\mathbb{R}$ by integer translations defined by the homomorphism $\tau\colon G\to \mathbb Z$. Indeed it is enough to take the horofunction $\pi\colon \mathbb T\to \mathbb{R}$ associated with $\omega$, which sends every vertex $v\in \mathsf{S}_{\ge n}$ to $n\in \mathbb{R}$, and maps the edge from $v$ to its successor to the interval $[n, n+1]$. It follows from Proposition \ref{prop.dynamic-class-horo} that for every element $g\in G$ the image $\varphi_{\s}(g)$ is a homothety if $\tau(g)\neq 0$, while it is totally bounded if $\tau(g)=0$. In particular every pseudohomothety in the image of $\varphi_{\s}$ is actually a homothety.
\begin{rem}
When a bi-infinite sequence $\s=(s_n)_{n\in \mathbb Z}$ as in \eqref{e-conditions-sequence} also satisfies $\lim_{n\to-\infty}s_n=a$ (that is, when the defining element $f_0$ has no fixed point on $X$), then the dynamical realization $\varphi_{\s}$ can also be increasingly horograded by the action of $G$ on $X$.
To see this, we proceed to define an ultrametric kernel $\delta:\mathsf{S}\times\mathsf{S}\to X\cup\{a\}$. This is given, for any distinct $\mathsf t,\mathsf t'\in\mathsf{S}$, by $\delta(\mathsf t,\mathsf t')=t_{m+1}=t'_{m+1}$, where $m=m(\mathsf t,\mathsf t')+1$ is the integer defined in \eqref{eq.mts}, and declare $\delta(\mathsf t,\mathsf t)=a$.
It follows directly from the definitions that $\delta$ is an ultrametric kernel, and that it is $G$-equivariant with respect to the actions of $G$ on $\mathsf{S}$ and $X$. Notice also that the $\delta$-balls are convex with respect to the total order relation $\prec$ on $\mathsf{S}$ and thus $\prec$ is $\delta$-convex. In order to apply Corollary \ref{cor.ultra} we need to show that the action of $G$ on $\mathsf{S}$ expands $\delta$-balls. As the action on $\mathsf S$ is transitive, it is enough to consider $\delta$-balls centered at $\s$. Fix $x\in X$, and consider the $\delta$-ball $B=B_\delta(\s,x)$. Since $\s$ converges to $a$ as $n\to-\infty$, we can find $n\in \mathbb Z$ such that $B_\delta(\s,s_n)\subseteq B$. On the other hand, we have $f_0.B_\delta(\s,s_n)=B_\delta(\s,s_{n+1})$. Since the $\delta$-balls $B_\delta(\s,s_n)$ define an exhaustion of $\mathsf{S}$, as $n$ runs over the integers, we deduce that the same holds for the collection of $\delta$-balls $f_0^n.B$, which proves that the action of $G$ on $\mathsf{S}$ expands $\delta$-balls as desired. Thus, from Corollary \ref{cor.ultra} we get that $\varphi_{\s}$ is increasingly horograded by the standard action of $G$ on $X$.
\end{rem}
\subsection{Orders of germ type and semidirect products}\label{ssec.germtype}
Here we explain a framework to construct minimal exotic actions of various classes of groups of homeomorphisms of intervals, including most groups of piecewise linear or projective homeomorphisms.
Set $X=(a, b)$ and let $G\subseteq \homeo_0(X)$ be a (countable) micro-supported group acting minimally on $X$. In order to run our construction, we require $G$ to satisfy the following condition. \begin{enumerate}[label=($\mathcal{G}$\arabic*)]
\item \label{i-germ-section} The group $\Germ(G, b)$ admits a \emph{section} inside $ \homeo_0(X)$, namely a subgroup $\Gamma \subseteq \homeo_0(X)$ such that $\Germ(\Gamma, b)=\Germ(G, b)$ and which projects bijectively to $\Germ(G, b)$.\footnote{The problem of when a group of germs admits a section as a group of homeomorphisms is very interesting. We refer the reader to \cite{Mann} for an example of a finitely generated group of germs which does not admit any such section.}
\end{enumerate}
Under assumption \ref{i-germ-section}, let $\Gamma\subseteq \homeo_0(X)$ be a section of $\Germ(G, b)$, and consider the overgroup $\widehat{G}=\langle G , \Gamma\rangle$. we will proceed by describing an action of $\widehat{G}$ and then restricting it to $G$. Note that $\Germ(\widehat{G}, b)=\Germ(G, b)$ induces the same group of left-germs at $b$ as $G$. The advantage of passing to $\widehat{G}$ is that it splits as a semidirect product
\[\widehat{G}=\widehat{G}_+\rtimes \Gamma.\]
where as usual $\widehat{G}_+\subseteq \widehat G$ is the subgroup of elements whose germ at $b$ is trivial.
Using this splitting, we can let the group $\widehat{G}$ act ``affinely'' on $\widehat{G}_+$: the subgroup $\widehat{G}_+$ acts on itself by left-multiplication, and $\Gamma$ acts on it by conjugation. Explicitly, if $g\in \widehat{G}$ and $h\in \widehat{G}_+$, writing $g=g_+\gamma_g$ with $g_+\in \widehat{G}_+$ and $\gamma_g\in \Gamma$, we set
\begin{equation} \label{e-affine-action} g\cdot h= g h\gamma_g^{-1}.\end{equation}
Note that we actually have $g\cdot h=g_+(\gamma_g h\gamma_g^{-1})$, from which it is straightforward to check that this defines indeed an action on $\widehat G_+$.
We want to find an order $\prec$ on $\widehat{G}_+$ which is invariant under the action of $G$, and then consider the dynamical realization of the action of $\widehat{G}$ on $(\widehat{G}_+, \prec)$. For this we look for a left-invariant order $\prec$ on $\widehat{G}_+$ which is also invariant under the conjugation action of $\Gamma$. We will say for short that such an order is \emph{$\Gamma$-invariant}.
Good candidates are the orders of germ type on $\widehat{G}_+$ described in \S \ref{s-germ-type}. Recall that an order of germ type is determined by a family of left orders $\{<^{(x)}\colon x\in X\}$, where for every $x\in X$, $<^{(x)}$ is a left-order on the group of germs $\Germ\left (\widehat{G}_{(a, x)}, x\right )$: the associated order of germ type on $\widehat{G}_+$ is the order $\prec$ whose positive cone is the subset
\[P=\left \{g\in \widehat{G}_+ \colon \mathcal{G}_{p_g}(g)\succ^{(p_g)} \mathsf{id}\right \},\]
where $p_g=\sup\{x\in X : g(x)\neq x\}$. However, not every order of germ type is $\Gamma$-invariant, and this is because for every $x\in X$ the stabilizer $\stab_{\widehat{G}}(x)$ of $x$ acts on $\widehat{G}_{(a, x)}$ by conjugation, and this action descends to an action of $\Germ (\widehat G, x )$ on $\Germ\left (\widehat{G}_{(a, x)}, x\right )$. In light of this, we are able to produce a $\Gamma$-invariant order of germ type on $\widehat G_+$ if and only if the following condition is satisfied.
\begin{enumerate}[label=($\mathcal{G}$\arabic*)]
\setcounter{enumi}{1}
\item \label{i-germ-invariant} For every $x\in X$, the group $\Germ\left (\widehat{G}_{(a, x)}, x\right )$ admits a left-invariant order $<^{(x)}$ which is invariant under conjugation by $\Germ (\Gamma, x )$.
\end{enumerate}
Indeed, suppose that $\{<^{(x)}: x\in X\}$ is a family such that the associated order of germ type is $\Gamma$-invariant, then each $\prec^{(x)}$ is as in \ref{i-germ-invariant}. Conversely if we choose such an order $<^{(x)}$ as in \ref{i-germ-invariant} for $x$ in a system of representatives of the $\Gamma$-orbits in $X$, then we can extend it uniquely by $\Gamma$-equivariance to a family $\{<^{(x)}\colon x\in X\}$ which defines a $\Gamma$-invariant order of germ type.
\begin{rem}
Here are two simple sufficient conditions for \ref{i-germ-invariant}.
\begin{enumerate}[label=($\mathcal{G}$2\roman*)]
\item\label{i-germ-3'}The group $\Gamma$ acts freely on $X$.
\item \label{i-germ-3''}For every $x\in X$, every non-trivial germ in $\Germ\left (\widehat{G}_{(a, x)}, x\right )$ has no fixed point accumulating on $x$ from the left (this does not depend on the choice of the element representing the germ).
\end{enumerate}
The fact that \ref{i-germ-3'} implies \ref{i-germ-invariant} is clear since in this case $\stab_\Gamma(x)$ is trivial. In contrast when \ref{i-germ-3''} holds, we can define an order $<^{(x)}$ on $\Germ\left (\widehat{G}_{(a, x)}, x\right )$ by setting $\mathcal{G}_x(g)>^{(x)} \mathsf{id}$ if and only if $g(y)>y$ for every $y\neq x$ in some left-neighborhood of $x$. Then this is a left-order on $\Germ\left (\widehat{G}_{(a, x)},x\right )$ which is invariant under conjugation by the whole stabilizer of $x$ in $\homeo_0(X)$. \end{rem}
Summing up, under conditions \ref{i-germ-section} and \ref{i-germ-invariant} we can consider a $\Gamma$-invariant order of germ type $\prec$ on $\widehat{G}_+$ and let $\widehat{G}$ act on $(\widehat{G}, \prec)$ by \eqref{e-affine-action}. By passing to the dynamical realization we obtain an action of $\widehat{G}$, and thus of $G$, on the real line.
This construction yields the following criterion for the existence of exotic actions.
\begin{prop}\label{p-exotic-germ}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be a finitely generated, micro-supported group acting minimally on $X$. Assume that $\Germ(G,b)$ admits a section $\Gamma\subseteq \homeo_0(X)$ (that is, condition \ref{i-germ-section} holds) such that $\widehat{G}=\langle G, \Gamma\rangle$ satisfies \ref{i-germ-invariant}.
Then there exists a minimal faithful action $\varphi\colon G\to \homeo_0(\mathbb{R})$ which is $\mathbb{R}$-focal and not conjugate to the standard action of $G$ on $X$.
\end{prop}
\begin{proof}
Choose a $\Gamma$-invariant order of germ type $\prec$ on $\widehat{G}_+$, defined from the family of orders $\{<^{(x)}\colon x\in X\}$. We let $\psi\colon G\to \homeo_0(\mathbb{R})$ be the dynamical realization of the action of $G$ on $(\widehat{G}_+, \prec)$. Set $N=[G_c, G_c]$, which by Proposition \ref{p-micro-normal} is the smallest non-trivial normal subgroup of $G$.
\begin{claim}
For every $x\in X$ the group $\psi(N_{(x, b)})$ acts on $\mathbb{R}$ without fixed points.
\end{claim}
\begin{proof}[Proof of claim]
Let $\iota\colon (\widehat{G}_+, \prec)\to (\mathbb{R}, +)$ be an equivariant good embedding associated with $\psi$ (Definition \ref{dfn.goodbehaved}). Observe that the subgroups $\widehat{G}_{(a, y)}$, for $y\in X$, are bounded convex subgroups of $(\widehat{G}_+, \prec)$ which form an increasing exhaustion of $\widehat{G}_+$, thus the convex hull of every $\iota(\widehat{G}_{(a, y)})$ is a bounded interval $I_y\subset \mathbb{R}$, giving an increasing exhaustion of $\mathbb{R}$. Now given any $x$ and $y$ in $X$, every $g\in N_{(x, b)}$ with $p_g>y$ satisfies $g\widehat{G}_{(a, y)}\neq \widehat{G}_{(a, y)}$, which in turn implies that $\psi(g)(I_y)\cap I_y=\varnothing$. Since $y$ is arbitrary, and the intervals $I_y$ exhaust $\mathbb{R}$, this implies that $\operatorname{\mathsf{Fix}}^{\psi}(N_{(x, b)})=\varnothing$.
\end{proof}
As $G$ is finitely generated, we can consider a canonical model $\varphi\colon G\to \homeo_0(\mathbb{R})$ of $\psi$, which is thus either minimal or cyclic.
Since $\varphi$ is semi-conjugate to $\psi$, the claim gives $\operatorname{\mathsf{Fix}}^\varphi(N_{(x, b)})=\varnothing$. Using Proposition \ref{p-micro-normal}, we deduce that $\varphi$ is faithful, and thus minimal. Moreover we see that it cannot be conjugate to the standard action of $X$, since $N_{(x, b)}$ does have fixed points on $X$.
Finally, as in the proof of the claim, it is not difficult to see that the collection
\[\mathcal S:=\{\psi(g)(I_y)\colon g\in G,y\in X\}\]
is an invariant CF-cover for $\psi$ (thus $\psi$ is $\mathbb{R}$-focal). Since $\varphi$ is semi-conjugate to $\psi$, Proposition \ref{p-focal-semiconj} implies that $\varphi$ is $\mathbb{R}$-focal. \qedhere
\end{proof}
\begin{rem}
With some additional work, one could show that $\varphi$ can be increasingly horograded by the natural action of $G$ on $X$. In particular the dynamics of element in the image of $\varphi$ can be determined by their dynamics on $X$ using Proposition \ref{prop.dynamic-class-horo}. We do not elaborate on this, as in Section \ref{sec.locandfoc} we will prove that this is the case for all exotic actions of a class of locally moving groups (although this class does not include all groups covered by Proposition \ref{p-exotic-germ}, the arguments in its proof can be adapted to the action constructed here).
\end{rem}
The criterion given by Proposition \ref{p-exotic-germ} applies to several classes of examples of micro-supported groups. Let us give a first illustration.
We say that the group of germs $\Germ(G, b)$ acts \emph{freely near} $b$ if every non-trivial germ has no fixed point accumulating on $b$ (this condition does not depend on the choice of the representative; cf.\ \ref{i-germ-3''}).
\begin{cor}\label{cor.germabelian}
For $X=(a, b)$, let $G\subseteq \homeo_0(X)$ be a finitely generated, micro-supported group acting minimally on $X$. Assume that $\Germ(G, b)$ is abelian and acts freely near $b$. Then there exists a faithful minimal action $\varphi\colon G\to \homeo_0(X)$ which is $\mathbb{R}$-focal and not conjugate to the standard action of $G$ on $X$.
\end{cor}
\begin{proof}
We first check that if $\Germ(G, b)$ is abelian and acts freely near $b$, then it admits a section $\Gamma\subset \homeo_0(X)$ which acts freely on $X$. To see this, using that $G$ is finitely generated, we can take finitely many elements $g_1, \ldots, g_r$ in $G$ whose germs at $b$ are non-trivial and generate $\Germ(G, b)$. Up to taking inverses, the condition that $\Germ(G, b)$ acts freely near $b$ allows to find $z\in X$ such that $g_i(x)>x$ and $g_ig_j(x)=g_jg_i(x)$ for every $x\in (z, b)$ and $i,j\in \{1,\ldots,r\}$. Choose an element $\gamma_1\in \homeo_0(X)$ which coincides with $g_1$ on $(z, b)$ and has no fixed points in $X$. Take $x_0\in (z, b)$ and consider the fundamental domain $I=[x_0,\gamma_1(x_0))$. As the elements $g_2,\ldots, g_r$ commute with $g_1=\gamma_1$ on $I$, they induce an action of $\mathbb Z^{r-1}$ on the circle $X/\langle \gamma_1\rangle= [x_0,\gamma_1(x_0))/_{x_0\sim\gamma_1(x_0)}$, which can be lifted to an action of $\mathbb Z^{r-1}$ on $X$ commuting with $\gamma_1$. In simpler terms, writing $I_n=\gamma_1^n(I)$, so that $X=\bigsqcup_{n\in \mathbb Z} I_n$, for every $i\in\{2,\ldots,r\}$ we can consider the homemorphism $\gamma_i\in \homeo_{0}(X)$ defined by
\begin{equation}\label{eq:section_germ}
\gamma_i(x)= \gamma_1^n g_i \gamma_1^{-n}(x)\quad\text{for }x\in I_n\text{ and }n\in \mathbb Z.
\end{equation}
The elements $\gamma_2,\ldots,\gamma_r$ define exactly the action of $\mathbb Z^{r-1}$ on $X$ which commutes with $\gamma_1$, as discussed above.
From the definition \eqref{eq:section_germ}, we see that every $\gamma_i$ coincides with $g_i$ on $[x_0,b)$, and in particular we have $\mathcal{G}_b(\gamma_i)=\mathcal{G}_b(g_i)$. This gives that $\Gamma=\langle \gamma_1,\ldots, \gamma_r\rangle$ is a section of $\Germ(G, b)$ acting freely on $X$. Thus conditions \ref{i-germ-section} and \ref{i-germ-3'} are satisfied, so that the conclusion follows from Proposition \ref{p-exotic-germ}. \qedhere
\end{proof}
A case in which the previous criterion applies is when $b<\infty$ and the group of germs $\Germ(G, b)$ coincides with a group of germs of linear homeomorphisms $x\mapsto \lambda(x-b)+b$. This is for instance the case whenever $G$ is a subgroup of the group $\PL(X)$ of finitary piecewise linear homeomorphisms of $X$. This case can be generalized as follows.
Recall from Definition \ref{d-pa} that given $X=(a, b)\subset \mathbb{R}$ an interval, we denote by $\operatorname{\mathsf{PDiff}}_{loc}^\omega(X)$ the group of all locally piecewise analytic, orientation-preserving homeomorphisms of $X$, with a countable discrete set of breakpoints in $X$. We also let $\PP_0(X)$ be the subgroup of $\operatorname{\mathsf{PDiff}}_{loc}^\omega(X)$ of piecewise projective homeomorphisms of $X$ with finitely many breakpoints, namely those that are locally of the form $x\mapsto \frac{px+q}{rx+s}$, with $ps-qr=1$.
\begin{cor}\label{c-exotic-pp}
For $X=(a,b)$, let $G\subseteq \operatorname{\mathsf{PDiff}}_{loc}^\omega(X)$ be a finitely generated micro-supported group acting minimally on $X$. Assume that either of the following conditions is satisfied:
\begin{enumerate}[label=(\alph*)]
\item \label{i-exotic-pp} $G$ is contained in the group $\PP_0(X)$ of piecewise projective homeomorphisms;
\item \label{i-exotic-pa} the group of germs $\Germ(G, b)$ admits a section $\Gamma$ contained in $\operatorname{\mathsf{PDiff}}_{loc}^\omega(X)$.
\end{enumerate}
Then there exists a faithful minimal action $\varphi\colon G\to \homeo_0(\mathbb{R})$ which is $\mathbb{R}$-focal and not conjugate to the action of $G$ on $X$.
\end{cor}
\begin{proof}
First of all observe that \ref{i-exotic-pp} implies \ref{i-exotic-pa}. To see this, assume first that $X=\mathbb{R}$; then $\Germ(G, +\infty)$ is a subgroup of the group of germs of the affine group $\Aff(\mathbb{R})=\{x\mapsto ax+b\}$, and thus admits a section inside $\Aff(\mathbb{R})\subseteq \PP_0(\mathbb{R})$. For general $X$, observe that if we fix $x_0\in X$, we can find $A, B\in \PSL(2, \mathbb{R})$ which fix $x_0$ and such that $A$ maps the interval $(a, x_0)$ to $(-\infty, x_0)$ and $B$ maps $(x_0, b)$ to $(x_0, +\infty)$. Then the map $H\colon X\to \mathbb{R}$, given by
\[
H(x)=\left\{\begin{array}{lr}
A(x)&\text{if }x\le x_0,\\
B(x)&\text{if }x>x_0,
\end{array}\right.
\]
conjugates $\PP(X)$ to $\PP(\mathbb{R})$, so that the we can conclude from the previous case.
Now assume that \ref{i-exotic-pa} holds, and choose a section $\Gamma\subseteq \operatorname{\mathsf{PDiff}}_0^\omega(X)$ of $\Germ(G, b)$. Then since non-trivial analytic maps have isolated fixed points, we see that $\widehat{G}$ satisfies \ref{i-germ-3''}, thus \ref{i-germ-invariant}, and we can apply Proposition \ref{p-exotic-germ}. \qedhere
\end{proof}
\begin{rem}
The conditions in Proposition \ref{p-exotic-germ} cannot be dropped: in \S\ref{s-no-actions} we will construct an example of a finitely generated locally moving group $G\subseteq \homeo_0(\mathbb{R})$ which admits no exotic action. Moreover this example satisfies \ref{i-germ-section} (but not \ref{i-germ-invariant}), and its natural action is by piecewise linear homeomorphisms with a countable set of singularities which admit a finite set of accumulation points inside $X$. This shows the sensitivity of Corollary \ref{c-exotic-pp} to its assumptions.
\end{rem}
\subsection{A construction of $\mathbb{R}$-focal actions for groups of PL homeomorphisms}\label{ss.Bieri-Strebelfocal} Let $X=(a,b)$ be an interval. Recall that we denote by $G(X;A,\Lambda)$ the Bieri-Strebel group of finitary PL-homeomorphisms associated to a non-trivial multiplicative subgroups $\Lambda \subset \mathbb{R}_*$ and a $\Lambda$-submodule $A\subset \mathbb{R}$ (see Definition \ref{d.BieriStrebel}). By Corollary \ref{c-exotic-pp}, we already know that the group $G(X; A, \Lambda)$ always admits an (exotic) minimal $\mathbb{R}$-focal action. In this subsection we present a generalization of that construction which is specific to the groups $G(X; \Lambda)$. The main interest of this generalisation is that in Section \ref{s-few-actions} we will show that for a vast class of Bieri-Strebel groups on $X=\mathbb{R}$ (including the groups $G(\lambda)$ from Theorem \ref{t-intro-Glambda}), all exotic actions arise through the construction presented here.
Throughout we fix $(\Lambda, A)$ and set $G:=G(X;A,\Lambda)$. We introduce the functions $j^{\pm}:G\times X\to\Lambda$ defined by \[j^+(g,x)=\prod_{y\ge x}\frac{D^-g(y)}{D^+g(y)}
\quad \text{and}\quad j^-(g,x)=\prod_{y\le x}\frac{D^+g(y)}{D^-g(y)},\]
that we call the \emph{right} (respectively, \emph{left}) \emph{jump cocycles} (cf.\ Example \ref{ex:Brown}).
Note that these are well defined as only finitely many terms in each product are different from $1$, and it is immediate to verify the cocycle relation
\begin{equation}\label{eq.cocycle}
j^{\pm}(g_2g_1,x)=j^{\pm}(g_2,g_1(x))\,j^\pm(g_1,x)\quad\text{for }x\in X\text{ and }g_1,g_2\in G.
\end{equation}
\begin{rem}\label{r.jump_cocycle}
For fixed $g\in G$, the function $j^+(g,-):X\to \Lambda$ (respectively, $j^-(g,-):X\to \Lambda$) has the following properties:
\begin{enumerate}
\item it is left (respectively, right) continuous,
\item it is piecewise constant, with discontinuity points in $A$,
\item it takes the constant value $1$ on a neighborhood of $b$ (respectively, $a$).
\end{enumerate}
\end{rem}
We next fix a preorder $\leq_\Lambda\in\LPO(\Lambda)$ and we denote by $\Lambda_0=[1]_{\leq_\lambda}$ its residue.
We then get a partition $G=P\sqcup H\sqcup P^{-1}$, with
\begin{equation}\label{eq.P}H=\left \{g\in G:j^+(g,x)\in\Lambda_0\ \forall x\in X\right \}\quad\text{and}\quad
P=\left \{g\in G\setminus H\colon j^+(g,x_{g,\Lambda_0})\in P_{\le_\Lambda}\right \},\end{equation}
where for given $g\in G\setminus H$, we are writing
\begin{equation}
\label{ekuation0}
x_{g,\Lambda_0}=\max\left \{x\in X\colon j^+(g,x)\notin \Lambda_0\right \}.
\end{equation}
It follows from the cocycle relation \eqref{eq.cocycle} that $P$ is a semigroup, $H$ is a subgroup and $HPH\subseteq P$. Thus, $P$ is the positive cone of a preorder of $G$ whose residue is $H$ (see Remark \ref{r.cones}).
\begin{dfn}\label{dfn.cocyclepreorder} For a given preorder $\le_\Lambda$ on $\Lambda$, we define the \emph{right jump preorder} on $G=G(X;A,\Lambda)$ as the preorder $\preceq_+$ whose positive cone is the semigroup $P$ in \eqref{eq.P}. Similarly, we define the \emph{left jump preorder} $\preceq_-$ as the preorder obtained from the analogue construction considering the left jump cocycle.
\end{dfn}
(This should be compared with the construction appeared in in \S \ref{s-example-PL-Q} for the group $\PL_{\mathbb Q}((0,1))$.)
The aim of this subsection is to show the following.
\begin{prop}\label{prop.cocicloRfocal} Let $X=(a,b)$ be an interval and let $G=G(X;A,\Lambda)$ be a countable Bieri--Strebel group.
a preorder $\leq_\Lambda\in\LPO(\Lambda)$, let $\preceq\in\LPO(G)$ be either the right or left jump preorder associated with $\leq_\Lambda$.
Assume in addition that one of the following conditions satisfied:
\begin{enumerate}
\item $a,b\in A\cup\{\pm\infty\}$, or
\item the residue $\Lambda_0=[1]_{\le_\Lambda}$ is non-trivial.
\end{enumerate}
Then, the dynamical realization of $\preceq$ is a faithful minimal $\mathbb{R}$-focal action.
Moreover, for two distinct preorders on $\Lambda$, the dynamical realizations of their associated right (respectively, left) jump preorders are not positively conjugate. In particular, when $\LPO(\Lambda)$ is uncountable, then $G$ admits uncountably many conjugacy classes of minimal $\mathbb{R}$-focal actions.
\end{prop}
In order to do this, we take a different approach to define the jump preorders. We only discuss the case of right jump preorder, the other case being totally analogous.
To start with, we note that the right jump cocycle $j^{+}$ induces a fibered action of $G$ on the trivial bundle $X\times \Lambda\to X$. Such action descends to a fibered action on the quotient $X\times \Lambda/\Lambda_0\to X$ and we can consider the induced action $\Phi$ on the space of sections $\mathsf{S}(X,\Lambda/\Lambda_0)=\{\mathsf t:X\to \Lambda/\Lambda_0\}$. More explicitly, for a given function $\mathsf{t}:X\to \Lambda/\Lambda_0$, and element $g\in G$, the action $\mathsf{t}\mapsto \Phi(g)(\mathsf{t})$ is defined by the expression
\begin{equation}\label{equ.varPhi}\Phi(g)(\mathsf{t}):x\mapsto j^+(g,g^{-1}(x))\,\mathsf{t}(g^{-1}(x))\pmod{\Lambda_0}.\end{equation}
It is routine to check from \eqref{eq.cocycle} that this is indeed an action of the group $G$. As usual, we will use the shorthand notation $g.\mathsf t$ for $\Phi(g)(t)$.
After the regularity properties of the cocycles (Remark \ref{r.jump_cocycle}), it is more appropriate to restrict such actions to the invariant subspace $\mathsf S$ of functions $\mathsf t:X\to \Lambda/\Lambda_0$ satisfying the analogous properties:
\begin{enumerate}
\item $\mathsf t$ is left continuous,
\item $\mathsf t$ is piecewise constant, with discontinuity points in $A$,
\item $\mathsf t$ takes the constant value $\Lambda_0$ on a neighborhood of $b$.
\end{enumerate}
The preorder $\le_\Lambda$ defines a total order $<_{\Lambda/\Lambda_0}$ on the quotient $\Lambda/\Lambda_0$, which is invariant by multiplication by elements of $\Lambda$. From this, we introduce a lexicographic total order $\prec_{\mathsf S}$ on the subspace $\mathsf S$: for distinct sections $\mathsf s,\mathsf t\in \mathsf S$, say that $\mathsf s\prec_{\mathsf S} \mathsf t$ if
\begin{equation*
\mathsf s(m)<_{\Lambda/\Lambda_0}\mathsf t(m),\quad\text{where } m=\max \left \{x\in X\colon \mathsf s(x)\neq \mathsf t(x)\right \}.
\end{equation*}
\begin{lem}
With notation as above, the action $\Phi$ of the group $G=G(X;A,\Lambda)$ on the space of sections $\mathsf S$ preserves the total order $\prec_{\mathsf S}$.
\end{lem}
\begin{proof}
The order $\prec_{\mathsf S}$ is invariant under pointwise multiplication by elements in the subspace $\mathsf S$, and also by precomposition of sections by homeomorphisms of the base $X$. Thus, from the expression \eqref{equ.varPhi} for the action $\Phi$, we immediately get the result.
\end{proof}
In what follows, we will denote by $\mathsf e$ the trivial section (namely, $\mathsf e(x)=\Lambda_0$ for every $x\in X$). Note that $\mathsf e$ belongs to $\mathsf S$.
We have the following.
\begin{lem}\label{l.cocyclepreorder} With notation as above, the pull-back of $\prec_{\mathsf S}$ by the map $g\in G\mapsto g.\mathsf e\in \mathsf S$ coincides with the right jump preorder.
Moreover, the restriction of the action $\Phi$ to the orbit of the trivial section $\mathsf e$ is faithful.
\end{lem}
\begin{proof}
For $g\in G$, we compute from \eqref{equ.varPhi} the image of the trivial section:
\begin{equation}\label{eq:image_trivial_section}
g.\mathsf{e}(x)=j^+(g,g^{-1}(x))\pmod{\Lambda_0}.
\end{equation}
Thus the subgroup $H$ and the semigroup $P$ defined at \eqref{eq.P} are respectively, the residue and the positive cone of the pull-back preorder.
In order to prove that the action is faithful, we will find an element $g\in [G_c,G_c]$ and $x\in X$ for which $j^+(g,x)\notin \Lambda_0$. From \eqref{eq:image_trivial_section}, this will give that $g$ does not fix the trivial section $\mathsf e$, and we can conclude using Proposition \ref{p-micro-normal}.
To do this, take an element $g_1\in G_c$ whose support is a interval $J=(c,d)\Subset X$, with $D^-g_1(d)\notin \Lambda_0$. Take an element $h\in G$ such that $h(d)<c$, and consider the commutator $g=[g_1,h]\in [G_c,G_c]$. Then $j^+(g,d)=D^-g_1(d)\notin \Lambda_0$, as desired.
\end{proof}
We need further preliminary results on the dynamics of the action $\Phi$.
\begin{lem}\label{l.move_trivial}
With notation as above, for any two distinct sections $\mathsf s,\mathsf t\in \mathsf S$ with $\mathsf s\prec_{\mathsf S}\mathsf t$, there exists $g\in G$ such that $\mathsf s\prec_{\mathsf S}g.\mathsf e\prec_{\mathsf S}\mathsf t$.
\end{lem}
\begin{proof}
Write $m=\max \left \{x\in X\colon \mathsf s(x)\neq \mathsf t(x)\right \}$, choose a point $m'<m$ in $A$ such that $\mathsf t$ is locally constant on $[m',m]$, and take an element $g\in G$ such that
\begin{enumerate}
\item $\mathsf t(x)=D^-g^{-1}(x)^{-1}\pmod{\Lambda_0}$ for every $x\in (m',b)$,
\item $t(m')>_{\Lambda/\Lambda_0} D^-g^{-1}(m')^{-1}\pmod{\Lambda_0}$,
\item $D^-g(x)=1$ for every $x$ in a sufficiently small neighborhood of $b$.
\end{enumerate}
Such an element (or better, its inverse $g^{-1}$) can be produced by integrating a representative of the section $\mathsf t^{-1}$ on $(m',b)$ (details are left to the reader).
Note that the last condition gives
\[g.\mathsf e(x)=j^+(g,g^{-1}(x))=D^-g^{-1}(x)^{-1}\pmod{\Lambda_0},\]
so that by the first two conditions we get $\mathsf s\prec_{\mathsf S}g.\mathsf e\prec_{\mathsf S}\mathsf t$.
\end{proof}
\begin{lem}\label{l.action_on_section_is_homothetic}
With notation as above, assume in addition $b\in A\cup \{+\infty\}$ or $\Lambda_0\neq \{1\}$. Then, for every choice of four sections $\s_1,\s_2,\mathsf t_1,\mathsf t_2\in \mathsf{S}$ with $\mathsf t_1\prec_{\mathsf S}\s_1\prec_{\mathsf S} \s_2\prec_{\mathsf{S}}\mathsf t_2$, there exists an element $g\in G$ such that
$g.\s_1\prec_{\mathsf S}\mathsf t_1\prec_{\mathsf S}\mathsf t_2 \prec_{\mathsf S} g.\s_2$.
\end{lem}
\begin{proof}
First of all, we argue that we can assume that $\s_1\prec_{\mathsf S}\mathsf e \prec_{\mathsf{S}}\s_2$. Indeed, for general $\s_1,\s_2,\mathsf t_1,\mathsf t_2\in \mathsf{S}$ with $\mathsf t_1\prec_{\mathsf S}\s_1\prec_{\mathsf S} \s_2\prec_{\mathsf{S}}\mathsf t_2$, after Lemma \ref{l.move_trivial} we can take an element $h\in G$ such that $\s_1\prec_{\mathsf S}h.\mathsf e\prec_{\mathsf S} \s_2$, or equivalently $h^{-1}.\s_1\prec_{\mathsf S}\mathsf e\prec_{\mathsf S} h^{-1}.\s_2$.
Assume there exists an element $k\in G$ such that
\[
kh^{-1}.\s_1\prec_{\mathsf S}h^{-1}.\mathsf t_1\prec_{\mathsf S}h^{-1}.\mathsf t_2\prec_{\mathsf{S}}kh^{-1}.\s_2,
\]
then the conjugate element $g=hkh^{-1}$ will make the job for $\s_1,\s_2,\mathsf t_1,\mathsf t_2\in \mathsf{S}$.
Now, for general $\mathsf t\in \mathsf S\setminus \{\mathsf e\}$, define $x_{\mathsf t}=\max\{x\in X\colon \mathsf t(x)\neq \Lambda_0\}$ and write
\[x_*=\min \{x_{\mathsf s_1},x_{\mathsf s_2}\},\quad y_*=\max \{x_{\mathsf t_1},x_{\mathsf t_2}\}.\]
Take an element $g\in G$ with the following properties:
\begin{enumerate}
\item $g(x_*)>y_*$ and
\item $j^+(g,g^{-1}(x))=1$ for every $x\in [g(x_*),b)$.
\end{enumerate}
To ensure the second condition, we can either choose $g\in G(X;A,\Lambda_0)$ (when the residue $\Lambda_0$ is non-trivial) or an element $g$ with no breakpoint in $[x_*,b)$ (in which case we have to assume that $b\in A\cup\{+\infty\}$).
\begin{rem}\label{r.action_is_homothetic}
We point out, for further use, that in the former case we can actually choose $g\in G(X;A,\Lambda_0)\cap [G_c,G_c]$.
\end{rem}
Then, for $i\in \{1,2\}$ we get from \eqref{equ.varPhi} the expression
\[g.\mathsf s_i(x)=\mathsf s_i(g^{-1}(x))\quad \text{for every }x\in [g(x_*),b).\]
In particular, we have
\[
g.\mathsf s_i(x)=\Lambda_0=\mathsf t_i(x)\quad \text{for every }x\in (g(x_{\mathsf s_i}),b),
\]
and
$g.\mathsf s_i(g(x_{\mathsf s_i}))=\mathsf s_i(x_{\mathsf s_i})$, while $\mathsf t_i(g(x_{\mathsf s_i}))=\Lambda_0$. As we are assuming $\mathsf s_2\prec_{\mathsf S}\mathsf e \prec_{\mathsf S} \s_2$, this gives
\[
g.\mathsf s_1(g(x_{\mathsf s_1}))<_{\Lambda/\Lambda_0}\Lambda_0=\mathsf t_1(g(x_{\mathsf s_1}))\quad\text{and}\quad
g.\mathsf s_2(g(x_{\mathsf s_2}))>_{\Lambda/\Lambda_0}\Lambda_0=\mathsf t_2(g(x_{\mathsf s_2})),
\]
so that we can conclude $g.\s_1\prec_{\mathsf S}\mathsf t_1\prec_{\mathsf S}\mathsf t_2 \prec_{\mathsf S} g.\s_2$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop.cocicloRfocal}]We will focus on the right jump preorder, the case for the left jump preorder being analogous.
For given $\mathsf{t}\in\mathsf S$ and $x\in X$, consider the subset
$$\mathsf{C}(\mathsf{t},x)=\{\mathsf{s}\in\mathsf S:\mathsf{s}(y)=\mathsf{t}(y)\text{ for every }y>x\}.$$ Then, we define $\mathcal{C}:=\{\mathsf{C}(\mathsf{t},x):\mathsf{t}\in\mathsf S,x\in X\}$. It is a straightforward verification that $\mathcal{C}$ is a $\Phi(G)$-invariant CF-cover of $\mathsf S$ by $\prec_{\mathsf{S}}$-convex subsets.
If we denote by $\varphi$ the dynamical realization of the action $\Phi$ on the orbit of the trivial section $\mathsf e$ (which is countable), and by $\iota$ its associated good embedding, we have that the family $\{\mathsf{hull}(\iota(C)):C\in\mathcal{C}\}$ is a $\varphi(G)$-invariant CF-cover (here $\mathsf{hull}(A)$ denotes the interior of the convex hull of $A\subset \mathbb{R}$).
Considering Lemma \ref{l.action_on_section_is_homothetic}, we get from the minimality criterion (Lemma \ref{lem.minimalitycriteria}) that $\varphi$ is minimal. Proposition \ref{prop.minimalimpliesfocal} then implies that $\varphi$ is an $\mathbb{R}$-focal action. Lemma \ref{l.cocyclepreorder} gives that $\varphi$ is faithful.
To prove the second part of the statement we show that the preorder $\leq_\Lambda$ can be read from the positive conjugacy class of $\varphi$. For this, given $g\in G_+$, let $m(g)$ denote the leftmost point of the support of $g$ (namely, $m(g)=\sup\{x\in X:g(x)\neq x\}$). Given $\lambda\in\Lambda$ we consider a sequence $(g_n)\subseteq G(X;A,\langle\lambda\rangle)\cap G_+$ satisfying:\begin{itemize}
\item $D^-g_n(m(g_n))=\lambda$ and
\item $m(g_n)\to b$ as $n\to+\infty$.
\end{itemize}
Note that this sequence exists since $G=G(X;A,\Lambda)$ is a Bieri--Strebel group.
Similarly as in the proof of Lemma \ref{l.action_on_section_is_homothetic}, we compute that for a given section $\mathsf t\in \mathsf S$, for every $n\in \mathbb N$ we have
\[
g_n.\mathsf t(x)=\mathsf t(x)\quad\text{for every }x\in (m(g_n),b)
\]
and
\[
g_n.\mathsf{t}(m(g_n))=\lambda\, \mathsf t(m(g_n))\pmod{\Lambda_0}.
\]
Assume first that $\lambda\in\Lambda_0$, so that $(g_n)\subseteq H=[1]_{\preceq_+}$. Since $\mathsf e\in\mathsf{Fix}^{\Phi}(H)$ it holds that $\iota(\mathsf e)$ is a common fixed point of the family $\{\varphi(g_n):n\in\mathbb N\}$. Consider now the case where $\lambda\gneq_\Lambda 1$. Note that in this case, for every $\mathsf{t}\in\mathsf{S}$ and $n\in \mathbb N$ large enough, the above computation gives $x_{g_n.\mathsf{t}}=m(g_n)$ and $(g_n.\mathsf{t})(x_{g_n.\mathsf{t}})\gneq_\Lambda 1$. In particular we have that for every $\mathsf{t}\in\mathsf{S}$, $g_n.\mathsf{t}\to+\infty$ as $n\to+\infty$ which implies that $\varphi(g_n)(\xi)\to+\infty$ for every $\xi\in\mathbb{R}$. Analogously, in the case $\lambda\lneq_\Lambda 1$ it holds that $\varphi(g_n)(\xi)\to-\infty$ for every $\xi\in\mathbb{R}$.
As such qualitative properties of the action $\varphi$ are invariant under positive conjugacy, we deduce that the positive conjugacy class of $\varphi$ determines the preorder $\le_\Lambda$.
\end{proof}
\begin{rem} Note that the existence of $\mathbb{R}$-focal actions for Bieri--Strebel groups also follows from Corollary \ref{cor.germabelian}. However, the examples in which $\leq_\Lambda$ has non-trivial residue cannot be obtained from Corollary \ref{cor.germabelian}. Indeed, in Theorem \ref{t-BBS} we show that all exotic actions of Bieri--Strebel groups of the form $G(\mathbb{R};A,\Lambda)$ (with some finitary assumption on $A$ and $\Lambda$) arise from cocyle preorders.
\end{rem}
\begin{rem}\label{r.opposite} Given a preorder $\leq\in\LPO(\Lambda)$ we can consider its opposite preorder, namely the preorder $\leq^{op}\in\LPO(\Lambda)$ such that $P_{\leq}=P_{\leq^{op}}^{-1}$. Note that in the case $\leq_1,\leq_2\in\LPO(\Lambda)$ are one opposite of the other, their associated right (respectively, left) jump preorders are also opposite to each other. Therefore their dynamical realizations are conjugate (although they are not positively conjugate).
\end{rem}
\begin{cor}\label{cor.hyperbieri} Let $G=G(X;A,\Lambda)$ be a countable Bieri--Strebel group and let $\leq_\Lambda\in\LPO(\Lambda)$ be a preorder with non-trivial residue. Consider $\preceq\in\LPO(G)$ the right or left jump preorder associated to $\leq_\Lambda$ and $\varphi$ its dynamical realization. Then, the action of $\varphi([G_c,G_c])$ on $\mathbb{R}$ is minimal.
\end{cor}
\begin{proof}
We focus on the right jump preorder, the other case being analogous. We keep notation as above.
We write $\Omega=[G_c,G_c].\mathsf e$ for the orbit of the trivial section. We argue similarly as in the proof of Proposition \ref{prop.cocicloRfocal}. After (the proof of) Lemma \ref{l.cocyclepreorder}, the dynamical realization of the action of $[G_c,G_c]$ on $\Omega$ is faithful. Also, by Lemma \ref{l.action_on_section_is_homothetic} and Remark \ref{r.action_is_homothetic} therein, we have that the stabilizer of $\mathsf e$ in $\Phi([G_c,G_c])$ is of homothetic type. As the action on $\Omega=[G_c,G_c].\mathsf e$ is transitive, we can apply Proposition \ref{p.minimalitycriteria} and get that the dynamical realization, which coincides with the restriction of $\varphi:G\to \homeo_0(\mathbb{R})$ to $[G_c,G_c]$, is minimal.
\end{proof}
\begin{rem}
We point out that, although Thompson's group $F$ does not satisfy the hypothesis of Corollary \ref{cor.hyperbieri}, we show in Theorem \ref{t-F-hyperexotic} how to construct exotic actions of $F$ with similar dynamical properties.
\end{rem}
\begin{rem}
Recal that after Proposition \ref{p-from-focal-to-trees}, every minimal $\mathbb{R}$-focal action of a countable group can be obtained as the dynamical realization of an action on a planar directed tree. The result above show that there are cases for which such action cannot be chosen to be \emph{simplicial}.
Indeed, assume that $G=G(X;A,\Lambda)$ and $\leq_\Lambda\in\LPO(\Lambda)$ satisfy the assumptions of Corollary \ref{cor.hyperbieri}. Let $\varphi$ denote the dynamical realization of a corresponding jump preorder. Then, by Proposition \ref{p-micro-normal}, every non-trivial normal subgroup of $G$ contains $[G_c,G_c]$ and therefore, by Corollary \ref{cor.hyperbieri}, the restriction of $\varphi$ to any non-trivial normal subgroup of $G$ is minimal. We conclude using Proposition \ref{p-focal-simplicial}.
\end{rem}
\subsection{A construction of micro-supported $\mathbb{R}$-focal actions}
Recall from Proposition \ref{prop.dicotomia} that if $G$ is a minimal micro-supported subgroup of $\homeo_0(\mathbb{R})$, then either $G$ is locally moving or its standard action is $\mathbb{R}$-focal.
In this subsection we give general construction of minimal micro-supported subgroups of $\homeo_0(\mathbb{R})$ whose action is $\mathbb{R}$-focal. We will use this construction to illustrate that the class of micro-supported subgroups of $\homeo_0(\mathbb{R})$ is much more flexible than the class of locally moving groups: we can find groups that admit uncountably many pairwise non-semi-conjugate faithful micro-supported actions (in contrast with Rubin's theorem for locally moving groups, see Corollary \ref{cor.unique}). Also many of these examples can even be chosen to be of class $C^1$ (in contrast with Theorem \ref{t-lm-C1}). These groups are described as groups of automorphisms of planar directed (simplicial) trees, by adapting the classical construction of Burger and Mozes \cite{BuMo}, and the related groups defined by Le Boudec \cite{LB-ae}.
\subsubsection{Groups with many micro-supported actions} We say that a pair $(A,a_0)$ is a \emph{marked alphabet} if $A$ is a set and $a_0\in A$. Then, denote by $\mathsf{S}\subseteq A^\mathbb Z$ the set of sequences with values in $A$ which take the constant value $a_0$ in all but finitely many terms. Following the notation as in \S\ref{subsubsec.simplicialfocal}, denote by $\mathsf{S}_n$ the truncations to $\mathbb Z_{j\geq n}$ of the elements in $\mathsf{S}$. Also, given a sequence $(t_j)_{j\geq n}$ say that $(t_j)_{j\geq n+1}$ is its successor. This defines a simplicial directed tree $(\mathbb{T}_A,\triangleleft)$ whose focus is defined by the successive truncations of the constant path $\s$ with value $a_0$.
By abuse of notation, we will identify $\mathbb T_A$ with its set of vertices $\bigsqcup_{n\in \mathbb Z}\mathsf{S}_n$.
Recall that for a vertex $v\in \mathbb T_A$, we denote by $E^-_v$ the set of edges below $v$ (opposite to $\omega$). For $v=(t_j)_{j\ge n}$, the set $E^-_v$ is naturally identified with the alphabet $A$, considering the labeling \[\dfcn{j_v}{E^-_v}{A}{[(t_j)_{j\geq {n-1}},(t_j)_{j\geq n}]}{t_{n-1}}.\]
Note also that every $g\in \Aut(\mathbb T, \triangleleft)$ induces a bijection between $E^-_v$ and $E^-_{g.v}$. We write $\sigma_{g, v}:= j_{g(v)}\circ g\circ j_v^{-1}$ for the induced permutation of $A$. Observe that we have the cocycle relation
\begin{equation}\label{eq:cocycle-BN}
\sigma_{g,h(v)}\,\sigma_{h,v}=\sigma_{gh,v}.
\end{equation}
\begin{dfn}
Let $(A, a_0)$ be a marked alphabet, $G\subset \Bij(A)$ be a group of permutations of $A$, and let $(\mathbb T_A, \triangleleft)$ be the directed tree defined above together with the labelings $j_v$, $v\in \mathbb T_A$. We define the \emph{generalized Brin--Navas group} $\mathsf{BN}(A;G)$ as the group of all elements $g\in \Aut(\mathbb T, \triangleleft)$ such that $\sigma_{g, v}\in G$ for all $v\in \mathbb T$ and $\sigma_{g, v}=\mathsf{id}$ for all but finitely many $v\in \mathbb T$. \end{dfn}
\begin{rem}The name comes from the fact that the Brin--Navas group considered in Example~\ref{e-BN} is isomorphic to the group $\mathsf{BN}(\mathbb Z;G)$ where $G\subset\Bij(\mathbb Z)$ is the group of translations of the integers.
\end{rem}
We need to define a suitable generating set. For this, given a vertex $v\in \mathbb T_A$ denote by $G_v\subset \mathsf{BN}(A;G)$ the subgroup of all $g\in \mathsf{BN}(A;G)$ which fix $v$ and such that $\sigma_{g, w}=\mathsf{id}$ for $w\neq v$ (clearly $G_v\cong G$). We then consider an extra generator defined as follows. Note that the shift map $\sigma:\mathsf S\to \mathsf S$, which sends a bi-infinite sequence $(t_j)_{j\in \mathbb Z}$ to $(t_{j-1})_{j\in \mathbb Z}$ naturally acts on the set of truncated sequences $\bigsqcup_{n\in \mathbb Z}\mathsf S_n$ preserving the successor relation. Thus it defines an automorphism
$f_0\in\Aut(\mathbb T_A,\triangleleft)$. It is direct to check that $f_0$ is a hyperbolic element in $\mathsf{BN}(A;G)$ whose axis consists on the geodesic $(w_n)_{n\in \mathbb Z}$ with $w_n=(a_0)_{j\geq n}$, and that moreover $\sigma_{f_0,v}=\mathsf{id}$ for every vertex $v\in \mathbb T_A$.
With this notation set, we have the following.
\begin{lem}\label{l-BN-generators}
Let $(A,a_0)$ be a marked alphabet, and assume that $G\subset \Bij(A)$ acts transitively on $A$. Then the group $\mathsf{BN}(A;G)$ is generated by $G_{w_0}$ and $f_0$. In particular it is finitely generated as soon as $G$ is so.
\end{lem}
\begin{proof} Write $\Gamma=\langle G_{w_0},f_0\rangle$ for the subgroup of $\Aut(\mathbb T_A,\triangleleft)$ generated by $G_{w_0}$ and $f_0$. We first observe the following.
\begin{claim}
For every vertices $v_1,v_2\in\mathbb T_A$, there exists $g\in\Gamma$ such that $g.v_1=v_2$ and $\sigma_{g,v}=\mathsf{id}$ for every vertex $v\triangleleft v_1$.
\end{claim}
\begin{proof}[Proof of claim]
First notice that, by composing with powers of $f_0$ and using the cocycle relation \eqref{eq:cocycle-BN}, we can assume that both $v_1$ and $v_2$ belong to $\mathsf{S}_0$.
Similarly, we can also assume that $v_1=w_0=(a_0)_{n\ge 0}$; write $v_2=(t_n)_{n\geq 0}$. Since $G$ acts transitively on $A$ there exists a sequence $(h_n)_{n\geq 0}$ in $G$ such that $h_n(a_0)=t_n$. Moreover, since $t_n=a_0$ for $n$ large enough, we can take $h_n$ to be the identity for $n$ large enough. By abuse of notation denote by $h_n\in G_{w_0}$ the element with $\sigma_{h_n,w_0}=h_n$. Thus, the product $g:=\prod_{n\geq 0}(f_0^nh_nf_0^{-n})$ is actually a finite product and thus defines an element of $\Gamma$, which moreover satisfies $\sigma_{g,v}=\mathsf{id}$ for every $v\in \bigsqcup_{n<0}\mathsf S_n$. It follows directly from the choices that $g.w_0=v_2$. This proves the claim.
\end{proof}
Take a vertex $v_0\in\mathbb T_A$, by the previous claim we can take $g\in \Gamma$ so that $g.v_0=w_0$ and $\sigma_{g,v}=\mathsf{id}$ for every $v\triangleleft v_0$. Then it is direct to check that $g^{-1}G_{w_0}g=G_{v_0}$. This shows that $G_{v_0}\subseteq \Gamma$ for every vertex $v_0\in\mathbb T_A$. Finally, given $g\in \mathsf{BN}(A;G)$ write
\[C(g):=|\{v\in \mathbb T_A:\sigma_{g,v}\neq \mathsf{id}\}|.\]
Notice that when $C(g)=0$ then $g$ is a power of $f_0$, hence it belongs to $\Gamma$. Take an element $g\in \mathsf{BN}(G)$ with $C(g)> 0$, and a vertex $v\in \mathbb T_A$ so that $\sigma_{g,v}\neq \mathsf{id}$. Then we can find $h\in G_{v}$ satisfying $\sigma_{h,v}=\sigma_{g,v}$. Then we have $C(gh^{-1})=C(g)-1$. By repeating this procedure finitely many times we can find $h'\in\Gamma$ so that $C(gh')=0$. We deduce that $g$ belongs to the group $\Gamma$.
\end{proof}
Notice that for every order $<$ on the alphabet $A$ there exists a unique planar order $\prec$ on $(\mathbb T_A,\triangleleft)$ for which the maps $j_v$ are order preserving. If in addition the subgroup $G\subset\Bij(A)$ preserves $<$, the group $\mathsf{BN}(A;G)$ preserves the associated planar order $\prec$. We call this action the induced \emph{planar directed tree representation} associated with $G$ and $<$
\begin{prop}\label{prop.BNfocal} Let $(A,a_0)$ be a marked alphabet, assume that $G\subset\Bij(A)$ acts transitively on $A$ and let $<$ be an $G$-invariant total order on $A$. Then, the dynamical realization of the planar directed tree representation associated with $G$ and $<$ is a minimal, faithful and micro-supported $\mathbb{R}$-focal action of $\mathsf{BN}(A;G)$.
Moreover, for two different $G$-invariant orders on $A$, the dynamical realizations of their corresponding planar directed tree representations are not positively semi-conjugate.
\end{prop}
\begin{proof} Let $\prec$ be the planar order associated with $<$, and let $\Phi:\mathsf{BN}(A;G)\to\Aut(\mathbb T_A,\triangleleft,\prec)$ be the corresponding action. Since $\Phi$ acts transitively on the vertices of $\mathbb T_A$ and $\mathsf{BN}(A;G)$ contains a hyperbolic element, the focality of $\Phi$ follows.
Let $\xi\in\partial^\ast\mathbb T_A$ be the end defined by the sequence $(w_n)_{n\leq 0}$, and let $\mathcal{O}_\xi$ be its orbit under the associated $\mathsf{BN}(A;G)$-action on $\partial^\ast\mathbb T_A$. Since $f_0$ is a hyperbolic element with axis $(w_n)_{n\in\mathbb Z}$, it acts on $(\mathcal{O}_\xi,\prec)$ as a homothety fixing $\xi$. This allows us to apply Lemma \ref{lem.minimalitycriteria} to deduce that the dynamical realization of $\mathsf{BN}(G)\to\Aut(\mathcal{O}_{\xi},\prec)$ is minimal. In other words, the dynamical realization of $\Phi$, that we denote by $\varphi$, is minimal. On the other hand, since $G_v$ is supported on the set of points below $v$, its associated action on $\mathcal{O}_\xi$ is supported on the shadow of $v$. This implies that the support of $\varphi(G_v)$ is relatively compact, and thus by Proposition \ref{p-micro-compact} we deduce that $\varphi$ is micro-supported. Finally, the faithfulness of $\varphi$ follows from that of $\Phi$.
Given $h\in G$ and vertex $v\in \mathbb T_A$ we denote by $h_v$ the element of $G_v$ satisfying $\sigma_{h_v,v}=h$. Also, let $p\in \mathbb{R}$ be the fixed point of $\varphi(f_0)$. Then, given $h\in G$ we have
\[\varphi(h_{w_0})(p)>p\quad\text{if and only if}\quad h(a_0)>a_0.\]
Since the action of $G$ on $A$ is transitive, this implies that we can read the total order $<$ from the action $\varphi$. In particular, dynamical realizations corresponding to different $G$-invariant orders on $A$ give rise to non-positively-conjugate actions. Finally, since these dynamical realizations are minimal and not positively conjugate, they are not positively semi-conjugate.
\end{proof}
Note that by the transitivity assumption in Proposition \ref{prop.BNfocal}, the planar directed tree representation is in fact determined by a choice of a left-invariant preorder on $G$, as an abstract group, and the second part in the statement says that two different left-invariant preorders on $G$ with same residue yield non-positively-semi-conjugate actions. As a particular case, one can consider a left-invariant order on $G$, in which case we can identify the marked alphabet with $(G,1_G)$, and the subgroup $G\subset \Bij(G)$ is the group of left translations. In such case, we will simply write $\mathsf{BN}(G)$ for the group $\mathsf{BN}(G;G)$.
Recall that if $G\subset \homeo_0(\mathbb{R})$ is a locally moving group, then every faithful locally moving action of $G$ on $\mathbb{R}$ is conjugate to its standard action (by Rubin's theorem, or by Corollary \ref{cor.unique}). The groups $\mathsf{BN}(G)$ show that the this is far from being true for micro-supported subgroups of $\homeo_0(\mathbb{R})$.
\begin{cor}\label{cor.brin-navas} Let $G$ be a finitely generated group whose space of left-invariant orders is uncountable. Then $\mathsf{BN}(G)$ has uncountably many, pairwise-non-conjugate, faithful, minimal and micro-supported actions on the line.
\end{cor}
\begin{proof} After the previous discussion, we can apply Proposition \ref{prop.BNfocal} to each left order in $\LO(G)$, which gives the desired result.
\qedhere
\end{proof}
\subsubsection{Groups with many differentiable micro-supported actions} Here we extend the result given by Corollary \ref{cor.brin-navas} by showing that for some finitely generated groups $G$, we can actually get many faithful micro-supported actions of class $C^1$ of the group $\mathsf{BN}(G)$ (compare with Theorem \ref{t-lm-C1}).
\begin{thm}\label{t.C1nonrigid} There exists a finitely generated group admitting uncountably faithful minimal micro-supported actions, which are pairwise not semi-conjugate (and not semi-conjugate to a non-faithful action), and each of which is semi-conjugate to a $C^1$ action.
More precisely, the generalized Brin--Navas group $\mathsf{BN}(\mathbb Z^2)$ satisfies such properties.
\end{thm}
\begin{rem}
In fact, it seems also possible to prove that the group $\mathsf{BN}(\mathbb Z^2)$ admits minimal faithful micro-supported actions which are \emph{conjugate} to a $C^1$ action and pairwise not semi-conjugate, but since this is more technical we will content ourselves of the previous statement (which suffices to disprove the analogue of Theorem \ref{t-lm-C1} for micro-supported groups).
\end{rem}
Theorem \ref{t.C1nonrigid} will be obtained as a consequence of Proposition \ref{prop.larguisima}, which is
a criterion to recognize faithful actions of $\mathsf{BN}(G)$ on the line. As a preliminary result, which may also help to follow the rather technical proof of Proposition \ref{prop.larguisima}, we work out a presentation of $\mathsf{BN}(G)$, analogue to the one for $\mathsf{BN}(\mathbb Z)$ appearing in Example \ref{e-BN}.
So let $G$ be a finitely generated group, and fix a presentation \[G=\langle g_1,\ldots,g_n\mid r_\nu(g_1,\ldots,g_n)\, (\nu\in\mathbb N)\rangle.\]
We will assume for simplicity that the generating set is symmetric and that for every distinct $i,j\in \{1,\ldots,n\}$, the generators $g_i$ and $g_j$ represent distinct non-trivial elements in $G$. The free product $G\ast \mathbb Z$ admits the presentation
\[G\ast\mathbb Z=\langle f,g_1,\ldots,g_n\mid r_\nu(g_1,\ldots,g_n)\, (\nu\in\mathbb N)\rangle.\]
It is convenient to introduce the following more redundant presentation of $G\ast \mathbb Z$, which corresponds to applying a (multiple) Tietze transformation to the previous one. For $m\in \mathbb Z$, write $\mathcal{G}_m=\{g_{i,m}:i\in\{1,\ldots,n\}\}$ and $\mathcal G=\bigcup_{m\in \mathbb Z}\mathcal G_m$.
Then, we get the presentation \begin{equation}\label{eq.presentation-gamma}G\ast\mathbb Z=\left\langle f_0,\mathcal{G}\,\middle\vert\, fg_{i,m}f^{-1}=g_{i,m+1} \text{ }(i\in \{1,\ldots,n\},m\in \mathbb Z),\text{ }r_\nu(g_{1,0},\ldots,g_{n,0})\, (\nu\in\mathbb N)\right\rangle.\end{equation}
Write $\mathcal{R}_0$ for the set of relators in the previous presentation. As in the proof of Lemma \ref{prop.BNfocal}, given $g\in G\subset\Bij(G)$ and $v\in\mathbb T_G$ we denote by $g_v$ the element in $\mathsf{BN}(G)$ fixing $v$ such that $\sigma_{g,v}=g$ and $\sigma_{g,w}=\mathsf{id}$ for every $w\neq v$. Then, we denote by $\Psi_0:G\ast\mathbb Z\to \mathsf{BN}(G)$ the morphism such that $\Psi_0(f)=f_0$ and $\Psi_0(g_{i,m})=f_0^m(g_i)_{w_0}f_0^{-m}$ for every $i\in\{1,\ldots,n\}$, and $m\in \mathbb Z$ (as before, we write $w_0=(1_G)_{n\ge 0}$).
In order to complete the set of relations for the desired presentation of $\mathsf{BN}(G)$, we need to study the support of some elements.
\begin{lem}\label{l.relations1} With notation as above,
fix $m\in \mathbb Z$. Then for every $g\in \langle\mathcal G_m\rangle \setminus \{1\}$ and $g_1,g_2\in \bigcup_{q<m} \mathcal G_q$ we have that the commutator $[g_1,gg_2g^{-1}]$ is in the kernel of $\Psi_0$.
\end{lem}
\begin{proof}
We will show that for any such choices of elements, we have that $g_1$ and $gg_2g^{-1}$ have disjoint support in $\mathbb T_G$ and hence commute. On the one hand, for every $k\in \mathbb Z$, and $h\in \mathcal G_k$, the support of $\Psi_0(h)$ is contained in $\mathbb T_G^{w_k}$, where $w_m=(1_G)_{n\ge k}$. On the other hand, for any $h\in\bigcup_{q<m}\mathcal{G}_q$, we have that the support of $\Psi_0(ghg^{-1})$ is contained in the subtree $\mathbb T_G^{v_{g,m-1}}$, where $v_{g,m-1}\in\mathbb T_G$ is the vertex corresponding to the sequence $(t_n)_{n\geq m-1}$ such that $t_0=1_G$ for $n\geq m$ and $t_{m-1}=g$. In particular, these two remarks apply respectively to the elements $g_1$ and $g_2$ from the statement, so that the images $\Psi_0(g_1)$ and $\Psi_0(gg_2g^{-1})$ have disjoint supports.
\end{proof}
Write $\mathcal{R}_1$ for the set of all the commutation relators from Lemma \ref{l.relations1} and set \begin{equation}\label{eq.presentation-gamma2}\Gamma=\langle f,\mathcal{G}\mid\mathcal{R}_0,\mathcal{R}_1\rangle.\end{equation}Then, we are ready to state the following:
\begin{prop}\label{lem.presentation} The map $\Psi:\Gamma\to \mathsf{BN}(G)$ induced by $\Psi_0$ is an isomorphism.
\end{prop}
Before proving Proposition \ref{lem.presentation} we need to fix some notation and state a technical lemma that will also be used later for Proposition \ref{prop.larguisima}, and whose proof is postponed. As in the proof of Lemma \ref{l.relations1}, given $g\in G$ and $m\in\mathbb Z$ denote by $v_{g,m}\in\mathbb T_G$ the vertex $(t_n)_{n\geq m}$ such that $t_n=1_G$ for $n>m$ and $t_m=g$; in particular $w_m=v_{1,m}$.
Note that after the conjugation relations in $\mathcal R_0$, the subgroup $H:=\langle \mathcal G\rangle$ is the normal closure of $\mathcal G_0$ in $\Gamma$, and the quotient $\Gamma/H$ is generated by the image of $f$. Given $\gamma\in H$, we denote by $\|\gamma\|_{\mathcal{G}}$ its word-length with respect to the generating system $\mathcal{G}$
\begin{lem}\label{sublem.technicalalternativo}\label{sublem.technical} Take a non-trivial element of $H$ written as $\gamma=\gamma_1\cdots\gamma_k$ with $\gamma_j\in\mathcal{G}_{m_j}$ for $j\in\{1,\ldots,k\}$, and write $M=\max\{m_j:j\in\{1,\ldots,k\}\}$. Assume that $\Psi(\gamma)(w_m)=w_m$ for some $m<M$.
Then, there exist $h_1,\ldots,h_l\in\left \langle\bigcup_{m<M}\mathcal{G}_{m}\right \rangle$ and pairwise distinct elements $f_1,\ldots,f_l\in\langle\mathcal{G}_M\rangle$, such that
\begin{equation}\label{eq.sublem}
\gamma=(f_1h_1f_1^{-1})\cdots (f_lh_lf_l^{-1})
\end{equation}
and $\|h_i\|_{\mathcal{G}}<k$ for every $i\in \{1,\ldots,l\}$.
\end{lem}
\begin{proof}[Proof of Proposition \ref{lem.presentation}] First notice that, by Lemma \ref{l-BN-generators}, $\Psi$ is surjective. To prove injectivity of $\Psi$, suppose by contradiction that $\ker \Psi$ is non-trivial. Consider the focal germ representation $\tau:\mathsf{BN}(G)\to\Germ(b)\cong \mathbb Z$ and notice that $\tau$ vanishes at $G_{w_0}=\Psi(\mathcal G_0)$ and that $\tau(f_0)$ is non-trivial. As the quotient of $\Gamma$ by the normal closure $H$ of $\mathcal G_0$ is generated by the image of $f$, we deduce that $\ker \Psi\subseteq H$.
In particular, the word-length $\|\cdot\|_{\mathcal G}$ is defined on the kernel of $\Psi$, so that we can consider a non-trivial element $\gamma\in \ker \Psi$ of minimal word-length.
We write $\gamma=\gamma_1\cdots\gamma_k$ with $k=\|\gamma\|_{\mathcal{G}}$ and $\gamma_j\in \mathcal G_{m_j}$ for every $j\in \{1,\ldots,k\}$. As $\Phi(\gamma)$ acts trivially, in particular it fixes every vertex of the form $w_m$, so that we can apply Lemma \ref{sublem.technical} to the factorization
$\gamma=\gamma_1\cdots\gamma_k$ and obtain a decomposition as in \eqref{eq.sublem} with $\|h_i\|_{\mathcal{G}}<\|\gamma\|_{\mathcal{G}}$ for $i\in\{1,\ldots,l\}$. Keeping the same notation as in Lemma \ref{sublem.technical}, we observe that the support of $\Psi(f_ih_if_i^{-1})$ is contained in $\mathbb T_G^{v_{f_i,M-1}}$ for $i\in\{1,\ldots,l\}$. Thus, since $f_i\neq f_j$ for $i\neq j$, we get that different factors in the factorization of $\gamma$ above have disjoint support. We deduce that every such factor must be in the kernel of $\Psi$, and therefore also every element $h_i$. However, as the word-length of every such element is less than $\|\gamma\|_{\mathcal{G}}$, the minimality assumption on $\gamma$ implies that every $h_i$ is trivial, contradicting the choice of $\gamma$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{sublem.technicalalternativo}]
In order to simplify notation, from here and until the end of the proof, we will write $\prod_{j\in E}\alpha(j):=\alpha(i_1)\cdots \alpha(i_k)$, for any function $\alpha:E\to\Gamma$, with $E=\{i_1,\ldots,i_k\}\subseteq\mathbb N$ with $i_1<\cdots<i_k$. We will also write
$\sigma_{g}$ instead of $\sigma_{\Psi(g),w_M}$, for every $g\in \Gamma$. Note that $\sigma_{g}=\mathsf{id}$ for every $g\in\langle\bigcup_{m<M}\mathcal{G}_m\rangle$.
For given $j\in \{1,\ldots,k\}$, we let $i_j$ be the index such that $\gamma_j=g_{i_j,m_j}$.
Notice that $\Psi(\gamma_j)(w_M)=w_M$ for every $j\in \{1,\ldots,k\}$, and
in particular $\Psi(\gamma)(w_M)=w_M$. Using the cocycle relation \eqref{eq:cocycle-BN}, we have
\begin{equation}\label{eq:sigma-prod}
\sigma_\gamma=\prod_{j=1}^k\sigma_{\gamma_j}.
\end{equation}
On the other hand, since $\Psi(\gamma)(w_s)=w_s$ for some $s<M$ and the action of $G\subset\Bij(G)$ on $G$ is free, we conclude that $\sigma_{\gamma}=\mathsf{id}$. Since only factors in $\mathcal G_M$ give non-trivial factors in the product \eqref{eq:sigma-prod}, writing $P=\{j\in\{1,\ldots,k\}:\gamma_j\in \mathcal G_M\}$, we get the equality
\[
\mathsf{id}=\prod_{j\in P}\sigma_{\gamma_j}.
\]
Note that for $j\in P$, the permutation $\sigma_j$ corresponds to the left translation by $g_{i_j}$, whence we get $\prod_{j\in P}g_{i_j}=1_G$. This implies
\begin{equation}\label{eq.identity} \prod_{j\in P}\gamma_j=1_{\Gamma}.\end{equation}
Next, for every $j\in \{1,\ldots,k\}$, consider the product in $\langle \mathcal G_M\rangle$ given by
\[f_j:=\prod_{l\in P,\,l\le j}\gamma_l.\]
Set $Q:=\{1,\ldots,k\}\setminus P$ and notice that after the relation \eqref{eq.identity} we can write
\begin{equation}\label{eq.telescopic}\gamma=\prod_{j\in Q}\left(f_j\gamma_jf_j^{-1}\right).
\end{equation}
Note that as $f_j\in \langle \mathcal{G}_M\rangle$ and $\gamma_j\in\langle\mathcal{G}_{m_j}\rangle$ with $m_j<M$ for every $j\in Q$, we deduce from Lemma \ref{l.relations1} that we can make commute any two factors $f_i\gamma_if_i^{-1}$ and $f_j\gamma_jf_j^{-1}$ in the product in \eqref{eq.telescopic}, as soon as $f_i\neq f_j$. Rearranging factors in this way, we can write \begin{equation}\label{eq.thefinal} \gamma=\prod_{j\in Q_0}\left(f_j\left (\prod_{l\in Q_0^j}\gamma_l\right )f_{j}^{-1}\right),\end{equation}
where $Q_0\subseteq Q$ is a section of the map $Q\to \Gamma$ given by $j\mapsto f_j$, and for $j\in Q_0$ we set $Q_0^j:=\{l\in Q_0:f_l=f_j\}$. We claim that the decomposition in \eqref{eq.thefinal} is the one that we are looking for. First notice that, by the choice of $Q_0$, $f_i\neq f_j$ whenever $i, j$ are different indices in $Q_0$. Secondly for every $j\in Q_0$, by definition of $Q_0$ and $Q_0^j$, the element $h_j:=\prod_{l\in Q_0^j}\gamma_l$ belongs to $\left \langle\bigcup_{m< M}\mathcal{G}_m\right \rangle$ and clearly satisfies $\|h_j\|_{\mathcal{G}}\le |Q_0^j|<k$.
\end{proof}
Consider now, for a left-invariant order $<$ on a group $G$, the directed tree representation $\Phi:\mathsf{BN}(G)\to\Aut(\mathbb T_G,\triangleleft,\prec)$ associated with $G$ and $<$; let $\varphi:\mathsf{BN}(G)\to\homeo_0(\mathbb{R})$ be its corresponding dynamical realization. Pursuing the discussion from the proof of Proposition \ref{prop.BNfocal}, we extract some properties of $\varphi$, which characterize it up to (positive) semi-conjugacy.
First recall that $G_{w_0}\subseteq \mathsf{BN}(G)$ is a subgroup supported on the subset of points below the vertex $w_0$. This implies that $J_0:=\suppphi(G_{w_0})$ is a relatively compact interval. Since $\Phi(f_0)$ is a hyperbolic element, the homeomorphism $\varphi(f_0)$ is a homothety (see Proposition \ref{prop.dynclasselements0}).
For $n\in \mathbb Z$, write $J_n:=f_0^{n}.J_0$; since $f_0^{-1}.w_0\triangleleft w_0$, we have the inclusion $J_{-1}=f_0^{-1}.J_0\Subset J_0$. Finally, since the action of $G$ on itself by left translations is free, we get that $G_{w_0}$ acts freely on $E_{w_0}^-$, which implies $g.J_{-1}\cap J_{-1}=\emptyset$ for every $g\in G_{w_0}\setminus\{1\}$. Moreover, the total order $<$ on $G$ (which coincides with the planar order $\prec^{w_0}$ on $E_{w_0}^-$) can be read from the action: \begin{center}$g>1$ $\Leftrightarrow$ $g.x>x$ for some $x\in J_{-1}$ (equivalently $\forall x\in J_{-1}$).\end{center} Summarizing we have the following:
\begin{enumerate}[label=(\alph*)]
\item\label{i-BrinNavas} $\suppphi(G_{w_0})=:J_0$ consists on a relatively compact interval;
\item\label{ii-BrinNavas} $\varphi(f_0)$ is a homotethy satisfying $J_{-1}\Subset J_0$, with $J_{-1}:=f_0^{-1}.J_0$;
\item\label{iii-BrinNavas} $g.J_{-1}\cap J_{-1}=\emptyset$ for every $g\in G_{w_0}\setminus \{1\}$;
\item\label{iv-BrinNavas} given $x\in J_{-1}$ and $g\in G_{w_0}$, it holds that $g.x>x$ if and only if $g>1$.
\end{enumerate}
For the next statement, recall that $\mathsf{BN}(G)$ is isomorphic to the group $\Gamma$ which, in turn, can be written as the quotient $\Gamma=G\ast\mathbb Z/\langle\langle\mathcal{R}_1\rangle\rangle$ (following the notation in \eqref{eq.presentation-gamma} and \eqref{eq.presentation-gamma2}, see Proposition \ref{lem.presentation}). We denote by $\pi:G\ast\mathbb Z\to\Gamma$ the corresponding projection. We also define the \emph{height} of an element $\gamma\in H=\langle\mathcal{G}\rangle$ as
\begin{equation}\label{eq.height}\mathsf{ht}(\gamma):=\inf\left \{n\in\mathbb Z:\gamma\in\left \langle\bigcup_{m\leq n}\mathcal{G}_m\right \rangle\right \}.\end{equation} Notice that the $\mathsf{ht}(\gamma)>-\infty$ for every $\gamma\in H\setminus\{1\}$. Indeed, if it were not the case, its image under the isomorphism $\Psi$ in Proposition \ref{lem.presentation} would be trivial, which is not the case.
\begin{prop}\label{prop.larguisima} Let $(G,<)$ be a finitely generated left-ordered group. Consider the free product $G\ast\mathbb Z$, and denote by $f$ a generator of its cyclic factor. Consider also an action $\varphi_0:G\ast\mathbb Z\to\homeo_0(\mathbb{R})$ and assume that it satisfies conditions \ref{i-BrinNavas} to \ref{iv-BrinNavas} above with $\langle \mathcal G_0\rangle$ and $f$ instead of $G_{w_0}$ and $f_0$, respectively. Then, $\varphi_0$ factors through the quotient $\pi:G\ast\mathbb Z\to\Gamma$ inducing an action $\varphi_1:\Gamma\to \homeo_0(\mathbb{R})$ which is (positively) semi-conjugate to the dynamical realization of the directed tree representation associated with $G\subset\Bij(G)$ and $<$.
\end{prop}
\begin{proof} After Proposition \ref{lem.presentation}, in order to show that $\varphi_0$ factors through the projection $\pi$ we need to check that the elements in $\mathcal{R}_1$ belong to the kernel of $\varphi_0$. That is, we need to check that the elements of the form $[g_{1},gg_{2}g^{-1}]$ are in the kernel of $\varphi_0$, whenever $g\in\langle\mathcal{G}_m\rangle\setminus\{1\}$ and $g_1,g_2\in \bigcup_{q<m}\mathcal{G}_q$. We closely follow the proof of Lemma \ref{l.relations1}.
On the one hand, for every $q<m$ and $h\in \mathcal G_q$, there exists $h'\in \mathcal G_0$ such that $h=f^qh'f^{-q}$, so that by condition \ref{i-BrinNavas}, we have that the support of $\varphi_0(h)$ is contained in $J_q:=f_0^q.J_0$, and thus in $J_m$ after condition \ref{ii-BrinNavas} (and the same argument for $q=m$).
On the other hand, for any $g\in \langle\mathcal{G}_m\rangle\setminus\{1\}$, condition \ref{iii-BrinNavas} gives that $g.J_{m-1}\cap J_{m-1}=\emptyset$. Thus, the support of $\varphi_0(gg_{2}g^{-1})$ is disjoint from $J_{m}$. Putting this all together we get that the support of $\varphi_0(g_{1})$ and that of $\varphi_0(gg_{2}g^{-1})$ are disjoint, as desired.
As in the statement, denote by $\varphi_1:\Gamma\to\homeo_0(\mathbb{R})$ the action induced by $\varphi_0$, and let $\mathcal{I}$ be the orbit of $J_0$ under $\varphi_1(\Gamma)$.
Let also $\Phi:\Gamma\to\Aut(\mathbb T_G,\triangleleft,\prec)$ be the directed tree representation associated with $G$ and $<$ and denote by $\varphi$ the dynamical realization of $\Phi$. Our goal is to show that $\varphi$ and $\varphi_1$ are semi-conjugate. This will be a direct consequence of the following statement.
\begin{claim}
The family of intervals $\mathcal I$ is a CF-cover, which determines a simplicial planar directed tree which is order-isomorphic to $(\mathbb T_G,\triangleleft,\prec)$, via a $\Gamma$-equivariant isomorphism.
\end{claim}
\begin{proof}[Proof of claim]
We consider the map
\[\dfcn{F}{\mathbb T_G}{\mathcal{I}}{g.w_0}{g.J_0}\]
and then prove that it gives the desired $\Gamma$-equivariant order-isomorphism.
To simplify notation, given $I_1,I_2\in\mathcal{I}$, we write $I_1<I_2$ if $\sup I_1\le \inf I_2$. Also, given two vertices $v_1,v_2\in\mathbb T_G$, we write $v_1\prec v_2$ if for every points $\xi_1\in\partial U_{v_1}$ and $\xi_2\in\partial U_{v_2}$ in the shadows, we have $\xi_1\prec\xi_2$.
We want to show that the map $F$ is well-defined, $\Gamma$-equivariant, and satisfies the following conditions:
\begin{enumerate}
\item\label{condFi} $v_1\triangleleft v_2$ implies $F(v_1)\subset F(v_2)$,
\item\label{condFii} $v_1\prec v_2$ implies $F(v_1)<F(v_2)$
\end{enumerate}
To see that $F$ is well-defined, we need to check that $\mathsf{Stab}^{\Phi}(w_0)\subseteq \mathsf{Stab}^{\varphi_1}(J_0)$.
Given $\gamma\in\langle\mathcal{G}\rangle$ consider its height $\mathsf{ht}(\gamma)$ defined as in \eqref{eq.height}. Notice that if $\mathsf{ht}(\gamma)\leq 0$ then the support of $\gamma$ is contained in $\bigcup_{r\leq 0}J_r$ and therefore $\gamma\in\mathsf{Stab}^{\varphi_1}(J_0)$. In order to show the inclusion between the stabilizers we claim that, every $\gamma\in\mathsf{Stab}^{\Phi}(w_0)$ can be written as
\begin{equation}\label{eq:decomp_stab}
\gamma=\gamma_1\gamma_2\quad\text{with }\gamma_1\in \mathsf{Stab}^{\varphi_1}(J_0)\text{ and }\mathsf{ht}({\gamma_2})<\mathsf{ht}({\gamma}).
\end{equation}
To obtain a decomposition as in \eqref{eq:decomp_stab}, first note that, in the case $\mathsf{ht}(\gamma)\leq 0$ we are done. Suppose it is not the case, and write $\gamma=\gamma_1\cdots\gamma_k$ with
\[\max\{\mathsf{ht}(\gamma_j):j\in \{1,\ldots,k\}\}=\mathsf{ht}(\gamma)>0.\]
Then, since $\gamma.w_0=w_0$ and $0<\mathsf{ht}(\gamma)$ we are in condition to apply Lemma \ref{sublem.technical} to the decomposition $\gamma=\gamma_1\cdots\gamma_k$. Thus, we can write $\gamma=(f_1h_1f_1^{-1})\cdots(f_lh_lf_l^{-1})$ with $f_1,\ldots,f_l\in\langle\mathcal{G}_{\mathsf{ht}(\gamma)}\rangle$ such that $f_i\neq f_j$ for $i\neq j$, and $\mathsf{ht}({h_i})<\mathsf{ht}({\gamma})$ for $i\in \{1,\ldots,l\}$. As discussed above, for every $i\in \{1,\ldots,l\}$ such that $f_i\neq 1_\Gamma$, we have that the support of $\varphi_1(f_ih_if_i^{-1})$ is disjoint from $J_{\mathsf{ht}(\gamma)-1}$ and, as a consequence, disjoint from $J_0$. Thus, if $f_i\neq 1_\Gamma$ for every $i\in\{1,\ldots,l\}$ we are done. Suppose it is not the case, and that for some $i$ we have $f_i=1_\Gamma$. In this case, by applying the commutation relations in $\mathcal{R}_1$ from Lemma \ref{l.relations1}, we can assume that $f_l=1_\Gamma$ and we set $\gamma_1=(f_1h_1f_1^{-1})\cdots(f_{l-1}h_{l-1}f_{l-1}^{-1})$ and $\gamma_2=h_{l}$.
Notice that, as we argued in the previous case, the element $\gamma_1$ fixes $J_0$. Finally notice that, by the choice from Lemma \ref{sublem.technical}, we have $\mathsf{ht}({h_l})<\mathsf{ht}(\gamma)$. This gives the desired decomposition as in \eqref{eq:decomp_stab}.
By applying the decomposition as in \eqref{eq:decomp_stab} finitely many times, we get a factorization $\gamma=\delta_1\cdots\delta_r$ where $\delta_i$ is in the stabilizer of $J_0$ for $i\in\{1,\ldots,r-1\}$, and $\mathsf{ht}({\delta_r})\leq 0$. Finally, since $\mathsf{ht}({\delta_r})\leq 0$, we also have that $\delta_r$ is in the stabilizer of $J_0$ and therefore the inclusion between the stabilizers follows. This gives that the map $F$ is well-defined, as wanted. Moreover, by definition of $F$, we also have that it is $\Gamma$-equivariant.
In order to prove condition \eqref{condFi}, first recall that $\mathsf{BN}(G)$ acts transitively on the vertices of $\mathbb T_G$ (see the claim in the proof of Lemma \ref{l-BN-generators}). Thus, for every vertices $v_1\triangleleft v_2$ in $\mathbb T_G$, there exists $\gamma\in \Gamma$ such that $\gamma.v_1=w_s$ for some $s\in \mathbb Z$ and thus $\gamma.v_2=w_r$ for some $r>s$.
Since the partial order $\triangleleft$ is preserved by the action $\Phi$ and the inclusion relation is preserved by the action induced by $\varphi_1$ on $\mathcal{I}$, using $\Gamma$-equivariance of $F$ we only need to check that condition (\ref{condFi}) holds when we take $v_1,v_2$ in the subset $\{w_n:n\in\mathbb Z\}$. For this consider $w_i\triangleleft w_j$, then $F(w_i)=J_i\subset J_j=F(w_j)$ as desired. To prove condition \eqref{condFii} first notice that, since condition \eqref{condFi} holds, it is enough to check the condition taking $v_1\prec v_2$ adjacent to $v_1\wedge v_2$. Notice that the relation $\prec$ on $\mathbb T_G$ is invariant under $\Phi$, and the relation $<$ on $\mathcal{I}$ is invariant under the action induced by $\varphi_1$. Following the same reasoning as for condition \eqref{condFi}, it is enough to check condition \eqref{condFii} taking $v_1,v_2$ adjacent to $w_0$. In that case, condition \eqref{condFi} follows from condition \ref{iv-BrinNavas} in the statement. Summarizing, conditions \eqref{condFi} and \eqref{condFii} are satisfied by the map $F$, as wanted.
\end{proof}
After the claim, we have that the directed tree representation $\Phi:\Gamma\to \Aut(\mathbb T_G,\triangleleft,\prec)$ is conjugate to a focal action representing $\varphi_1$ (technically speaking, its positive semi-conjugacy class, as $\varphi_1$ need not be minimal and we may be forced to consider a minimal action semi-conjugate to $\varphi_1$). This implies that $\varphi$ and $\varphi_1$ are positively semi-conjugate.
\end{proof}
\begin{proof}[Proof of Theorem \ref{t.C1nonrigid}] Given any irrational $\alpha\in\mathbb{R}\setminus\mathbb Q$, denote by $\tau^\alpha:\mathbb Z^2\to\homeo_0(\mathbb{R})$ the action by translations so that $\tau^\alpha((1,0))(x)=x+1$ and $\tau^\alpha((0,1))(x)=x+\alpha$. By the Pixton--Tsuboi examples (see \cite{PixtonTsuboi}), for each $\alpha\in\mathbb{R}\setminus\mathbb Q$, there exists an action $\varphi^\alpha:\mathbb Z^2\to\Diff^1_0(\mathbb{R})$ such that:\begin{itemize}
\item $\varphi^\alpha$ is supported on $(0,1)$;
\item the restriction $\varphi^\alpha\restriction_{(0,1)}$ is semi-conjugate to $\tau^\alpha$ and
\item $\varphi^\alpha\restriction_{(0,1)}$ has an exceptional minimal set $\Lambda_\alpha\subseteq (0,1)$.
\end{itemize}
Then, for any irrational $\alpha\in\mathbb{R}\setminus\mathbb Q$, consider an affine expanding homothety $f_\alpha:\mathbb{R}\to\mathbb{R}$ with fixed point in $((0, 1)\setminus \Lambda_\alpha$ so that $f_\alpha^{-1}((0,1))\cap \Lambda_\alpha=\emptyset$. Consider the free product $\mathbb Z^2\ast\mathbb Z$ and denote by $f_0$ a generator of the cyclic factor. Then, we define the action $\varphi^\alpha_0:\mathbb Z^2\ast\mathbb Z\to\Diff^1_0(\mathbb{R})$ so that $\varphi^\alpha_0$ coincides with $\varphi^\alpha$ on the $\mathbb Z^2$ factor and $\varphi^\alpha_0(f_0)=f_\alpha$. It is direct to check that the action $\varphi^\alpha_0$ satisfies conditions \ref{i-BrinNavas} to \ref{iv-BrinNavas} in the statement of Proposition \ref{prop.larguisima} with $G=\mathbb Z^2$ and with $<$ being the left-order $<_\alpha$ induced by $\tau^\alpha$. Then, applying Proposition \ref{prop.larguisima} we conclude that $\varphi^\alpha_0$ induces an action $\Psi^\alpha:\mathsf{BN}(\mathbb Z^2)\to \Diff^1_0(\mathbb{R})$ which is semi-conjugate to the dynamical realization of the planar directed tree representation associated with $\mathbb Z^2\subseteq\Bij(\mathbb Z^2)$ and $<_\alpha$, which is minimal, faithful, and micro-supported (in particular, any action sei-conjugate to $\Psi^\alpha$ must be faithful). On the other hand, by Proposition \ref{prop.BNfocal}, different orders $<_{\alpha_1}$ and $<_{\alpha_2}$ give rise to planar directed tree representations with non-conjugate dynamical realizations, therefore, $\Psi^{\alpha_1}$ and $\Psi^{\alpha_2}$ are not semi-conjugate.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,118 |
Q: Using ruby variable in a PowerShell script with winrm EDIT:
I would like to count files/folders of storage containers via virtual machine manager PowerShell cmdlet.
I went over similar questions, but still am struggling with syntax.
I have a ruby script that is executing a PowerShell script on a remote server.
I want to use a ruby variable within the Powershel script.
For example
path_str = "\\XXX\YYY\" #This is the ruby var
PSoutput = shell.run(" #This part is executing the PS script
$Files = Get-ChildItem -Recurse -File -Path #{path_str} | Measure-Object | %{$_.Count}" | stdout
How do I use the ruby variable path_str with the PS script?
I have tried
# {path_str}
\" " + path_str + " \"
Double quotes and single quotes
Nothing worked for me.
A: There are a few things that I see causing the issues.
*
*# is a reserved character in powershell. When you use #, anything after that is a comment.
*You are assigning the output of Get-ChildItem to $files. There will be no output from shell.run() to assign to PSoutput because output from cmdlet is getting assigned to $files.
*Get-ChildItem is a powershell specific command, not a command line / dos command that you can execute from within shell, without first calling powershell executable. (this, I am a little doubtful on but quite sure its correct).
What you can do from ruby is, should work,
PSoutput = system("powershell get-childitem -Recurse -File -Path " + #{path_str} + " | Measure-Object | % {$_.Count}")
Once the command executes, you should have a total number of files under the #path_str directory.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,689 |
\section{Introduction}
In this paper, we consider a smooth symplectic manifold $(X,\omega)$
of dimension $2n$. Let $(L,h^L)$ be a Hermitian line bundle on $X$
with a Hermitian connection
$\nabla^L : \mathscr{C}^\infty(X,L)\to
\mathscr{C}^\infty(X,T^*X\otimes L)$.
We assume that $L$ satisfies the prequantization condition:
\begin{equation}\label{e1.1}
\frac{i}{2\pi}R^L=\omega,
\end{equation}
where $R^L=(\nabla^L)^2$ is the curvature of the connection $\nabla^L$.
Let $(E, h^E)$ be a Hermitian vector bundle on $X$
with Hermitian connection $\nabla^E$ and its curvature $R^E$.
Let $g^{TX}$ be a Riemannian metric on $X$ and $\nabla^{TX}$ be
the Levi-Civita connection of $(X, g^{TX})$.
Let $J_0 : TX\to TX$ be a skew-adjoint operator such that
\begin{equation}\label{e1.2}
\omega(u,v)=g^{TX}(J_0u,v), \quad u,v\in TX.
\end{equation}
Consider the operator $J : TX\to TX$ given by
\begin{equation}\label{e1.3}
J=J_0(-J^2_0)^{-1/2}.
\end{equation}
Then $J$ is an almost complex structure compatible
with $\omega$ and $g^{TX}$, that is, $g^{TX}(Ju,Jv)=g^{TX}(u,v)$,
$\omega(Ju,Jv)=\omega(u,v)$ for any $u,v\in TX$ and
$\omega(u,Ju)>0$ for any $u\in TX$, $u\neq0$.
Let $\nabla^{L^p\otimes E}: \mathscr{C}^\infty(X,L^p\otimes E)\to
\mathscr{C}^\infty(X, T^*X \otimes L^p\otimes E)$ be the connection
on $L^p\otimes E$ induced by $\nabla^{L}$ and $\nabla^E$.
Denote by $\Delta^{L^p\otimes E}$ the induced
Bochner-Laplacian acting on $\mathscr{C}^\infty(X,L^p\otimes E)$ by
\begin{equation}\label{e1.4}
\Delta^{L^p\otimes E}=\big(\nabla^{L^p\otimes E}\big)^{\!*}\,
\nabla^{L^p\otimes E},
\end{equation}
where $\big(\nabla^{L^p\otimes E}\big)^{\!*}:
\mathscr{C}^\infty(X,T^*X\otimes L^p\otimes E)\to
\mathscr{C}^\infty(X,L^p\otimes E)$ denotes the formal adjoint of
the operator $\nabla^{L^p\otimes E}$.
The \textbf{\emph{renormalized Bochner-Laplacian}} is a differential
operator $\Delta_p$ acting on $\mathscr{C}^\infty(X,L^p\otimes E)$ by
\begin{equation}\label{e:Delta_p}
\Delta_p=\Delta^{L^p\otimes E}-p\tau,
\end{equation}
where $\tau\in \mathscr{C}^\infty(X)$ is given by
\begin{equation}\label{tau}
\tau(x)=-\pi \operatorname{Tr}[J_0(x)J(x)],\quad x\in X.
\end{equation}
The renormalized Bochner-Laplacian was introduced by
Guillemin and Uribe in \cite{Gu-Uribe}.
When $(X,\omega)$ is a K\"{a}hler manifold, it is twice the
corresponding Kodaira-Laplacian on functions
$\Box^{L^p}=\bar\partial^{L^p*}\bar\partial^{L^p}$.
The asymptotic of the spectrum of the operator
$\Delta_p$ as $p\to \infty$ was studied in
\cite{BVa89,B-Uribe,Gu-Uribe, ma-ma02,MM07}.
In this paper, we will suppose that $(X, g^{TX})$ is complete
and $R^L$, $R^E$, $J$, $g^{TX}$ have bounded geometry
(i.e., they and their derivatives of any order are uniformly
bounded on $X$ in the norm induced by $g^{TX}$, $h^L$
and $h^E$, and the injectivity radius of $(X, g^{TX})$ is positive).
We will also assume that
\begin{equation}\label{mu}
\mu_0=\inf_{\substack{x\in X \\ u\in T_xX\setminus\{0\}}}
\frac{iR^L_x(u,J(x)u)}{|u|_{g^{TX}}^2}>0.
\end{equation}
Note that $\lambda(x)=
\inf_{u\in T_xX\setminus\{0\}}iR^L_x(u,J(x)u)/|u|_{g^{TX}}^2$
is the smallest eigenvalue of $iR^L_x(\cdot,J(x)\,\cdot)$
with respect to $g^{TX}_x$, for $x\in X$. Thus \eqref{mu}
is a condition of uniform positivity of $R^L$ with respect to $g^{TX}$.
Since $(X,g^{TX})$ is complete, the Bochner-Laplacian and
the renormalized Bochner-Laplacian $\Delta_p$
are essentially self-adjoint, see Theorem \ref{esa}.
We still denote by $\Delta_p$ the unique self-adjoint extension of
$\Delta_p:\mathscr{C}^\infty_c(X,L^p\otimes E)\to
\mathscr{C}^\infty_c(X,L^p\otimes E)$
acting on compactly supported smooth sections,
and by $\sigma(\Delta_p)$ its spectrum in
$L^2(X,L^p\otimes E)$.
First, we state the following spectral gap property for the operator
$\Delta_p$ which is a direct consequence of \cite[Lemma 1]{ma-ma15}.
\begin{thm}\label{t:gap0}
Let $(X,\omega)$ be a symplectic manifold with a prequantum line bundle
$(L,\nabla^L,h^L)$. Let $g^{TX}$ be a complete
Riemannian metric on $X$ and let $J$ be the almost complex
structure defined by \eqref{e1.3}. Let
$(E,\nabla^E, h^E)$ be an auxiliary vector bundle on $X$.
We assume that $R^L$, $R^E$, $J$, $g^{TX}$ have bounded geometry
and \eqref{mu} holds.
Then there exists a constant $C_L>0$ such that for any $p\in \mathbb N$
the spectrum of the renormalized Bochner-Laplacian \eqref{e:Delta_p}
satisfies
\begin{equation}\label{e1.8}
\sigma(\Delta_p)\subset \big[\!-C_L,C_L\big]\cup
\big[2p\mu_0-C_L,+\infty\big).
\end{equation}
\end{thm}
When $X$ is compact and $E$ is the trivial line bundle,
this theorem (with a not precised constant $\mu_0$)
is the main result of Guillemin and Uribe \cite{Gu-Uribe}.
For a general vector bundle $E$, it was proved by Ma and Marinescu
\cite[Corollary 1.2]{ma-ma02}, cf.\ also \cite[Theorem 8.3.1]{MM07},
with the geometric constant $\mu_0$ given by \eqref{mu}.
The analogous theorem for the spin$^c$ Dirac operator
on a manifold of bounded geometry is stated in \cite[Lemma 1]{ma-ma15}.
Theorem~\ref{t:gap0} can be directly derived from this result,
following the proof of \cite[Corollary 1.2]{ma-ma02} (see
also \cite[Corollary 4.7]{ma-ma02} for the case of a covering of
a compact manifold),
thus we will not repeat this proof here.
For a Borel set $B\subset\mathbb{R}$,
we denote by $\mathcal{E}(B,\Delta_p)$ the spectral projection
corresponding to the subset $B$.
Consider the spectral space $\mathcal H_p\subset L^2(X,L^p\otimes E)$
of $\Delta_p$
corresponding to $[-C_L,C_L]$,
\begin{equation}\label{e:hp}
\mH_p:=\operatorname{Range}\,\mE([-C_L,C_L],\Delta_p).
\end{equation}
If $X$ is compact, the spectrum of $\Delta_p$ is discrete
and $\mH_p$ is the subspace spanned by the eigensections of $\Delta_p$
corresponding to eigenvalues
in $[-C_L,C_L]$. Let
\begin{equation}\label{php}
P_{\mathcal H_p}:=\mE([-C_L,C_L],\Delta_p)
:L^2(X,L^p\otimes E)\longrightarrow \mathcal H_p,
\end{equation}
be the orthogonal projection.
Let $\pi_1$ and $\pi_2$ be the projections of $X\times X$ on
the first and second factor. The Schwartz kernel of the operator
$P_{\mathcal H_p}$ with respect to the Riemannian volume form
$dv_{X}$ is a smooth section
$P_{p}(\cdot,\cdot)\in \mathscr{C}^\infty(X\times X,
\pi_1^*(L^p\otimes E)\otimes \pi_2^*(L^p\otimes E)^*)$,
see \cite[Remark 1.4.3]{MM07}. It is called the
\textbf{\emph{generalized Bergman kernel}} of $\Delta_p$
in \cite{ma-ma08}, since it generalizes the Bergman kernel
on complex manifolds.
\begin{thm}\label{t:mainPp}
Under the assumptions of Theorem \ref{t:gap0},
there exists $c>0$ such that for any $k\in \mathbb N$,
there exists $C_k>0$ such that for any $p\in \mathbb N$,
$x, x^\prime \in X$, we have
\begin{equation}\label{e1.9}
\big|P_p(x, x^\prime)\big|_{\mathscr{C}^k}\leq C_k p^{n+\frac{k}{2}}
e^{-c\sqrt{p} \,d(x, x^\prime)}.
\end{equation}
\end{thm}
Here $d(x,x^\prime)$ is the geodesic distance and
$|P_p(x, x^\prime)|_{\mathscr{C}^k}$ denotes the pointwise
$\mathscr{C}^k$-seminorm of the section $P_p$ at a point
$(x, x^\prime)\in X\times X$, which is the sum of the norms induced
by $h^L, h^E$ and $g^{TX}$ of the derivatives up to order $k$ of
$P_p$ with respect to the connection $\nabla^{L^p\otimes E}$ and
the Levi-Civita connection $\nabla^{TX}$ evaluated at $(x, x^\prime)$.
For the Bergman kernel of the spin$^c$ Dirac operator associated to
a positive line bundle on a symplectic manifold of bounded geometry,
the same type of
exponential estimate is proved in \cite[Theorem 1]{ma-ma15}
(see also the references therein for the previous results).
In \cite{ma-ma15}, the authors use the methods of
\cite{DLM04a,MM07,ma-ma08} based on the spectral gap property
of the spin$^c$ Dirac operator,
finite propagation speed arguments
for the wave equation, the heat semigroup and rescaling of
the spin$^c$ Dirac operator
near the diagonal,
which is inspired by the analytic localization technique of
Bismut-Lebeau \cite{BL}. It is important in \cite{ma-ma15}
that the eigenvalues of the associated Laplacian are either $0$
or tend to $+\infty$. In the current situation,
the renormalized Bochner-Laplacian has possibly different bounded
eigenvalues, which makes difficult to use the heat semigroup technique.
So we replace the heat semigroup technique by a different approach,
which was developed by the first author in \cite{bergman}:
We follow essentially the general strategy
of \cite{DLM04a,MM07,ma-ma08} but use weighted
estimates with appropriate exponential weights as in \cite{Kor91}
instead of the use of the heat semigroup and finite propagation speed
arguments.
As an application of our proof of Theorem~\ref{t:mainPp},
we obtain the relation between the generalized Bergman kernel
on a Galois covering of a compact symplectic manifold and
the generalized Bergman kernel on the base as an analogue of
\cite[Theorem 2]{ma-ma15} for the Bergman kernel of the
spin$^c$ Dirac operator.
\begin{thm}\label{t:covering}
Let $(X, \omega)$ be a compact symplectic manifold.
Let $(L,\nabla^L, h^L)$, $(E, \nabla^E, h^E)$, $g^{TX}$
be given as above. Consider a Galois covering $\pi : \widetilde X\to X$ and
let $\Gamma$ be the group of deck transformations.
Denote by $\widetilde \omega$,
$(\widetilde L,\nabla^{\widetilde L}, h^{\widetilde L})$,
$(\widetilde E, \nabla^{\widetilde E}, h^{\widetilde E})$,
$g^{T\widetilde X}$ be the lifts
of the above data to $\widetilde X$. Let $\widetilde \Delta_p$
be the renormalized Bochner-Laplacian acting on
$\mathscr{C}^\infty(\widetilde X, \widetilde L^p\otimes \widetilde E)$
and $\widetilde P_{p}(\cdot,\cdot)$ be the generalized Bergman kernel
of $\widetilde \Delta_p$. There exists $p_1\in \mathbb N$
such that for any $p>p_1$ we have for any $x,x^\prime\in \widetilde X$,
\begin{equation}\label{e:sum}
\sum_{\gamma\in \Gamma}\widetilde P_p(\gamma x,x^\prime)=
P_p(\pi(x), \pi(x^\prime)).
\end{equation}
\end{thm}
This type of results has a long history.
In the category of complex manifolds it appeared in connection
with the theory of automorphic forms and Poincar\'e series
in the works of Selberg and Godement.
Earle \cite{Ea} gave a proof when $\widetilde X$
is a bounded symmetric domain
(under some hypothesis on the variation of Bergman kernels).
The second and third authors proved \eqref{e:sum}
for the Bergman kernels
associated to the spin$^c$ Dirac operator on a symplectic manifold,
in particular, in the K\"ahler case \cite[Theorem 2]{ma-ma15}.
Lu and Zelditch \cite{Lu-Z} independently proved \eqref{e:sum}
for the Bergman kernels on K\"ahler manifolds when $E=\mathbb{C}$.
As another application of the technique developed in this paper,
we extend the results on the full off-diagonal asymptotic expansion
of the generalized Bergman kernels of the renormalized Bochner-Laplacians
associated to high tensor powers of a positive line bundle
over a compact symplectic manifold, obtained in \cite{lu-ma-ma,bergman},
to the case of manifolds of bounded geometry and slightly improve
the remainder estimate in the asymptotic expansions, proving
an exponential estimate $\mathcal O(e^{-c_0\sqrt{p}})$
instead of $\mathcal O(p^{-\infty})$ (see Theorem~\ref{t:main} below).
Finally, we study the theory of Berezin-Toeplitz quantization
on symplectic orbifolds by using as quantum spaces the spectral
spaces $\mH_p$, especially we show that the
set of Toeplitz operators forms an algebra.
Ma and Marinescu obtained first Berezin-Toeplitz quantization
on symplectic orbifolds by using as quantum spaces
the kernel of the spin$^c$ Dirac operator,
in particular, on compact complex orbifolds
\cite[Theorems 6.13, 6.16]{ma-ma08a}.
Let us note also that
Hsiao and Marinescu \cite{HM}
constructed a Berezin-Toeplitz quantization for eigenstates of
small eigenvalues in the case of complex manifolds.
For a comprehensive introduction to this subject
see \cite{ma:ICMtalk,MM07,MM11}.
The paper is organized as follows. In Section~\ref{norm},
we collect some necessary background information on differential operators
and Sobolev spaces on manifolds of bounded geometry.
In Section~\ref{maintheorem}, we remind some results on
weighted estimates on manifolds of bounded geometry and
prove Theorem~\ref{t:mainPp} and~\ref{t:covering}.
Section~\ref{expansions} is devoted to the full off-diagonal
asymptotic expansions. In Section~\ref{pbs4} we study
Berezin-Toeplitz quantization on symplectic orbifolds.
\section{Preliminaries on differential operators and Sobolev spaces}
\label{norm}
In this section, we collect some necessary background information
on differential operators and Sobolev spaces on manifolds of bounded
geometry. We refer the reader to \cite{Kor91,ma-ma15}
for more information. The novel point is that our constructions are adapted
to a particular sequence of vector bundles $L^p\otimes E, p\in \mathbb N$.
This concerns with a specific choice of the Sobolev norm as well as with
a slightly refined form of the Sobolev embedding theorem.
We will keep the setting described in Introduction.
\subsection{Differential operators}
Let $\mathcal F$ be a vector bundle over $X$. Suppose that $\mathcal F$
is Euclidean or Hermitian depending on whether it is real or complex
and equipped with a metric connection $\nabla^{\mathcal F}$.
The Levi-Civita connection $\nabla^{TX}$ on $(X,g^{TX})$ and
the connection $\nabla^{\mathcal F}$ define a metric connection
$\nabla^{\mathcal F} : \mathscr{C}^\infty(X, (T^*X)^{\otimes j}
\otimes \mathcal F)\to \mathscr{C}^\infty(X, (T^*X)^{\otimes (j+1)}
\otimes \mathcal F)$
on each vector bundle $(T^*X)^{\otimes j} \otimes \mathcal F$ for
$j\in \mathbb N$, that allows us to introduce the operator
$$\big(\nabla^{\mathcal F}\big)^{\!\ell} :
\mathscr{C}^\infty(X, \mathcal F)
\to \mathscr{C}^\infty(X, (T^*X)^{\otimes \ell} \otimes \mathcal F)$$
for every $\ell\in \mathbb N$.
Any differential operator $A$ of order $q$ acting in
$\mathscr{C}^\infty(X,\mathcal F)$ can be written as
\begin{equation}\label{e2.1}
A=\sum_{\ell=0}^q a_\ell\cdot \big(\nabla^{\mathcal F}\big)^{\!\ell},
\end{equation}
where $a_\ell \in \mathscr{C}^\infty(X, (TX)^{\otimes \ell})$
and the endomorphism
$\cdot : (TX)^{\otimes \ell}\otimes ((T^*X)^{\otimes \ell}
\otimes \mathcal F) \to \mathcal F$ is given by the contraction.
If $\mathcal F$ has bounded geometry, we denote by
$\mathscr{C}^k_b(X,\mathcal F)$ the space of sections
$u\in \mathscr{C}^k(X,\mathcal F)$ such that
\begin{equation}\label{e2.2}
\|u\|_{\mathscr{C}^k_b}=\sup_{x\in X, \ell\leq k}
\Big|\big(\nabla^{\mathcal F}\big)^{\!\ell} u(x)\Big| <\infty,
\end{equation}
where $|\cdot|_x$ is the norm in
$(T^*_xX)^{\otimes \ell} \otimes \mathcal F_x$
defined by $g^{TX}$ and $h^\mathcal F$.
We also denote by
$BD^q(X,\mathcal F)$ the space of differential operators $A$
of order $q$ in $\mathscr{C}^\infty_c(X,\mathcal F)$ with
coefficients $a_\ell$ in $\mathscr{C}^\infty_b(X, (TX)^{\otimes \ell})$.
Usually, we will deal with families of differential operators of the
form $$\{A_p\in BD^q(X,L^p\otimes E), p\in \mathbb N\}.$$
We will say that such a family $\{A_p\in BD^q(X,L^p\otimes E),
p\in \mathbb N\}$ is bounded in $p$, if
\begin{equation}\label{e2.3}
A_p=\sum_{\ell=0}^q a_{p,\ell}\cdot \Big(\frac{1}{\sqrt{p}}
\nabla^{L^p\otimes E}\Big)^\ell,\quad a_{p,\ell}
\in \mathscr{C}^\infty_b(X, (TX)^{\otimes \ell}),
\end{equation}
and, for any $\ell=0,1,\ldots,q$, the family
$\{a_{p,\ell}, p\in \mathbb N\}$
is bounded in the Frechet space
$\mathscr{C}^\infty_b(X, (TX)^{\otimes \ell})$.
An example of a bounded in $p$ family of differential operators
is given by $\{\frac{1}{p}\Delta_p:p\in \mathbb N\}$.
\subsection{Sobolev spaces}
Denote by $dv_{X}$ the Riemannian volume form of $(X,g^{TX})$.
The $L^2$-norm on $L^2(X,L^p\otimes E)$ is given by
\begin{equation}\label{e2.5}
\|u\|^2_{p,0}=\int_{X}|u(x)|^2dv_{X}(x), \quad u\in L^2(X,L^p\otimes E).
\end{equation}
For any integer $m>0$, we introduce the norm $\|\cdot\|_{p,m}$
on $\mathscr{C}^\infty_c(X,L^p\otimes E)$ by the formula
\begin{equation}\label{e2.6}
\|u\|^2_{p,m}=\sum_{\ell=0}^m \int_{X} \left|\Big(\frac{1}{\sqrt{p}}
\nabla^{L^p\otimes E}\Big)^\ell u(x)\right|^2 dv_{X}(x),
\quad u\in H^m(X,L^p\otimes E).
\end{equation}
The completion of $\mathscr{C}^\infty_c(X,L^p\otimes E)$
with respect to $\|\cdot\|_{p,m}$ is the Sobolev space
$H^m(X,L^p\otimes E)$ of order $m$.
Denote by $\langle\cdot,\cdot\rangle_{p,m}$ the corresponding inner
product on $H^m(X,L^p\otimes E)$. For any integer $m<0$,
we define the norm in the Sobolev space $H^m(X,L^p\otimes E)$ by duality.
For any bounded linear operator
$A : H^m(X,L^p\otimes E)\to H^{m^\prime}(X,L^p\otimes E)$,
$m,m^\prime\in \mathbb Z$, we will denote its operator norm by
$\|A\|^{m,m^\prime}_p$.
One can easily derive the following mapping properties of
differential operators in Sobolev spaces.
\begin{prop}\label{p:Sobolev-mapping}
Any operator $A\in BD^q(X,L^p\otimes E)$ defines a bounded operator
\[A:H^{m+q}(X,L^p\otimes E)\longrightarrow H^{m}(X,L^p\otimes E)\]
for any $m\in \mathbb N$. Moreover, if a family
$\{A_p\in BD^q(X,L^p\otimes E), p\in \mathbb N\}$
is bounded in $p$, then for any $m\in \mathbb N$, there exists
$C_m>0$ such that, for all $p\in \mathbb N$,
\begin{equation}\label{e2.7}
\|A_pu\|_{p,m}\leq C_m\|u\|_{p,m+q},\quad
u\in H^{m+q}(X,L^p\otimes E).
\end{equation}
\end{prop}
\subsection{Sobolev embedding theorem}
We will need a refined form of the Sobolev embedding theorem adapted
to the sequence $L^p\otimes E, p\in \mathbb N$.
\begin{prop}[\cite{ma-ma15}, Lemma 2]\label{p:Sobolev}
For any $k, m\in \mathbb N$ with $m>k+n$, we have an embedding
\begin{equation}\label{e2.16}
H^m(X,L^p\otimes E)\subset \mathscr{C}^k_b(X,L^p\otimes E).
\end{equation}
Moreover, there exists $C_{m,k}>0$ such that, for any $p\in \mathbb N$
and $u\in H^m(X,L^p\otimes E)$,
\begin{equation}\label{e2.17}
\|u\|_{\mathscr{C}^k_b}\leq C_{m,k}p^{(n+k)/2}\|u\|_{p,m}.
\end{equation}
\end{prop}
For any $x\in X$ and $v\in (L^p\otimes E)_x$, we define the delta-section
$\delta_v\in \mathscr{C}^{-\infty}(X,L^p\otimes E)$ as a linear functional
on $\mathscr{C}^{\infty}_c(X,L^p\otimes E)$ given by
\begin{equation}\label{e2.18}
\langle \delta_v, \varphi\rangle
=\langle v, \varphi(x)\rangle_{h^{L^p\otimes E}},
\quad \varphi \in \mathscr{C}^{\infty}_c(X,L^p\otimes E).
\end{equation}
\begin{prop}\label{p:delta}
For any $m>n$ and $v\in L^p\otimes E$,
$\delta_v\in H^{-m}(X,L^p\otimes E)$ with the following norm estimate
\begin{equation}\label{e2.19}
\sup_{|v|=1}p^{-n/2}\|\delta_v\|_{p,-m}<\infty.
\end{equation}
\end{prop}
\begin{proof}
By Proposition~\ref{p:Sobolev} and the definition of the Sobolev norm,
we have
\begin{equation}\label{e2.20}
\|\delta_v\|_{p,-m}\leq C\sup_{\phi\in H^m(X,L^p\otimes E)}
\frac{\langle \delta_v,\phi\rangle}{\|\phi\|_{p,m}}\leq Cp^{n/2}|v|.
\qedhere
\end{equation}
\end{proof}
\subsection{The renormalized Bochner-Laplacian}
Let us first note the following basic result.
\begin{thm}\label{esa}
Let $(X,\omega)$ be a symplectic manifold with a prequantum line bundle
$(L,\nabla^L,h^L)$. Let $g^{TX}$ be a complete
Riemannian metric on $X$ and let
$(E,\nabla^E, h^E)$ be an auxiliary vector bundle.
\noindent
(i)
The space $\mathscr{C}^\infty_c(X,L^p\otimes E)$ is dense in the graph
norm of the maximal extension of $\nabla^{L^p\otimes E}$ and
$\mathscr{C}^\infty_c(X,T^*X\otimes L^p\otimes E)$
is dense in the graph norm of the maximal extension of
$\big(\nabla^{L^p\otimes E}\big)^{\!*}$.
\noindent
(ii) The Hilbert space adjoint of the maximal extension
of $\nabla^{L^p\otimes E}$ coincides with the maximal extension
of $\big(\nabla^{L^p\otimes E}\big)^{\!*}$.
\noindent
(iii) The Bochner-Laplacian
$\Delta^{L^p\otimes E}
=\big(\nabla^{L^p\otimes E}\big)^{\!*}\,\nabla^{L^p\otimes E}$
acting on $\mathscr{C}^\infty_c(X,L^p\otimes E)$ is essentially selfadjoint.
In particular, the renormalized Bochner-Laplacian
$\Delta_p$ acting on $\mathscr{C}^\infty_c(X,L^p\otimes E)$
is essentially selfadjoint.
\end{thm}
\begin{proof}
Assertion (i) is a form of
the Andreotti-Vesentini Lemma \cite[Lemma 3.3.1]{MM07}.
The proof is obtained
by replacing $\overline\partial^E$ in \cite[Lemma 3.3.1]{MM07}
with $\nabla^{L^p\otimes E}$.
Assertions (ii) and (iii) are obtained by adapting in the same way the
proofs of \cite[Corollary 3.3.3]{MM07} and \cite[Corollary 3.3.4]{MM07},
respectively (valid for
$\big(\overline\partial^E\big)^*$ and the Kodaira-Laplacian $\Box^E$).
\end{proof}
Now we establish some additional properties of the
family $\{\frac{1}{p}\Delta_p, p\in \mathbb N\}$ of differential operators,
which is bounded in $p$.
\begin{thm}\label{uniformD}
There exist constants $C_2, C_3>0$ such that for any $p\in\mathbb N$,
$u, u^\prime\in \mathscr{C}^\infty_c(X,L^p\otimes E)$,
\begin{equation}\label{e:deltap1}
\Big\langle \frac{1}{p}\Delta_p u, u\Big\rangle_{p,0}
\geq \|u\|^2_{p,1}-C_2\|u\|^2_{p,0}\,,
\end{equation}
\begin{equation}\label{e:deltap2}
\left|\Big\langle \frac{1}{p}\Delta_p u, u^\prime\Big\rangle_{p,0}\right|
\leq C_3\|u\|_{p,1}\|u^\prime\|_{p,1}\,.
\end{equation}
\end{thm}
\begin{proof}
These estimates follow immediately from the identity
\begin{equation}\label{e2.9}
\Big\langle \frac{1}{p}\Delta_p u, u\Big\rangle_{p,0}
=\left\|\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}u\right\|^2_{p,0}
-\langle \tau u, u\rangle_{p,0}. \qedhere
\end{equation}
\end{proof}
Let $\delta$ be the counterclockwise oriented circle in $\mathbb C$
centered at $0$ of radius $\mu_0$.
\begin{thm}\label{Thm1.7}
There exists $p_0\in \mathbb N$ such that for any $\lambda\in \delta$
and $p\geq p_0$ the operator $\lambda-\frac{1}{p}\Delta_p$ is invertible
in $L^2(X,L^p\otimes E)$, and there exists $C>0$ such that for
all $\lambda\in \delta$ and $p\geq p_0$ we have
\begin{equation}\label{e:Thm1.7}
\left\|\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-1}\right\|^{0,0}_p\leq C,
\quad
\left\|\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-1}\right\|^{-1,1}_p\leq C.
\end{equation}
\end{thm}
\begin{proof}
We will closely follow the proof of
\cite[Theorem 4.8]{DLM04a} or \cite[Theorem 1.7]{ma-ma08}
(cf.\ also the proof of \cite[Theorem 11.27]{BL}).
The first estimate follows from Theorem~\ref{t:gap0} and
the spectral theorem. By \eqref{e:deltap1}, we have, for
$\lambda_0\leq -C_2$,
\begin{equation}\label{e2.11}
\Big\langle \Big(\,\frac{1}{p}\Delta_p-\lambda_0\Big)u,
u\Big\rangle_{p,0}\geq \|u\|^2_{p,1},
\end{equation}
therefore, the resolvent $\Big(\lambda_0-\frac{1}{p}\Delta_p\Big)^{-1}$
exists and
\begin{equation}\label{e:Thm1.71}
\left\|\Big(\lambda_0-\frac{1}{p}\Delta_p\Big)^{-1}\right\|^{-1,1}_p
\leq 1.
\end{equation}
Now we can write, for $\lambda\in \delta$ and $\lambda_0\leq -C_2$,
\begin{equation}\label{e:Thm1.72}
\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-1}
=\Big(\lambda_0-\frac{1}{p}\Delta_p\Big)^{-1}
-(\lambda-\lambda_0)\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-1}
\Big(\lambda_0-\frac{1}{p}\Delta_p\Big)^{-1}.
\end{equation}
Thus for $\lambda\in \delta$, we get from
the first estimate of \eqref{e:Thm1.7},
\eqref{e:Thm1.71} and \eqref{e:Thm1.72},
\begin{equation}\label{e:Thm1.73}
\left\|\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-1}\right\|^{-1,0}_p
\leq 1+C|\lambda-\lambda_0|.
\end{equation}
Changing the last two factors in \eqref{e:Thm1.72} and applying
\eqref{e:Thm1.73}, we get
\begin{equation}\label{e2.15}
\left\|\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-1}\right\|^{-1,1}_p
\leq 1+|\lambda-\lambda_0|
(1+C|\lambda-\lambda_0|).
\end{equation}
The proof of Theorem \ref{Thm1.7} is completed.
\end{proof}
\section{Proof of main results}\label{maintheorem}
This section is devoted to the proofs of Theorems~\ref{t:mainPp} and
\ref{t:covering}. First, we describe a class of exponential weight functions
as in \cite{Kor91}. Then we prove norm estimates in weighted Sobolev
spaces for the resolvent $\big(\lambda-\frac{1}{p}\Delta_p\big)^{-m}$.
Here we follow general constructions of \cite{DLM04a,MM07,ma-ma08},
which are inspired by the analytic localization technique of
Bismut-Lebeau \cite[\S 11]{BL}.
Next, we derive pointwise exponential estimates for the
Schwartz kernel of the operator
$\big(\lambda-\frac{1}{p}\Delta_p\big)^{-m}$
and its derivatives of an arbitrary order, using a refined form of
the Sobolev embedding theorem stated in Proposition~\ref{p:Sobolev}.
Finally, we use the formula as in \cite[(1.55)]{ma-ma08}
\begin{equation}\label{bergman-integral}
P_{\mathcal H_p}=\frac{1}{2\pi i}
\int_\delta \lambda^{m-1}
\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}d\lambda, \quad m\geq 1,
\end{equation}
that allows us to complete the proofs of Theorems~\ref{t:mainPp}
and \ref{t:covering}.
\subsection{Weight functions}
Recall that $d$ denotes the distance function on $X$.
By \cite[Proposition 4,1]{Kor91}, there exists a ``smoothed distance''
function $\widetilde{d}\in \mathscr{C}^\infty(X\times X)$, satisfying
the following conditions:
\medskip\par
(1) there is a constant $r>0$ such that
\begin{equation}\label{e:3.11}
\big\vert \widetilde{d}(x,y) - d (x,y)\big\vert < r,\quad x, y\in X;
\end{equation}
(2) for any $k>0$, there exists $C_k>0$ such that, for any multi-index
$\beta$ with $|\beta|=k$,
\begin{equation}\label{e:3.12}
\big\vert \partial^\beta_{x}\widetilde{d}(x,y)\big\vert < C_{k},\quad
x, y\in X,
\end{equation}
where the derivatives are taken with respect to normal coordinates
defined by the exponential map at $x$.
Actually, we will work with a sequence of smoothed distance functions
$\widetilde d_p$, $p\in\mathbb N$, to remove small distances effects of
smoothing. As one can easily see from the proof of
\cite[Proposition 4.1]{Kor91}, for any $\gamma\in (0,1]$,
there exists a function $\widetilde{d}\in \mathscr{C}^\infty(X\times X)$,
satisfying \eqref{e:3.11} with $r=\gamma$ and \eqref{e:3.12} with
$C_k=c_k \gamma^{1-k}$, $c_k>0$ is independent of $\gamma$.
Let us briefly describe its construction.
Let $a^X$ be the injectivity radius of $(X,g^{TX})$.
We denote by $B^{X}(x,r)$
and $B^{T_{x}X}(0,r)$ the open balls in $X$ and $T_xX$ with center $x$
and radius $r$, respectively. For any $x_0\in X$, we identify
$B^{T_{x_0}X}(0,a^X)$ with $B^{X}(x_0,a^X)$ via the exponential map
$\exp^X_{x_0} : T_{x_0}X\to X$. One can show that,
for $\varepsilon\in (0,a^X)$, the geodesic distance on
$B^{X}(x_0,\varepsilon)$ is equivalent to the Euclidean distance on
$B^{T_{x_0}X}(0,\varepsilon)$ uniformly on $x_0\in X$: there exists
$C>0$ such that for any $x_0\in X$ and
$Z,W\in B^{T_{x_0}X}(0,\varepsilon)$,
\begin{equation}\label{e:equiv-dist}
C^{-1}d^{T_{x_0}X}(Z,W) \leq d (\exp^X_{x_0}(Z),
\exp^X_{x_0}(W))\leq Cd^{T_{x_0}X}(Z,W).
\end{equation}
By \cite[Lemma 2.3]{Kor91}, for $\varepsilon <a^X/2$,
there exists a covering $\{B^X(x_j, \varepsilon)\}_{j\in \NN}$
of $X$ and $N\in \mathbb N$ such that every intersection of $N+1$
balls $B^X(x_j, 2\varepsilon)$ is empty. Furthermore,
by \cite[Lemma 2.4]{Kor91}, there exists a partition of unity
$\sum_{j=1}^\infty\phi_j = 1$ subordinated to this covering such that
${\rm supp}\ \phi_j\subset B^X(x_j, \varepsilon)$ for any $j$ and,
for any $k\in \mathbb N$, there exists $C_k$ such that
$|\partial^\alpha \phi_j(x)|<C_k$ for any $j$, $x\in B^X(x_j, \varepsilon)$
and $|\alpha|<k$, where the derivatives are computed with respect to
the normal coordinates on $B^X(x_j, \varepsilon)$. Choose a function
$\theta\in \mathscr{C}^\infty_c(\mathbb R^{2n})$
such that $\theta(x) = 0$
if $|x|>1$, $\theta(x) \geq 0$ for any $x\in \mathbb R^{2n}$ and
$\int_{\mathbb R^{2n}} \theta(x)\,dx= 1$. For any $\delta>0$, put
\begin{equation}\label{e:3.14}
\theta_\delta(x)=\delta^{-2n}\theta(\delta^{-1}x),
\quad x\in \mathbb R^{2n}.
\end{equation}
The function $\widetilde d$ is defined by
\begin{equation}\label{e:3.13}
\widetilde{d}(x,y)=\sum_{j=1}^\infty \phi_j(x) \int_{\mathbb R^{2n}}
\theta_\delta\big((\exp^X_{x_j})^{-1}(x)-z\big)
d\big(\exp^X_{x_j}(z),y\big)\,dz.
\end{equation}
Using the formula
\begin{equation}\label{e:3.7}
d(x,y)=\sum_{j=1}^\infty \phi_j(x) \int_{\mathbb R^{2n}}
\theta_\delta((\exp^X_{x_j})^{-1}(x)-z)d(x,y)\,dz,
\end{equation}
the triangle inequality, the fact that ${\rm supp}\,
\theta_\delta\subset B(0,\delta)$ and \eqref{e:equiv-dist}, we get
\begin{equation}\label{e:3.8}
\big|\widetilde{d}(x,y)-d(x,y)\big| \leq \sum_{j=1}^\infty \phi_j(x)
\int_{\mathbb R^{2n}} \theta_\delta((\exp^X_{x_j})^{-1}(x)-z)
d(x,\exp^X_{x_j}(z))\,dz\leq C\delta.
\end{equation}
Choosing $\delta<C^{-1}\gamma$, we obtain \eqref{e:3.11}
with $r=\gamma$. The second property of $\widetilde{d}$ is proved
by differentiating the formulas \eqref{e:3.13} and \eqref{e:3.7} with respect to $x$ and using \eqref{e:3.14}.
We will use such a function $\widetilde{d}$ for
$\gamma=\frac{1}{\sqrt{p}}$, $p\in \mathbb N^{*}$,
denoting it by $\widetilde d_p$. So it satisfies the conditions:
(1) we have
\begin{equation}\label{(1.1)}
\big\vert \widetilde{d}_p(x,y) - d (x,y)\big\vert < \frac{1}{\sqrt{p}}\;,
\quad \text{ for any } x, y\in X;
\end{equation}
(2) for any $k>0$, there exists $c_k>0$ such that, for any multi-index
$\beta$ with $|\beta|= k$,
\begin{equation}
\label{dist}
\left| \Big(\frac{1}{\sqrt{p}}\Big)^{k} \partial^\beta_{x}
\widetilde{d}_p(x,y)\right| < \frac{c_{k}}{\sqrt{p}}\:,\quad
\text{ for any } x, y\in X.
\end{equation}
For any $\alpha\in \mathbb R$, $p\in \mathbb N^{*}$ and $y\in X$,
we introduce a weight function
$f_{\alpha,p,y} \in \mathscr{C}^{\infty}(X)$ by
\begin{equation}\label{e:3.9}
f_{\alpha,p,y}(x) = e^{\alpha \widetilde{d}_{p,y}(x)},\quad
\text{ for } x \in X,
\end{equation}
where $\widetilde{d}_{p,y}$ is a smooth function on $X$ given by
\begin{equation}\label{e:3.10}
\widetilde{d}_{p,y}(x) = \widetilde{d}_p(x,y), \quad \text{ for } x\in X.
\end{equation}
We don't introduce explicitly the weighted Sobolev spaces associated
with $f_{\alpha,p,y}$. Instead, we will work with the operator families
for $p\in \mathbb N^{*},
\alpha\in \mathbb R, y\in X$,
\begin{align}\label{e:3.15}
A_{p;\alpha,y}=f_{\alpha,p,y}A_pf^{-1}_{\alpha,p,y}:
\mathscr{C}^\infty_c(X,L^p\otimes E)\to
\mathscr{C}^{-\infty}(X,L^p\otimes E)
\end{align}
defined by an operator family
$\{A_p: \mathscr{C}^\infty_c(X,L^p\otimes E)\to
\mathscr{C}^{-\infty}(X,L^p\otimes E), p\in \mathbb N^{*}\}$.
In particular, the desired exponential estimate of the Bergman kernel
will be derived from the fact that the operator
$f_{\alpha,p,y}P_{\mathcal H_p}f^{-1}_{\alpha,p,y}$
is a smoothing operator in the scale of Sobolev spaces.
\subsection{Weighted estimates for the renormalized Bochner-Laplacian}
Observe that, for $v\in \mathscr{C}^\infty(X,TX)$,
\begin{equation}\label{e:3.16}
\nabla^{L^p\otimes E}_{\alpha,y;v}
:=f_{\alpha,p,y}\nabla^{L^p\otimes E}_{v}f^{-1}_{\alpha,p,y}
=\nabla^{L^p\otimes E}_{v}-\alpha v(\widetilde{d}_{p,y}).
\end{equation}
Therefore, $\nabla^{L^p\otimes E}_{\alpha,y;v}\in BD^1(X,L^p\otimes E)$.
Moreover, for any $a>0$ and $v\in \mathscr{C}^\infty_b(X,TX)$,
the family
\[\left\{\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{\alpha,y;v} :
y\in X, |\alpha|<a\sqrt{p}\right\}\]
is a family of operators from
$BD^1(X,L^p\otimes E)$, uniformly bounded in $p$.
This immediately implies that, if $Q\in BD^q(X,L^p\otimes E)$,
then, for any $\alpha\in \mathbb R$ and $y\in X$, the operator
$f_{\alpha,p,y}Qf^{-1}_{\alpha,p,y}$ is in $BD^q(X,L^p\otimes E)$.
Moreover, for any $a>0$, the family
$\{f_{\alpha,p,y}Qf^{-1}_{\alpha,p,y} :
y\in X, |\alpha|<a\sqrt{p}\}$ is a family of operators from
$BD^q(X,L^p\otimes E)$, uniformly bounded in $p$.
Now the operator $\Delta_{p;\alpha,y}:=
f_{\alpha,p,y}\Delta_{p}f^{-1}_{\alpha,p,y}$ has the form
\begin{equation}\label{Lta}
\Delta_{p;\alpha,y}=\Delta_{p}+\alpha A_{p;y}+\alpha^2B_{p;y},
\end{equation}
where $A_{p;y}\in BD^1(X,L^p\otimes E)$ and $B_{p;y}
\in BD^0(X,L^p\otimes E)$. Moreover, for any $a>0$, the families
$\{\frac{1}{\sqrt{p}}A_{p;y} : p\in \mathbb N^{*}, y\in X\}$ and
$\{B_{p;y} : p\in \mathbb N^{*}, y\in X\}$ are uniformly bounded in $p$.
If $\{e_j, j=1,\ldots,2n\}$ is a local frame in $TX$ on a domain
$U\subset X$, and functions $\Gamma^{i}_{jk}\in \mathscr{C}^\infty(U),
i,j,k=1,\ldots,2n,$ are defined by
$\nabla^{TX}_{e_j}e_k=\sum_{i}\Gamma^{i}_{jk}e_i$, then we have
\begin{equation}\label{Delta-p}
\Delta_{p}=-g^{jk}(Z)\left[\nabla^{L^p\otimes E}_{e_j}
\nabla^{L^p\otimes E}_{e_k}- \Gamma^{\ell}_{jk}(Z)
\nabla^{L^p\otimes E}_{e_\ell}\right]-p\tau(Z),
\end{equation}
where $(g^{jk}(Z))_{j,k}$ is the inverse of the matrix $(\langle e_{i},e_{j}\rangle(Z))_{i,j}$
and
\begin{equation}\label{LtaZ}
\Delta_{p;\alpha,y}=-g^{jk}(Z)\left[\nabla^{L^p\otimes E}_{\alpha,y;e_j}
\nabla^{L^p\otimes E}_{\alpha,y;e_k}
- \Gamma^{\ell}_{jk}(Z)\nabla^{L^p\otimes E}_{\alpha,y;e_\ell}\right]
-p\tau(Z).
\end{equation}
In particular,
\begin{align}\label{ReLta}
\operatorname{Re}\, \Delta_{p;\alpha,y}=\Delta_{p}
- \alpha^2\sum_{j,k=1}^{2n} g^{jk}(Z) e_j(\widetilde{d}_{p,y})
e_k(\widetilde{d}_{p,y})
=\Delta_{p} - \alpha^2|\nabla\widetilde{d}_{p,y}|_{g(Z)}^2.
\end{align}
From \eqref{LtaZ}, we easily get
\begin{align}\begin{split}\label{Apy}
&A_{p;y}=-\sum_{j,k=1}^{2n}g^{jk}(Z)\Big(2e_j(\widetilde{d}_{p,y})
\nabla^{L^p\otimes E}_{e_k} +e_j(e_k(\widetilde{d}_{p,y}))
-\Gamma^{\ell}_{jk}(Z)e_\ell(\widetilde{d}_{p,y})\Big),\\
&B_{p;y}
=-\sum_{j,k=1}^{2n}g^{jk}(Z)e_j(\widetilde{d}_{p,y})
e_k(\widetilde{d}_{p,y}).
\end{split} \end{align}
By Proposition~\ref{p:Sobolev-mapping}, for any $m\in \mathbb N$,
there exists $C_m>0$ such that, for any $p\in \mathbb N^{*}$, $y\in X$
and $u\in H^{m}(X,L^p\otimes E)$,
\begin{equation}\label{e:Sobolev-mapping}
\|A_{p;y}u\|_{p,m-1}\leq C_mp^{1/2}\|u\|_{p,m},\quad \|B_{p;y}u\|_{p,m}
\leq C_m\|u\|_{p,m}.
\end{equation}
We have the following extension of Theorem~\ref{uniformD}.
\begin{thm}\label{t3.1}
There exist constants $C_0, C_2, C_3>0$ such that for any
$p\in \mathbb N^{*}$, $\alpha \in \mathbb R$, $y\in X$ and
$u, u^\prime\in \mathscr{C}^\infty_c(X,L^p\otimes E)$,
\begin{align}\begin{split}\label{e:3.17}
&\operatorname{Re}\, \Big\langle \frac{1}{p}\Delta_{p;\alpha,y} u,
u\Big\rangle_{p,0}\geq \|u\|^2_{p,1}
-\Big(C_2+C_0\frac{\alpha^2}{p}\Big)\|u\|^2_{p,0},\\[3pt]
&\left|\Big\langle \frac{1}{p}\Delta_{p;\alpha,y} u,
u^\prime\Big\rangle_{p,0}\right|
\leq C_3\left(\|u\|_{p,1}\|u^\prime\|_{p,1}
+ \left(\frac{|\alpha|}{\sqrt{p}}\|u\|_{p,1}
+\frac{\alpha^2}{p}\|u\|_{p,0}\right)\|u^\prime\|_{p,0}\right).
\end{split} \end{align}
\end{thm}
\begin{proof} Using Theorem~\ref{uniformD}, \eqref{Lta}, \eqref{ReLta}
and \eqref{e:Sobolev-mapping}, we get
\begin{equation}\label{e:3.18}
\begin{split}
\operatorname{Re}\, \Big\langle \frac{1}{p} \Delta_{p;\alpha,y} u,
u\Big\rangle_{p,0}
&\geq \Big\langle \frac{1}{p} \Delta_{p} u, u\Big\rangle_{p,0}
-C_0\frac{\alpha^2}{p} \|u\|^2_{p,0}\\[3pt]
&\geq \|u\|^2_{p,1}
-\Big(C_2+C_0\frac{\alpha^2}{p}\Big)\|u\|^2_{p,0}\,,
\end{split}
\end{equation}
and
\begin{equation}\label{e:3.19}
\begin{split}
&\left|\Big\langle \frac{1}{p} \Delta_{p;\alpha,y} u,
u^\prime\Big\rangle_{p,0}\right|\\[3pt]
&\leq \left|\Big\langle \frac{1}{p} \Delta_{p} u,
u^\prime\Big\rangle_{p,0}\right|
+ |\alpha| \left|\Big\langle \frac{1}{p} A_{p,y} u,
u^\prime\Big\rangle_{p,0}\right|
+ \alpha^2 \left|\Big\langle \frac{1}{p} B_{p,y} u,
u^\prime\Big\rangle_{p,0}\right|\\[3pt]
&\leq C_3\left(\|u\|_{p,1}\|u^\prime\|_{p,1}
+\frac{|\alpha|}{\sqrt{p}}\|u\|_{p,1}\|u^\prime\|_{p,0}
+\frac{\alpha^2}{p} \|u\|_{p,0}\|u^\prime\|_{p,0}\right).
\end{split}
\end{equation}
\end{proof}
\subsection{Weighted estimates for the resolvent}\label{s3.3}
Theorems \ref{Thm1.7W}- \ref{Thm1.9} are the weighted analogues of
\cite[Theorems 4.8-4.10]{DLM04a}, \cite[Theorems 1.7-1.9]{ma-ma08}
which are inspired by \cite[\S 11]{BL}.
Now we extend Theorem~\ref{Thm1.7} to the setting of weighted spaces.
\begin{thm}\label{Thm1.7W}
There exist $c>0$, $C>0$ and $p_0\in \mathbb N$ such that, for all
$\lambda\in \delta$, $p\geq p_0$, $|\alpha|<c\sqrt{p}$, $y\in X$,
the operator $\lambda-\frac{1}{p}\Delta_{p;\alpha,y}$ is invertible in
$L^2(X,L^p\otimes E)$, and
we have
\begin{align}\label{e:3.20}
\left\|\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-1}
\right\|^{0,0}_p\leq C, \quad
\left\|\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-1}
\right\|^{-1,1}_p\leq C.
\end{align}
\end{thm}
\begin{proof}
Let us denote in this proof
$$R(\lambda,\tfrac1p\Delta_p):=\Big(\lambda-\frac{1}{p}\Delta_{p}\Big)^{-1},
\quad R\big(\lambda,\tfrac1p\Delta_{p;\alpha,y}\big):=\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}
\Big)^{-1}.$$
By Theorem~\ref{Thm1.7}, \eqref{Lta} and \eqref{e:Sobolev-mapping},
it follows that, for all $\lambda\in \delta$, $p\in \mathbb N^{*}$,
$\alpha\in \mathbb R$ and $y\in X$, we have
\begin{equation}\label{e:3.22}
\begin{split}
&\left\|(\Delta_{p;\alpha,y}-\Delta_{p})
R(\lambda,\tfrac1p\Delta_p)\right\|^{-1,0}_p
= \left\|\Big(\alpha A_{p,y}+\alpha^2B_{p,y}\Big)
R(\lambda,\tfrac1p\Delta_p)\right\|^{-1,0}_p\\[3pt]
&\leq C\left(|\alpha|\sqrt{p}
\left\|R(\lambda,\tfrac1p\Delta_p)\right\|^{-1,1}_p
+\alpha^2\left\|R(\lambda,\tfrac1p\Delta_p)
\right\|^{-1,0}_p\right)
\leq C\Big(|\alpha|\sqrt{p}+\alpha^2\Big).
\end{split}
\end{equation}
Choose $c>0$ such that $C(c+c^2)<\frac 12$. Then, if
$|\alpha|<c\sqrt{p}$, we have
\begin{equation}\label{e:d}
\left\|\Big(\frac{1}{p}\Delta_{p;\alpha,y}-\frac{1}{p}\Delta_{p}\Big)
R(\lambda,\tfrac1p\Delta_p)\right\|^{0,0}_p
\leq \left\|\Big(\frac{1}{p}\Delta_{p;\alpha,y}-\frac{1}{p}\Delta_{p}\Big)
R(\lambda,\tfrac1p\Delta_p)\right\|^{-1,0}_p<\frac 12\,\cdot
\end{equation}
Therefore, for all $\lambda\in \delta$, $p\in \mathbb N^{*}$,
$\alpha\in \mathbb R$, $|\alpha|<c\sqrt{p}$, and $y\in X$,
the operator $\lambda-\frac{1}{p}\Delta_{p;\alpha,y}$ is invertible in
$L^2$, and we have
\begin{equation}\label{e:res}
R\big(\lambda,\tfrac1p\Delta_{p;\alpha,y}\big)
=R(\lambda,\tfrac1p\Delta_p)\\
+R(\lambda,\tfrac1p\Delta_p)
\sum_{j=1}^{\infty}\Big(\Big(\frac{1}{p}\Delta_{p;\alpha,y}
-\frac{1}{p}\Delta_{p}\Big)R\big(\lambda,\tfrac1p\Delta_{p;\alpha,y}\big)\Big)^j.
\end{equation}
Therefore, by \eqref{e:Thm1.7}, \eqref{e:d}
and \eqref{e:res}, we get
\begin{equation}\label{e:3.26}
\begin{split}
&\left\|R\big(\lambda,\tfrac1p\Delta_{p;\alpha,y}\big)\right\|^{-1,1}_p\leq
\left\|R(\lambda,\tfrac1p\Delta_p)\right\|^{-1,1}_p\\
&+\left\| R(\lambda,\tfrac1p\Delta_p)
\right\|^{0,1}_p\sum_{j=1}^{\infty}\left\| \Big(\Big(\frac{1}{p}
\Delta_{p;\alpha,y}-\frac{1}{p}\Delta_{p}\Big)
R\big(\lambda,\tfrac1p\Delta_{p;\alpha,y}\big)\Big)^{j-1}
\right\|^{0,0}_p
\left\| \Big(\frac{1}{p}\Delta_{p;\alpha,y}
-\frac{1}{p}\Delta_{p}
\Big)R\big(\lambda,\tfrac1p\Delta_{p;\alpha,y}\big)
\right\|^{-1,0}_p\\
&\hspace{5mm}\leq C + C \sum_{j=1}^{\infty}2^{-j} = 2C.
\end{split}
\end{equation}
Since $\|\cdot \|^{0,0}_{p}\leq \|\cdot \|^{-1,1}_{p}$,
\eqref{e:3.26} entails \eqref{e:3.20}.
\end{proof}
In the sequel, we will keep notation $c$ for the constant given by
Theorem \ref{Thm1.7W}, which will be usually related with the interval
$(-c\sqrt{p}, c\sqrt{p})$ of admissible values of the parameter $\alpha$.
\begin{remark}\label{r:conj}
Observe that, for any $\lambda\in \delta$, $p\geq p_0$,
$\alpha\in \mathbb R$ and $y\in X$, the operators
$(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}$ and
$(\lambda-\frac{1}{p}\Delta_{p})^{-1}$ are related by the identity
\begin{equation}\label{LaL}
\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-1}
=f_{\alpha,p,y}\Big(\lambda-\frac{1}{p}\Delta_{p}\Big)^{-1}
f^{-1}_{\alpha,p,y},
\end{equation}
which should be understood in the following way. If $\alpha<0$, then,
for any $s\in \mathscr{C}^\infty_c(X, L^p\otimes E)$, the expression
$f_{\alpha,p,y}(\lambda-\frac{1}{p}\Delta_{p})^{-1}f^{-1}_{\alpha,p,y}s$
makes sense and defines a function in $L^2(X, L^p\otimes E)$.
Thus, we get a well-defined operator
\begin{align}\label{e:3.30}
f_{\alpha,p,y}\Big(\lambda-\frac{1}{p}
\Delta_{p}\Big)^{-1}f^{-1}_{\alpha,p,y}:
\mathscr{C}^\infty_c(X, L^p\otimes E)\to L^2(X, L^p\otimes E),
\end{align}
and one can check that $f_{\alpha,p,y}
(\lambda-\frac{1}{p}\Delta_{p})^{-1}f^{-1}_{\alpha,p,y}u
=(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}u$ for any
$u\in \mathscr{C}^\infty_c(X, L^p\otimes E)$. So \eqref{LaL}
means that the operator
$f_{\alpha,p,y}(\lambda-\frac{1}{p}\Delta_{p})^{-1}f^{-1}_{\alpha,p,y}$
extends to a bounded operator in $L^2(X, L^p\otimes E)$,
which coincides with $(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}$.
If $\alpha>0$, then, for any $u\in L^2(X, L^p\otimes E)$, the expression
$f_{\alpha,p,y}(\lambda-\frac{1}{p}\Delta_{p})^{-1}f^{-1}_{\alpha,p,y}u$
makes sense as a distribution on $X$. Thus, we get a well-defined operator
\begin{align}\label{e:3.31}
f_{\alpha,p,y}\Big(\lambda-\frac{1}{p}
\Delta_{p}\Big)^{-1}f^{-1}_{\alpha,p,y}:
L^2(X, L^p\otimes E) \to \mathscr{C}^{-\infty}(X, L^p\otimes E).
\end{align}
So \eqref{LaL} means that this operator is indeed a bounded operator in
$L^2(X, L^p\otimes E)$, which coincides with
$(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}$.
\end{remark}
\begin{thm}\label{Thm1.9}
For any $p\in \mathbb N^{*}$, $p\geq p_0$, $\lambda\in \delta$,
$m\in \mathbb N$, $y\in X$ and $|\alpha|<c\sqrt{p}$ with the constant
$c$ as in Theorem \ref{Thm1.7W}, the resolvent
$(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}$ maps
$H^m(X,L^p\otimes E)$ to $H^{m+1}(X,L^p\otimes E)$.
Moreover, for any $m\in \mathbb N$, there exists $C_m>0$
such that for any $p\geq p_0$, $\lambda\in \delta$, $y\in X$ and
$|\alpha|<c\sqrt{p}$,
\begin{equation}\label{e:mm+1}
\left\|\Big(\lambda-\frac{1}{p}
\Delta_{p;\alpha,y}\Big)^{-1}\right\|_{p}^{m,m+1} \leq C_{m}.
\end{equation}
\end{thm}
\begin{proof}
The first statement of the theorem is a consequence of a general fact
about operators on manifolds of bounded geometry. The operator
$(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}$ is
a pseudodifferential operator of order $-2$, so it maps
$H^m(X,L^p\otimes E)$ to $H^{m+2}(X,L^p\otimes E)$.
It remains to prove the norm estimate \eqref{e:mm+1}.
To prove \eqref{e:mm+1}, first, we introduce normal coordinates
near an arbitrary point $x_0\in X$. As above, we will identify the
balls $B^{T_{x_0}X}(0,a^X)$ and $B^{X}(x_0,a^X)$ via the exponential
map $\exp^X_{x_0} : T_{x_0}X\to X$. Furthermore, we choose
a trivialization of the bundle $L$ and $E$ over $B^{X}(x_0,a^X)$,
identifying the fibers $L_Z$ and $E_Z$ of $L$ and $E$ at
$Z\in B^{T_{x_0}X}(0,a^X)\cong B^{X}(x_0,a^X)$ with $L_{x_0}$
and $E_{x_0}$ by parallel transport with respect to the connection
$\nabla^L$ and $\nabla^E$ along the curve
$\gamma_Z : [0,1]\ni u \to \exp^X_{x_0}(uZ)$.
Denote by $\nabla^{L^p\otimes E}$ and $h^{L^p\otimes E}$
the connection and the Hermitian metric on the trivial line bundle with
fiber $(L^p\otimes E)_{x_0}$ induced by this trivialization.
Let $\Gamma^L$, $\Gamma^E$ be the connection forms of
$\nabla^L$ and $\nabla^E$ with respect to some fixed frames for
$L$, $E$ which is parallel along the curve
$\gamma_Z : [0,1]\ni u \to \exp^X_{x_0}(uZ)$ under our trivialization of
$B^{T_{x_0}X}(0,\varepsilon)$.
Then we have
\begin{align}\label{e:3.32}
\nabla^{L^p}_U=\nabla_U+p\Gamma^L(U)+\Gamma^E(U).
\end{align}
For any $x_0\in X$, fix an orthonormal basis $e_1,\ldots,e_{2n}$ in
$T_{x_0}X$. We still denote by
$\{e_{j}\}_{j=1}^{2n}$ the constant vector fields $e_j(Z)=e_j$
on $B^{T_{x_0}X}(0,\varepsilon)$.
One can show that the restriction
of the norm $\|\cdot\|_{p,m}$ to
$\mathscr{C}^\infty_c(B^{T_{x_0}X}(0,\varepsilon), L^p\otimes E)
\cong \mathscr{C}^\infty_c(B^{X}(x_0,\varepsilon), L^p\otimes E)$
is equivalent uniformly on $x_0\in X$ and $p\in \mathbb N^{*}$
to the norm $\|\cdot \|^\prime_{p,m}$ given for
$u\in \mathscr{C}^\infty_c(B^{T_{x_0}X}(0,\varepsilon), L^p\otimes E)$
by
\begin{equation}\label{localSobolev}
\|u\|^\prime_{p,m}=\Big(\sum_{\ell=0}^m\sum_{j_1,\ldots,j_\ell=1}^{2n}
\int_{T_{x_0}X} \Big(\frac{1}{\sqrt{p}}\Big)^\ell|
\nabla^{L^p\otimes E}_{e_{j_1}}\cdots
\nabla^{L^p\otimes E}_{e_{j_\ell}}u|^2dZ\Big)^{1/2}.
\end{equation}
That is, there exists $C_m>0$ such that, for any $x_0\in X$,
$p\in \mathbb N^{*}$
we have
\begin{equation}\label{pm-prime}
C_m^{-1}\|u\|^\prime_{p,m}\leq \|u\|_{p,m}\leq C_m\|u\|^\prime_{p,m}\,,
\end{equation}
for any $u\in \mathscr{C}^\infty_c(B^{T_{x_0}X}(0,\varepsilon),
L^p\otimes E)
\cong \mathscr{C}^\infty_c(B^{X}(x_0,\varepsilon), L^p\otimes E)$.
By choosing an appropriate covering of $X$ by normal coordinate charts,
we can reduce our considerations to the local setting. Without loss
of generality, we can assume that
$u\in \mathscr{C}^\infty_c(B^{T_{x_0}X}(0,\varepsilon), L^p\otimes E)$
for some $x_0\in X$ and the Sobolev norm of $u$ is given by
the norm $\|u\|^\prime_{p,m}$ given by \eqref{localSobolev}.
(Later on, we omit `prime' for simplicity.)
We have to show the estimate \eqref{e:mm+1}, uniform on $x_0$.
That is, we claim that, for any $m\in \mathbb N$, there exists $C_m>0$
such that for any $p\in \mathbb N^{*}$, $p\geq p_0$,
$\lambda\in \delta$, $y\in X$, $|\alpha|<c\sqrt{p}$ and $x_0\in X$,
\begin{equation}\label{e:local}
\left\|\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-1}u
\right\|_{p,m+1}\leq C_{m}\left\|u\right\|_{p,m},\quad
u\in \mathscr{C}^\infty_c(B^{T_{x_0}X}(0,\varepsilon), L^p\otimes E).
\end{equation}
\begin{prop}\label{p:iterated}
For any $1\leq j_1 \leq \ldots \leq j_k\leq 2n$, the iterated commutator
\begin{equation}\label{e:comm}
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_1}},
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_2}},\ldots,
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_k}},
\frac{1}{p}\Delta_{p;\alpha,y}\right]\ldots \right]\right]
\end{equation}
defines a family of second order differential operators, bounded uniformly
on $p\in \mathbb N^{*}$, $y\in X$, $|\alpha|<c\sqrt{p}$ and $x_0\in X$.
In particular, there exists $C>0$ such that for $p\in \mathbb N^{*}$,
$|\alpha|<c\sqrt{p}$, $y\in X$ and
$u,u^\prime\in \mathscr{C}^\infty_c(B^{T_{x_0}X}(0,\varepsilon),
L^p\otimes E)$
\begin{multline}\label{e:3.35}
\left|\Big\langle \left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_1}},
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_2}},\ldots,
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_k}},
\frac{1}{p}\Delta_{p;\alpha,y}\right]\ldots \right]\right]u,
u^\prime \Big\rangle_{p,0}\right| \\
\leq C\|u\|_{p,1}\|u^\prime\|_{p,1}.
\end{multline}
\end{prop}
\begin{proof}
By \eqref{Lta}, \eqref{LtaZ} and \eqref{Apy},
the operator $\frac{1}{p}\Delta_{p;\alpha,y}$ has the form
\begin{equation}\label{e:3.36}
\begin{split}
\frac{1}{p}\Delta_{p;\alpha,y}=\sum_{i,j}a^{ij}_{p;\alpha,y}(Z)
\Big(\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_i}\Big)
\Big(\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_j}\Big)\\
+\sum_{\ell}b^{\ell}_{p;\alpha,y}(Z)\frac{1}{\sqrt{p}}
\nabla^{L^p\otimes E}_{e_\ell}+c_{p;\alpha,y}(Z),
\end{split}
\end{equation}
where
\begin{align*}
a_{p;\alpha,y}^{ij}(Z)=&-g^{ij}(Z),\\
b_{p;\alpha,y}^\ell(Z)=&\frac{1}{\sqrt{p}}\sum_{j,k=1}^{2n} g^{jk}(Z)
\Gamma^\ell_{jk}(Z)-\frac{2\alpha}{\sqrt{p}} \sum_{j=1}^{2n}
g^{j\ell}(Z)e_j(\widetilde{d}_{p,y}),\\
c_{p;\alpha,y}(Z)=&-\tau(Z)-\frac{\alpha}{p}\sum_{j,k=1}^{2n}
g^{jk}(Z)\Big(e_j(e_k(\widetilde{d}_{p,y}))- \Gamma^{\ell}_{jk}(Z)
e_\ell(\widetilde{d}_{p,y})\Big)\\
&-\frac{\alpha^2}{p}\sum_{j,k=1}^{2n}g^{jk}(Z)e_j(\widetilde{d}_{p,y})
e_k(\widetilde{d}_{p,y}).
\end{align*}
It is easy to see that if $f_{p;\alpha,y}$ is $a_{p;\alpha,y}^{ij}$,
$b_{p;\alpha,y}^\ell$ or $c_{p;\alpha,y}$, the iterated commutator
\[
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_1}},
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_2}},\ldots,
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_k}}, f_{p;\alpha,y}
\right]\ldots \right]\right]
\]
is a smooth function on $B^{T_{x_0}X}(0,\varepsilon)$ with sup-norm,
uniformly bounded on $p\in \mathbb N^{*}$, $y\in X$,
$|\alpha|<c\sqrt{p}$ and $x_0\in X$.
Recall the commutator relations
\begin{align}\label{e:3.37}
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_i},
\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_j}\right]
=\frac{1}{p}R^{L^p\otimes E}(e_i,e_j)=R^{L}(e_i,e_j)
+\frac{1}{p}R^{E}(e_i,e_j).
\end{align}
Using these facts, one can see that the
iterated commutator \eqref{e:comm}
has the same structure as $\frac{1}{p}\Delta_{p;\alpha,y}$.
This easily completes the proof.
\end{proof}
Now the proof of \eqref{e:local} is completed as in
\cite[Theorem 4.10]{DLM04a} or \cite[Theorem 1.9]{ma-ma08}. For any
$1\leq j_1 \leq \ldots \leq j_\ell\leq 2n$ with $\ell=1,\ldots, m$,
we can write the operator
\begin{align}\label{e:3.38}
\Big(\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_1}}\Big)
\Big(\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_2}}\Big)\ldots
\Big(\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_\ell}}\Big)
(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}
\end{align}
as a linear combination of operators of the type
\begin{multline}\label{e:3.39}
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_1}},
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_2}},\ldots,
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_k}},
(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}\right]\ldots \right]\right]
\times \\ \times \Big(\frac{1}{\sqrt{p}}
\nabla^{L^p\otimes E}_{e_{j_{k+1}}}\Big)\ldots
\Big(\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_\ell}}\Big)
\end{multline}
and of the operator
\begin{align}\label{e:3.40}
\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-1}
\Big(\frac{1}{\sqrt{p}}
\nabla^{L^p\otimes E}_{e_{j_1}}\Big) \Big(\frac{1}{\sqrt{p}}
\nabla^{L^p\otimes E}_{e_{j_2}}\Big)\ldots \Big(\frac{1}{\sqrt{p}}
\nabla^{L^p\otimes E}_{e_{j_\ell}}\Big).
\end{align}
Each commutator
\[
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_1}},
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_2}},\ldots,
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{j_k}},
(\lambda-\frac{1}{p}\Delta_{p;\alpha,y})^{-1}\right]\ldots \right]\right]
\]
is a linear combination of operators of the form
\begin{equation}\label{e:RR}
\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-1}R_1
\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-1}R_2 \ldots
R_k\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-1},
\end{equation}
where the operators $R_1,\ldots, R_k$ are of the form
\begin{align}\label{e:3.41}
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{i_1}},
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{i_2}},\ldots,
\left[\frac{1}{\sqrt{p}}\nabla^{L^p\otimes E}_{e_{i_l}},
\frac{1}{p}\Delta_{p;\alpha,y}\right]\ldots \right]\right].
\end{align}
By Proposition \ref{p:iterated}, each operator $R_j$ defines
a bounded operator from $H^1$ to $H^{-1}$ with the norm,
uniformly bounded on $p\in \mathbb N^{*}$,
$|\alpha|<c\sqrt{p}$, $y\in X$.
Therefore, by Theorem \ref{Thm1.7W}, each operator \eqref{e:RR} defines
a bounded operator from $L^2$ to $H^{1}$ with the norm, uniformly
bounded on $p\in \mathbb N^{*}$, $p\geq p_0$,
$|\alpha|<c\sqrt{p}$, $y\in X$.
This immediately completes the proof.
\end{proof}
\subsection{Pointwise exponential estimates for the resolvents}
\label{norm-Bergman}
In this section, we derive the pointwise estimates for the Schwartz
kernel $R^{(m)}_{\lambda,p}(\cdot,\cdot)\in \mathscr{C}^{-\infty}
(X\times X, \pi_{1}^{*}(L^p\otimes E)\otimes
\pi_{2}^{*}(L^p\otimes E)^*)$
of the operator $\big(\lambda-\frac{1}{p}\Delta_{p}\big)^{-m}$.
Recall that $p_{0}\in \mathbb N^{*}$, $c>0$ are given in
Theorems \ref{Thm1.7}, \ref{Thm1.7W}.
\begin{thm}\label{p:west}
For any $m,k\in \mathbb N$ with $m>2n+k+1$, for any
$p\in \mathbb N^{*}$, $p\geq p_0$, and $\lambda\in\delta$, we have
$R^{(m)}_{\lambda,p}(\cdot,\cdot)\in
\mathscr{C}^k(X\times X, \pi_1^*(L^p\otimes E)
\otimes \pi_2^*(L^p\otimes E)^*)$
and, for any $c_1\in (0,c)$, there exists $C_{m,k}>0$ such that
for any $p\in \mathbb N^{*}$, $p\geq p_0$, $\lambda\in\delta$,
$x, x^\prime \in X$, we have
\begin{align}\label{e:3.43}
\big|R^{(m)}_{\lambda,p}(x, x^\prime)\big|_{C^k}
\leq C_{m,k} p^{n+\frac{k}{2}} e^{-c_1\sqrt{p} d (x, x^\prime)}.
\end{align}
\end{thm}
\begin{proof}
By \eqref{LaL}, for any $m\in \mathbb N$, $p\in \mathbb N^{*}$,
$p\geq p_0$, $\lambda\in\delta$, $y\in X$ and $|\alpha|<c\sqrt{p}$,
we have
\begin{align}\label{e:3.44}
f_{\alpha,p,y} \Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}
f^{-1}_{\alpha,p,y}=\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}
\Big)^{-m}.
\end{align}
As in Remark \ref{r:conj}, one can show that this formula gives a
well-defined operator from $\mathscr{C}^\infty_c(X,L^p\otimes E)$
to $\mathscr{C}^{-\infty}(X,L^p\otimes E)$.
Since $\Delta_p$ is formally self-adjoint with respect to $\|\cdot\|_{p,0}$,
we have $\Delta_{p;\alpha,y}^*=\Delta_{p;-\alpha,y}$. Using this
fact and Theorem \ref{Thm1.9}, we easily get that, for any
$m_1\in \mathbb Z$, there exists $C_{m,m_1}>0$ such that,
for all $p\in \mathbb N^{*}$, $p\geq p_0$, $\lambda\in\delta$, $y\in X$
and $|\alpha|<c\sqrt{p}$, we have
\begin{equation}\label{res-est}
\left\|\Big(\lambda - \frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-m}
\right\|^{m_1,m_1+m}_p\leq C_{m,m_1}.
\end{equation}
The Schwartz kernel $R^{(m)}_{\lambda,p;\alpha, y}(\cdot,\cdot)\in
\mathscr{C}^k (X\times X, \pi^*_1(L^p\otimes E)\otimes
\pi_2^*(L^p\otimes E)^*)$ of the operator
$\big(\lambda-\frac{1}{p}\Delta_{p;\alpha,y}\big)^{-m}$
is related to the Schwartz kernel $R^{(m)}_{\lambda,p}(\cdot,\cdot)$
of the operator $\big(\lambda-\frac{1}{p}\Delta_{p}\big)^{-m}$
by the formula
\begin{align}\label{e:3.45}
R^{(m)}_{\lambda,p;\alpha, y}(x,x^\prime)
=e^{\alpha \widetilde{d}_{p,y}(x)}R^{(m)}_{\lambda,p}(x,x^\prime)
e^{-\alpha \widetilde{d}_{p,y}(x^\prime)}, \quad x, x^\prime \in X.
\end{align}
For $x,x^\prime,y\in X$ and $v\in (L^p\otimes E)_{x^\prime}$,
we can write
\begin{align}\label{e:3.46}
e^{\alpha \widetilde{d}_{p,y}(x)} R^{(m)}_{\lambda,p}(x,x^\prime)
e^{-\alpha \widetilde{d}_{p,y}(x^\prime)}v=\Big(\Big(\lambda
-\frac{1}{p}\Delta_{p;\alpha,y}\Big)^{-m}\delta_v \Big)(x)
\in (L^p\otimes E)_{x}.
\end{align}
In particular, putting $y=x^\prime$, we get
for $x,x^\prime\in X$ and $v\in (L^p\otimes E)_{x^\prime}$,
\begin{align}\label{e:3.47}
e^{\alpha \widetilde{d}_{p,x^\prime}(x)} R^{(m)}_{\lambda,p}(x,x^\prime)
e^{-\alpha \widetilde{d}_{p,x^\prime}(x^\prime)}v
=\Big(\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,x^\prime}\Big)^{-m}
\delta_v \Big)(x)\in (L^p\otimes E)_{x}.
\end{align}
By \eqref{(1.1)}, it follows that, for $0<\alpha<c\sqrt{p}$ and
$x^\prime\in X$, we have an estimate
$e^{\alpha \widetilde{d}_{p,x^\prime}(x^\prime)}\leq e^c$.
It is in this place that we need to use the smoothed distance function
$\widetilde{d}$, depending on $p$. Assuming $0<\alpha<c\sqrt{p}$,
by Propositions \ref{p:Sobolev}, \ref{p:delta} and \ref{p:west}, we get,
for $m>2n+1$, that $R^{(m)}_{\lambda,p}(x,x^\prime)$
is continuous and
\begin{equation}\label{e:3.48}
\begin{split}
\sup_{x,x^\prime\in X} e^{\alpha \widetilde{d}_{p,x^\prime}(x)}
|R^{(m)}_{\lambda,p}(x,x^\prime)|
&\leq e^c \sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,x^\prime}\Big)^{-m}
\delta_v \right\|_{\mathscr{C}^0_b}\\[3pt]
&\leq C_1p^{n/2} \sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,x^\prime}\Big)^{-m}
\delta_v \right\|_{p,n+1}\\[3pt]
&\leq C_2p^{n/2} \sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|\delta_v \right\|_{p,n+1-m}\leq C_3p^{n}.
\end{split}
\end{equation}
Similarly, for any $Q_1\in BD^{k_1}(X,L^p\otimes E)$ and
$Q_2\in BD^{k_2}(X,L^p\otimes E)$, $k_1+k_2=k$, we get with
$m>2n+k+1$,
\begin{align}\begin{split}\label{e:3.49}
\sup_{x,x^\prime\in X} e^{\alpha \widetilde{d}_{p,x^\prime}(x)}
| &(Q_{1}\otimes Q_{2})R^{(m)}_{\lambda,p}(x,x^\prime)|\\
&\leq e^c \sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|F_{\alpha,p,x^\prime}Q_1\Big(\lambda-\frac{1}{p}\Delta_{p}
\Big)^{-m}Q_2^*F^{-1}_{\alpha,p,x^\prime}\delta_v
\right\|_{\mathscr{C}^0_b}\\[3pt]
&= e^c \sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|Q_{1;\alpha,p,x^\prime}\Big(\lambda-\frac{1}{p}
\Delta_{p;\alpha,x^\prime}\Big)^{-m}Q_{2;\alpha,p,x^\prime}^*
\delta_v \right\|_{\mathscr{C}^0_b}\\[3pt]
&\leq C_1p^{n/2} \sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|Q_{1;\alpha,p,x^\prime}\Big(\lambda-\frac{1}{p}
\Delta_{p;\alpha,x^\prime}\Big)^{-m}Q_{2;\alpha,p,x^\prime}^*
\delta_v \right\|_{p,n+1}
\end{split}\end{align}
\begin{align}\begin{split}\nonumber
&\hspace{30mm}\leq C_2p^{(n+k_1)/2}
\sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|\Big(\lambda-\frac{1}{p}\Delta_{p;\alpha,x^\prime}\Big)^{-m}
Q_{2;\alpha,p,x^\prime}^*\delta_v \right\|_{p,n+k_1+1}\\[3pt]
&\hspace{30mm}\leq C_3p^{(n+k_1)/2}
\sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|Q_{2;\alpha,p,x^\prime}^*\delta_v \right\|_{p,n-m+k_1+1}\\[3pt]
&\hspace{30mm}\leq C_4p^{(n+k)/2}
\sup_{v\in (L^p\otimes E)_{x^\prime}, |v|=1}
\left\|\delta_v \right\|_{p,n-m+k+1}\leq C_5p^{n+k/2}.
\end{split}\end{align}
For any $x_0\in X$, fix an orthonormal basis $e_1,\ldots,e_{2n}$ in
$T_{x_0}X$. As above, we extend it to
a frame $e_1,\ldots,e_{2n}$ on $B^{X}(0,\varepsilon)$
as constant vector fields on $T_{x_{0}}X$.
One can show (see, for instance, \cite[Proposition 1.5]{Kor91})
that the norm $\|\cdot\|_{\mathscr{C}^k_b}$ on
$\mathscr{C}^k_b(X, L^p\otimes E)$ is equivalent uniformly on
$p\in \mathbb N^{*}$ to the norm $\|\cdot\|^\prime_{\mathscr{C}^k_b}$
given for $ u\in \mathscr{C}^k_b(X, L^p\otimes E)$ by
\begin{align}\label{e:3.50}
\|u\|^\prime_{\mathscr{C}^k_b}=\sup_{x_0\in X}\sup_{0\leq \ell \leq k}
\left\| \nabla^{L^p\otimes E}_{e_{j_1}}\cdots
\nabla^{L^p\otimes E}_{e_{j_\ell}}u(x_0)\right\|.
\end{align}
That is, there exists $C_k>0$ such that, for any $p\in \mathbb N^{*}$
and $u\in \mathscr{C}^k_b(X, L^p\otimes E)$, we have
\begin{equation}\label{Ckb-prime}
C_k^{-1}\|u\|_{\mathscr{C}^k_b}\leq \|u\|^\prime_{\mathscr{C}^k_b}
\leq C_k\|u\|_{\mathscr{C}^k_b}.
\end{equation}
Let $\phi\in \mathscr{C}^\infty_c(\mathbb R^{2n})$ be any function
supported in the ball $B(0,\varepsilon)$ such that $\phi\equiv 1$ on
$B(0,\varepsilon/2)$. Consider the function
$\phi_{x_0}\in \mathscr{C}^\infty_c(B^{X}(x_0,\varepsilon))$,
corresponding to $\phi$ under the isomorphisms
$\mathscr{C}^\infty_c(B(0,\varepsilon))\cong
\mathscr{C}^\infty_c(B^{T_{x_0}X}(0,\varepsilon))
\cong \mathscr{C}^\infty_c(B^{X}(x_0,\varepsilon))$
induced by the basis $e_1,\ldots,e_{2n}$ and the exponential map
$\exp^X_{x_0}$. The family $\{\phi_{x_0}, x_0\in X\}$ is bounded in
$\mathscr{C}^\infty_b(X)$.
By \eqref{Ckb-prime}, it follows that there exists $C_k>0$ such that,
for any $x,x^\prime\in X$, we have
\begin{align}\label{e:3.51}
|R^{(m)}_{\lambda,p}(x,x^\prime)|_{\mathscr{C}^k}
\leq C_k\sup_{Q_1,Q_2} |(Q_{1}\otimes Q_{2})
R^{(m)}_{\lambda,p}(x,x^\prime)|,
\end{align}
with the supremum taken over all pairs $(Q_1,Q_2)$, where
$Q_1\in BD^{k_1}(X,L^p\otimes E)$ and
$Q_2\in BD^{k_2}(X,L^p\otimes E)$, $k_1+k_2\leq k$, have the form
\begin{align}\label{e:3.52}
Q_1=\nabla^{L^p\otimes E}_{e_{i_1}}\cdots
\nabla^{L^p\otimes E}_{e_{i_{k_1}}}\phi_x,\quad
Q_2=\nabla^{L^p\otimes E}_{e_{j_1}}\cdots
\nabla^{L^p\otimes E}_{e_{j_{k_2}}}\phi_{x^\prime},
\end{align}
and $i_1,\ldots, i_{k_1},j_1,\ldots, j_{k_2}\in \{1,\ldots,2n\}$,
that immediately completes the proof of \eqref{e:3.43}.
\end{proof}
\subsection{Proof of Theorems \ref{t:mainPp} and~\ref{t:covering}}
In this section, we complete the proofs of Theorems \ref{t:mainPp}
and~\ref{t:covering}.
\begin{proof}[Proof of Theorem \ref{t:mainPp}]
Let $k\in \mathbb N$. Take an arbitrary $m>2n+k+1$.
By \eqref{bergman-integral}, we have
\begin{equation}\label{bergman-integral1}
P_{p}(x,x^\prime)=\frac{1}{2\pi i}
\int_\delta \lambda^{m-1}R^{(m)}_{\lambda,p}(x, x^\prime)\,d\lambda.
\end{equation}
By Theorem~\ref{p:west}, it implies immediately the
$\mathscr{C}^k$-estimate in Theorem \ref{t:mainPp} with $c>0$
given by that theorem.
\end{proof}
\begin{proof}[Proof of Theorem \ref{t:covering}] First, we observe that,
for $\widetilde \tau\in \mathscr{C}^\infty(\widetilde X)$ and
$\tau\in \mathscr{C}^\infty(X)$ given by \eqref{tau}, we have
$\widetilde \tau=\pi^*\tau$. Next, the quantities $\widetilde\mu_0$
and $\mu_0$ defined by \eqref{mu} are equal. In particular,
$\widetilde\mu_0>0$.
By Theorem~\ref{t:gap0}, there exists constant $C_L>0$ such that
for any $p\in \mathbb N^{*}$
\begin{align}\label{e:3.55}
\sigma(\Delta_p)\cup \sigma(\widetilde \Delta_p)\subset [-C_L,C_L]
\cup [2p\mu_0-C_L,+\infty).
\end{align}
Recall that $\delta$ denotes the counterclockwise oriented circle in
$\mathbb C$ centered at $0$ of radius $\mu_0$.
For any $p,m\in \mathbb N$, $p\geq p_0$, and $\lambda\in\delta$,
denote by $\widetilde R^{(m)}_{\lambda,p}\in
\mathscr{C}^{-\infty}(\widetilde X\times \widetilde X, \pi_1^*
(\widetilde L^p\otimes \widetilde E)\otimes
\pi_2^*(\widetilde L^p\otimes \widetilde E)^*)$
and $R^{(m)}_{\lambda,p}\in \mathscr{C}^{-\infty}(X\times X,
\pi_1^*(L^p\otimes E)\otimes \pi_2^*(L^p\otimes E)^*)$
the Schwartz kernels of the operators
$\big(\lambda-\frac{1}{p}\widetilde \Delta_{p}\big)^{-m}$
and $\big(\lambda-\frac{1}{p}\Delta_{p}\big)^{-m}$, respectively.
Recall that, for any $x^\prime$, they satisfy the identities
\begin{equation}\label{delta-eq0}
\Big(\lambda-\frac{1}{p}\widetilde \Delta_{p}\Big)^{m}
\widetilde R^{(m)}_{\lambda,p}(\cdot,x^\prime)
=\delta_{x^\prime},\quad \Big(\lambda-\frac{1}{p}\Delta_{p}\Big)^{m}
R^{(m)}_{\lambda,p}(\cdot,x^\prime)=\delta_{x^\prime}.
\end{equation}
Moreover, $u=R^{(m)}_{\lambda,p}(\cdot,x^\prime)$ is the unique
distributional solution of the equation
\begin{equation}\label{delta-eq}
\Big(\lambda-\frac{1}{p}\Delta_{p}\Big)^{m}u=\delta_{x^\prime}.
\end{equation}
Let $m>2n+1$. Then, by elliptic regularity and Sobolev embedding theorem,
$\widetilde R^{(m)}_{\lambda,p}$ and $R^{(m)}_{\lambda,p}$
are continuous.
We claim that there exists $p_1\in\mathbb N$ such that for any
$p>p_1$ and $x,x^\prime\in \widetilde X$,
\begin{equation}\label{av-res}
\sum_{\gamma\in \Gamma}\widetilde
R^{(m)}_{\lambda,p}(\gamma x,x^\prime)
= R^{(m)}_{\lambda,p}(\pi(x), \pi(x^\prime)).
\end{equation}
By \cite{Milnor}, there exists $K>0$ such that
$\sum_{\gamma\in \Gamma}e^{-ad(\gamma x, x^\prime)}<+\infty$
for any
$a>K$ and $x,x^\prime\in \widetilde X$. Put $p_1\geq K^2/c^2+p_0$.
Then, by Theorem~\ref{p:west}, for any $p>p_1$ and
$x,x^\prime\in \widetilde X$, the series in
the left-hand side of \eqref{av-res}
is absolutely convergent with respect to $\mathscr{C}^0$-norm
and its sum
\begin{align}\label{e:3.56}
\widetilde S^{(m)}_{\lambda,p}(x,x^\prime)
=\sum_{\gamma\in \Gamma}\widetilde R^{(m)}_{\lambda,p}
(\gamma x,x^\prime)
\end{align}
is a $\Gamma$-invariant continuous section on
$\widetilde X\times \widetilde X$.
So we can write
\begin{align}\label{e:3.57}
\widetilde S^{(m)}_{\lambda,p}(x,x^\prime)
=S^{(m)}_{\lambda,p}(\pi(x), \pi(x^\prime))
\text{ with } S^{(m)}_{\lambda,p}\in \mathscr{C}^0
(X\times X, \pi_1^*(L^p\otimes E)\otimes \pi_2^*(L^p\otimes E)^*).
\end{align}
Moreover, by \eqref{delta-eq0}, for any $x^\prime \in \widetilde X$,
$\widetilde S^{(m)}_{\lambda,p}(\cdot,x^\prime)$ satisfies the identity
\begin{equation}\label{e:3.58}
\Big(\lambda-\frac{1}{p}\widetilde \Delta_{p}\Big)^{m} \,
\widetilde S^{(m)}_{\lambda,p}(\cdot,x^\prime)=\delta_{x^\prime}.
\end{equation}
Therefore, for any $x^\prime\in X$,
$S^{(m)}_{\lambda,p}(\cdot,x^\prime)$ satisfies the identity
\begin{equation}\label{e:3.59}
\Big(\lambda-\frac{1}{p}\Delta_{p}\Big)^{m}\,
S^{(m)}_{\lambda,p}(\cdot,x^\prime)=\delta_{x^\prime}.
\end{equation}
By the uniqueness of the solution of \eqref{delta-eq}, it follows that
$S^{(m)}_{\lambda,p}=R^{(m)}_{\lambda,p}$. This completes the
proof of \eqref{av-res}.
Since, by Theorem~\ref{p:west}, the series in the left-hand side of
\eqref{av-res} is absolutely convergent with respect to the
$\mathscr{C}^0$-norm uniformly in $\lambda\in \delta$, we can integrate
it term by term over $\delta$. Using \eqref{bergman-integral1} and
\eqref{av-res},
for any $p>p_1$ and $x,x^\prime\in \widetilde X$, we get
\begin{equation}\label{e:3.61}
\begin{split}
\sum_{\gamma\in \Gamma}\widetilde P_p(\gamma x,x^\prime)
&=\frac{1}{2\pi i}\sum_{\gamma\in \Gamma} \int_\delta \lambda^{m-1}
\widetilde R^{(m)}_{\lambda,p}(\gamma x, x^\prime)\,d\lambda\\
&= \frac{1}{2\pi i}
\int_\delta \lambda^{m-1}R^{(m)}_{\lambda,p}(\pi(x), \pi(x^\prime))\,
d\lambda= P_p(\pi(x), \pi(x^\prime)).
\end{split}
\end{equation}
The proof of Theorem \ref{t:covering} is completed.
\end{proof}
\section{Full off-diagonal asymptotic expansion} \label{expansions}
In this section, we study the full off-diagonal asymptotic expansion
in the geometric situation of our paper described in the Introduction.
In the case of a compact K\"ahler manifold the asymptotic
expansion of the Bergman kernel
$P_p(x,x)$ restricted to the diagonal was
initiated by Tian \cite{Tian}, who proved the expansion up to first order.
Catlin \cite{Catlin} and Zelditch \cite{Zelditch98} proved the asymptotic
expansion of $P_p(x,x)$ up to arbitrary order, see \cite{MM07} for the
numerous applications of these results.
On the other hand, the off-diagonal expansion of the Bergman kernel has
many applications.
In the case of complex manifolds the expansion of $P_p(x,x')$
holds in a fixed neighborhood of the diagonal (independent of $p$),
see \cite{DLM04a}, \cite[Theorem 4.2.1]{MM07}. Such kind of expansion
is called full off-diagonal expansion.
As already noted in \cite[Problem 6.1, p.\ 292]{MM07}
the proof of the full off-diagonal expansion holds also for
complex manifolds with bounded geometry.
In the case of the Bergman kernel associated to the
renormalized Bochner-Laplacian considered
in the present paper, it was shown
in \cite[Theorem 1.19]{ma-ma08} that the off-diagonal expansion holds in
a neighborhood of size $1/\sqrt{p}$ of the diagonal. This is called
near off-diagonal expansion.
Moreover, it was shown in \cite[p.\,329]{MM07}
that the Bergman kernel is $\mathcal{O}(p^{-\infty})$ outside
a neighborhood of size $p^{-\theta}$, for any $\theta\in(0,1/2)$.
These estimates are used in the proof of the Kodaira embedding theorem
for symplectic manifolds \cite[Theorem 3.6]{ma-ma08}.
In \cite{lu-ma-ma} a less precise estimate than
\cite[Theorem 1.19]{ma-ma08}
was obtained in a neighborhood
of size $p^{-\theta}$, $\theta\in(0,1/2)$, which is however enough to
derive the Berezin-Toeplitz quantization for the quantum spaced $\mH_p$
in \cite{ioos-lu-ma-ma}.
Finally, in \cite{bergman} the full off-diagonal expansion
for the Bergman kernel associated to the renormalized
Bochner-Laplacian was proved by combining \cite[\S 1]{ma-ma08}
and weight function trick in \cite{Kor91}.
In this section we extend the results on the full off-diagonal asymptotic
expansion to the case of manifolds of bounded geometry. Moreover,
using the technique of weighted estimates of the previous sections,
we slightly improve the remainder in the asymptotic expansions.
We will keep the setting described in Introduction.
Consider the fiberwise product $TX\times_X TX=\{(Z,Z^\prime)
\in T_{x_0}X\times T_{x_0}X : x_0\in X\}$. Let
$\pi : TX\times_X TX\to X$ be the natural projection given by
$\pi(Z,Z^\prime)=x_0$. The kernel $P_{q,p}(x,x^\prime)$
(of the operator $(\Delta_{p})^{q} P_{\mathcal{H}_{p}}$) induces
a smooth section $P_{q,p,x_0}(Z,Z^\prime)$ of the vector bundle
$\pi^*(\operatorname{End}(E))$ on $TX\times_X TX$ defined for all
$x_0\in X$ and $Z,Z^\prime\in T_{x_0}X$ with $|Z|, |Z^\prime|<a_X$.
We will follow the arguments of \cite{bergman} and \cite{ma-ma08}.
We will use the normal coordinates near an arbitrary point $x_0\in X$
introduced in the proof of Theorem~\ref{Thm1.9}. Let $\{e_j\}$ be
an oriented orthonormal basis of $T_{x_0}X$. It gives rise to an
isomorphism $X_0:=\mathbb R^{2n}\cong T_{x_0}X$.
Consider the trivial bundles $L_0$ and $E_0$ with fibers $L_{x_0}$
and $E_{x_0}$ on $X_0$. Recall that we have the Riemannian metric
$g$ on $B^{T_{x_0}X}(0,a^X)$ as well as the connections
$\nabla^L$, $\nabla^E$ and the Hermitian metrics $h^L$, $h^E$
on the restrictions of $L_0$, $E_0$ to $B^{T_{x_0}X}(0,a^X)$
induced by the identification $B^{T_{x_0}X}(0,a^X)\cong B^{X}(x_0,a^X)$.
In particular, $h^L$, $h^E$ are the constant metrics
$h^{L_0}=h^{L_{x_0}}$, $h^{E_0}=h^{E_{x_0}}$.
For $\varepsilon \in (0,a^X/4)$, one can extend these geometric objects
from $B^{T_{x_0}X}(0,\varepsilon)$ to $X_0\cong T_{x_0}X$
in the following way.
Let $\rho : \mathbb R\to [0,1]$ be a smooth even function such
that $\rho(v)=1$ if $|v|<2$ and $\rho(v)=0$ if $|v|>4$.
Let $\varphi_\varepsilon : \mathbb R^{2n}\to \mathbb R^{2n}$
be the map defined by $\varphi_\varepsilon(Z)=\rho(|Z|/\varepsilon)Z$.
We equip $X_0$ with the metric $g_0(Z)=g(\varphi_\varepsilon(Z))$.
Set $\nabla^{E_0}=\varphi^*_\varepsilon\nabla^E$. Define the Hermitian
connection $\nabla^{L_0}$ on $(L_0,h^{L_0})$
by (cf. \cite[(4.23)]{DLM04a}, \cite[(1.21)]{ma-ma08})
\begin{align}\label{e:4.1}
\nabla^{L_0}\left|_Z\right.=\varphi_\varepsilon^*\nabla^L
+\frac 12(1-\rho^2(|Z|/\varepsilon))R^L_{x_0}(\mathcal R,\cdot),
\end{align}
where $\mathcal R(Z)=\sum_j Z_je_j\in \mathbb R^{2n}\cong T_ZX_0$.
By \cite[(4.24)]{DLM04a} or \cite[(1.22)]{ma-ma08},
if $\varepsilon$ is small enough, then the curvature $R^{L_0}$
is positive and satisfies the following estimate for any $x_0\in X$,
\begin{align}\label{e:4.2}
\inf_{u\in T_{x_0}X\setminus\{0\}}\frac{iR^{L_0}(u,J^{L_0}u)}
{|u|_g^2}\geq \frac{4}{5}\mu_0.
\end{align}
Here $J^{L_0}$ is the almost complex structure on $X_0$ defined by
$g_0$ and $iR^{L_0}$. From now on, we fix such an $\varepsilon>0$.
Let $dv_{TX}$ be the Riemannian volume form of
$(T_{x_0}X, g^{T_{x_0}X})$. Let $\kappa$ be the smooth positive
function on $X_0$ defined by the equation
\begin{equation}\label{e:kappa}
dv_{X_0}(Z)=\kappa(Z)dv_{TX}(Z), \quad Z\in X_0.
\end{equation}
Let $\Delta_p^{X_0}=\Delta^{L_0^p\otimes E_0}-p\tau_0$ be
the associated renormalized Bochner-Laplacian acting on
$\mathscr{C}^\infty(X_0,L_0^p\otimes E_0)$. Then
(cf. \cite[(1.23)]{ma-ma08}) there exists a constant $C_{L_0}>0$
such that for any $p$
\begin{equation}\label{gap0}
\sigma(\Delta_p^{X_0})\subset [-C_{L_0},C_{L_0}]\cup
\left[\frac{8}{5}p\mu_0-C_{L_0},+\infty\right).
\end{equation}
Consider the subspace $\mathcal H^0_p$ in
$\mathscr{C}^\infty(X_0,L_0^p\otimes E_0)\cong
\mathscr{C}^\infty(\mathbb R^{2n}, E_{x_0})$
spanned by the eigensections of $\Delta^{X_0}_p$
corresponding to eigenvalues in
$[-C_{L_0},C_{L_0}]$. Let $P_{\mathcal H^0_p}$
be the orthogonal projection onto
$\mathcal H^0_p$. The smooth kernel of
$(\Delta^{X_0}_p)^qP_{\mathcal H^0_p}$
with respect to the Riemannian volume form $dv_{X_0}$ is denoted by
$P^0_{q,p}(Z,Z^\prime)$. As proved in \cite[Proposition 1.3]{ma-ma08},
the kernels $P_{q,p,\,x_0}(Z,Z^\prime)$ and $P^0_{q,p}(Z,Z^\prime)$
are asymptotically close on $B^{T_{x_0}X}(0,\varepsilon)$ in the
$\mathscr{C}^\infty$-topology, as $p\to \infty$.
In the next theorem,
we improve the $\mathcal O(p^{-\infty})$-estimate for the norms
of the difference between $P_{q,p,x_0}(Z,Z^\prime)$ and
$P^0_{q,p}(Z,Z^\prime)$ of \cite[Proposition 1.3]{ma-ma08},
proving an exponential decay estimate. This is essentially the main
new result of this section.
\begin{thm}\label{prop13}
There exists $c_0>0$ such that, for any $k\in \mathbb N$,
there exists $C_{k}>0$ such that for $p\in \mathbb N^{*}$, $x_0\in X$
and $Z,Z^\prime\in B^{X_0}(0,\varepsilon)$,
\begin{align}\label{e:4.5}
\big|P_{q,p,x_0}(Z,Z^\prime)-P^0_{q,p}(Z,Z^\prime)
\big|_{\mathscr{C}^k}\leq C_{k}e^{-c_0\sqrt{p}}.
\end{align}
\end{thm}
\begin{proof}
As in Section~\ref{norm-Bergman}, $R^{(m)}_{\lambda,p}\in
\mathscr{C}^{-\infty}(X\times X, \pi_1^*(L^p\otimes E)\otimes
\pi_2^*(L^p\otimes E)^*)$ denotes the Schwartz kernel of the operator
$\Big(\lambda-\frac{1}{p}\Delta_{p}\Big)^{-m}$. We also denote by
\begin{align}\label{e:4.6}
R^{X_0,(m)}_{\lambda,p}\in \mathscr{C}^{-\infty}(X_0\times X_0,
\pi_1^*(L_0^p\otimes E_0)\otimes \pi_2^*(L^p_0\otimes E_0)^*)
\end{align}
the Schwartz kernel of the operator $\Big(\lambda-\frac{1}{p}
\Delta^{X_0}_{p}\Big)^{-m}$. For $m>2n+1$, the distributional sections
$R^{(m)}_{\lambda,p}(x,x^\prime)$ and
$R^{X_0,(m)}_{\lambda,p}(Z,Z^\prime)$ are continuous sections.
Recall that $d$ denotes the geodesic distance on $X$ and $d^{X_0}$
the geodesic distance on $X_0$. By construction, $d$ coincides with
$d^{X_0}$ on $B^X(x_0, 2\varepsilon)\times B^{X}(x_0, 2\varepsilon)
\cong B^{X_0}(0, 2\varepsilon)\times B^{X_0}(0, 2\varepsilon)$.
Let $\psi \in \mathscr{C}^\infty_c(B^{X_0}(0,4\varepsilon))$,
$\psi\equiv 1$ on $B^{X_0}(0,3\varepsilon)$.
Consider a function $\chi\in \mathscr{C}^\infty(\mathbb R)$ such that
$\chi(r)=1$ for $|r|\leq \varepsilon$ and $\chi(r)=0$ for
$|r|\geq 2\varepsilon$.
For any $\alpha\in \mathbb R$ and $W\in X_0$, we introduce
a weight function $\phi^{X_0}_{\alpha,W} \in \mathscr{C}^{\infty}(X_0)$
by
\begin{align}\label{e:4.7}
\phi^{X_0}_{\alpha,W}(Z) = \exp \left[{\alpha
\chi(d ^{X_0}(Z,W))}\right],\quad Z \in X_0.
\end{align}
Consider the operators
\begin{align}\label{e:4.8}
\Delta^{X_0}_{p;\alpha,W}:=\phi^{X_0}_{\alpha,W}
\Delta^{X_0}_{p}(\phi^{X_0}_{\alpha,W})^{-1}.
\end{align}
With the same arguments as in Theorem \ref{Thm1.9},
one can show that
\begin{thm}\label{Thm1.9a}
There exist $c_0>0$ and $p_0\in \mathbb N^{*}$ such that, for any
$p\in \mathbb N^{*}$, $p\geq p_0$, $\lambda\in \delta$, $W\in X_0$
and $|\alpha|<c_0\sqrt{p}$,
the operator $\lambda-\frac{1}{p}\Delta^{X_0}_{p;\alpha,W}$
is invertible in $L^2$. Moreover, for any $m\in \mathbb N$,
the resolvent
$\big(\lambda-\frac{1}{p}\Delta^{X_0}_{p;\alpha,W}\big)^{-1}$
maps $H^m$ to $H^{m+1}$ with the following norm estimates:
\begin{equation}\label{e:mm+1-again}
\left\|\Big(\lambda-\frac{1}{p}\Delta^{X_0}_{p;\alpha,W}\Big)^{-1}
\right\|_{p}^{m,m+1}\leq C,
\end{equation}
where $C>0$ is independent of $p\in \mathbb N^{*}$, $p\geq p_0$,
$\lambda\in \delta$, $W\in X_0$ and $x_0\in X$.
\end{thm}
For any $Z,W\in B^{X_0}(0,\varepsilon/2)$, we have
$d^{X_0}(Z,W)<\varepsilon$ and, therefore,
$\phi^{X_0}_{\alpha,W}(Z) =e^{\alpha}$.
For $m>2n+1$, we have
\begin{equation}\label{e:diff}
\begin{split}
&\sup_{Z,W\in B^{X_0}(0,\varepsilon/2)} |R^{(m)}_{\lambda,p}(Z,W)
-R^{X_0, (m)}_{\lambda,p}(Z,W)| \\
&\leq e^{-\frac 12c_0\sqrt{p}}\sup_{Z,W\in B^{X_0}(0,\varepsilon/2)}
\left|\phi_{\frac 12c_0\sqrt{p},W}^{X_0}(Z) \Big(\psi(Z)
R^{(m)}_{\lambda,p}(Z,W)-R^{X_0, (m)}_{\lambda,p}(Z,W)\psi(W)\Big)
\right|\\
&\leq e^{-\frac 12c_0\sqrt{p}}
\sup_{\substack{W\in B^{X_0}(0,\varepsilon/2),\\
v\in (L_0^p\otimes E_0)_W, |v|=1}}
\left\|\phi_{\frac 12c_0\sqrt{p},W}^{X_0}
\Big(\psi \Big(\lambda-\frac{1}{p}\Delta_{p}\Big)^{-m}
-\Big(\lambda-\frac{1}{p}\Delta^{X_0}_{p}\Big)^{-m}\psi\Big)
\delta_v \right\|_{\mathscr{C}^0_b}\\
&\leq Cp^{n/2}
e^{-\frac 12c_0\sqrt{p}}
\sup_{\substack{W\in B^{X_0}(0,\varepsilon/2),\\
v\in (L_0^p\otimes E_0)_W, |v|=1}}
\left\|\phi_{\frac 12c_0\sqrt{p},W}^{X_0}
\Big(\psi \Big(\lambda-\frac{1}{p}\Delta_{p}\Big)^{-m}
-\Big(\lambda-\frac{1}{p}\Delta^{X_0}_{p}\Big)^{-m}\psi\Big)
\delta_v \right\|_{p,n+1}.
\end{split}
\end{equation}
By construction, we have for any
$u \in \mathscr{C}^\infty_c(B^{X_0}(0,2\varepsilon))$,
\begin{align}\label{e:4.10}
\Delta_pu(Z)=\Delta^{X_0}_pu(Z)
\end{align}
Then, for any $u$, we have
\begin{equation}\label{e:4.11}
\Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{m}
\psi\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}u =
\left[\Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{m},
\psi\right]\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}u+\psi u
\end{equation}
and
\begin{multline}\label{e:4.12}
\psi\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}u
- \Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{-m}\psi u\\
=\Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{-m}
\left[\Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{m},
\psi\right]\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}u.
\end{multline}
Now, for any $p\in \mathbb N^{*}$, $\lambda\in \delta$ and $W\in X_0$,
by Theorem~\ref{Thm1.9a} and \eqref{e:4.12}, we have for $m>2n+1$,
$v\in (L_0^p\otimes E_0)_W, |v|=1$,
\begin{multline}\label{e:4.13}
\left\|\phi^{X_0}_{\frac 12c_0\sqrt{p},W}
\Big(\psi\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}
- \Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{-m}\psi\Big)
\delta_v\right\|_{p,n+1}\\
\leq C \left\|\phi^{X_0}_{\frac 12c_0\sqrt{p},W}
\left[\Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{m},
\psi\right]\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}
\delta_v\right\|_{p,n+1-m}.
\end{multline}
Since the operator
$\left[\Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{m}, \psi\right]$
vanishes on $B^{X_0}(0,3\varepsilon)$, for any
$W\in B^{X_0}(0,\varepsilon)$ we have
$d^{X_0}(W,Z)>2\varepsilon$ and, therefore,
$\phi^{X_0}_{\frac 12c_0\sqrt{p},W}(Z)=1$ on the support of
$\left[\Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{m}, \psi\right]$.
Hence, for $W\in B^{X_0}(0,\varepsilon)$, by \eqref{e:4.13}, we get
\begin{equation}\label{e:4.14}
\begin{split}
&\left\|\phi^{X_0}_{\frac 12c_0\sqrt{p},W}
\Big(\psi\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}
- \Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{-m}\psi\Big)
\delta_v\right\|_{p,n+1}\\[3pt]
&\leq C \left\|
\left[\Big(\lambda-\frac{1}{p}\Delta^{X_0}_p\Big)^{m},
\psi\right]\!\Big(\lambda-\frac{1}{p}\Delta_p\Big)^{-m}\delta_v
\right\|_{p,n+1-m}\\[3pt]
&\leq C \left\|\Big(\lambda
-\frac{1}{p}\Delta_p\Big)^{-m}\delta_v\right\|_{p,n+1}
\leq C \left\|\delta_v\right\|_{p,n+1-m}\leq C p^{n/2}.
\end{split}
\end{equation}
and, finally, by \eqref{e:diff} and \eqref{e:4.14}, we get
\begin{equation}\label{e:4.15}
\sup_{Z,W\in B^{X_0}(0,\varepsilon/2)} \big|R^{(m)}_{\lambda,p}(Z,W)
-R^{X_0, (m)}_{\lambda,p}(Z,W)\big| \leq C_1p^{n}
e^{-\frac 12c_0\sqrt{p}} \leq C_2 e^{-\frac 14c_0\sqrt{p}}.
\end{equation}
Using \eqref{bergman-integral1}, we complete the proof for $k=0$.
The proof of the case of arbitrary $k$ can be given similarly to
the proof of Theorem \ref{p:west}.
\end{proof}
Now we can proceed as in \cite{bergman,ma-ma08}. We only observe
that all the constants in the estimates in \cite{bergman,ma-ma08} depend
on finitely many derivatives of $g^{TX}$, $h^L$, $\nabla^L$, $h^E$,
$\nabla^E$, $J$ and the lower bound of $g^{TX}$. Therefore, by
the bounded geometry assumptions, all the estimates are uniform on
the parameter $x_0\in X$. We will omit the details and give only the
final result, stating the full off-diagonal asymptotic expansion of the
generalized Bergman kernel $P_{q,p}$ as $p\to \infty$
(see Theorem~\ref{t:main} below).
The almost complex structure $J_{x_0}$ induces a decomposition
$T_{x_0}X\otimes_{\mathbb R}\mathbb
C=T^{(1,0)}_{x_0}X\oplus T^{(0,1)}_{x_0}X$, where
$T^{(1,0)}_{x_0}X$ and $T^{(0,1)}_{x_0}X$ are the eigenspaces
of $J_{x_0}$ corresponding to eigenvalues $i$ and $-i$ respectively.
Denote by $\det_{\mathbb C}$ the determinant function of the complex
space $T^{(1,0)}_{x_0}X$. Put
\begin{align}\label{e:4.17}
\mathcal J_{x_0}=-2\pi i J_0.
\end{align}
Then $\mathcal J_{x_0} : T^{(1,0)}_{x_0}X\to T^{(1,0)}_{x_0}X$
is positive, and $\mathcal J_{x_0} : T_{x_0}X\to T_{x_0}X$ is
skew-adjoint \cite[(1.81)]{ma-ma08} (cf. \cite[(4.114)]{DLM04a}).
We define a function $\mathcal P=\mathcal P_{x_0}\in
\mathscr{C}^\infty(T_{x_0}X\times T_{x_0}X)$ by
\begin{equation}\label{e:Bergman}
\mathcal P(Z, Z^\prime)
=\frac{\det_{\mathbb C}\mathcal J_{x_0}}{(2\pi)^n}
\exp\Big(-\frac 14\langle (\mathcal J^2_{x_0})^{1/2}(Z-Z^\prime),
(Z-Z^\prime)\rangle +\frac 12 \langle \mathcal J_{x_0} Z, Z^\prime
\rangle \Big).
\end{equation}
It is the Bergman kernel of the second order differential operator
$\mathcal L_0$ on $\mathscr{C}^\infty(T_{x_0}X,\mathbb{C})$ given by
\begin{equation}\label{e:L0}
\mathcal L_0=-\sum_{j=1}^{2n} \Big(\nabla_{e_j}
+\frac 12 R^L_{x_0}(Z,e_j)\Big)^2-\tau (x_0),
\end{equation}
where $\{e_j\}_{j=1,\ldots,2n}$ is an orthonormal base in $T_{x_0}X$.
Here, for $U\in T_{x_0}X$, we denote by $\nabla_U$ the ordinary operator
of differentiation in the direction $U$ on the space
$\mathscr{C}^\infty(T_{x_0}X, \mathbb{C})$. Thus, $\mathcal P$
is the smooth kernel (with respect to $dv_{TX}$) of the orthogonal
projection in $L^2(T_{x_0}X, \mathbb{C})$
to the kernel of $\mathcal L_0$.
Let $\kappa$ be the function defined in \eqref{e:kappa}.
\begin{thm}\label{t:main}
There exists $\varepsilon\in (0,a^X)$ such that, for any
$j,m,m^\prime\in \mathbb N$, $j\geq 2q$, there exist positive
constants $C$, $c$ and $M$ such that for any $p\geq 1$, $x_0\in X$
and $Z,Z^\prime\in T_{x_0}X$, $|Z|, |Z^\prime|<\varepsilon$, we have
\begin{multline}\label{e:main}
\sup_{|\alpha|+|\alpha^\prime|\leq m}
\Bigg|\frac{\partial^{|\alpha|+|\alpha^\prime|}}
{\partial Z^\alpha\partial Z^{\prime\alpha^\prime}}
\Bigg(\frac{1}{p^n}P_{q,p,x_0}(Z,Z^\prime)
-\sum_{r=2q}^jF_{q,r,x_0}(\sqrt{p} Z, \sqrt{p}Z^\prime)
\kappa^{-\frac 12}(Z)\kappa^{-\frac 12}(Z^\prime)p^{-\frac{r}{2}+q}
\Bigg)\Bigg|_{\mathscr{C}^{m^\prime}(X)}\\
\leq Cp^{-\frac{j-m+1}{2}+q}(1+\sqrt{p}|Z|+\sqrt{p}|Z^\prime|)^M
\exp(-c\sqrt{\mu_0p}|Z-Z^\prime|)+\mathcal O(e^{-c_0\sqrt{p}}),
\end{multline}
where
\begin{align}\label{e:4.19}
F_{q,r,x_0}(Z,Z^\prime)=J_{q,r,x_0}(Z,Z^\prime)\mathcal
P_{x_0}(Z,Z^\prime),
\end{align}
$J_{q,r,x_0}(Z,Z^\prime)$ are polynomials in $Z, Z^\prime$,
depending smoothly on $x_0$, with the same parity as $r$ and
$\operatorname{deg} J_{q,r,x_0}\leq 3r$.
\end{thm}
Here $\mathscr{C}^{m^\prime}(X)$ is the
$\mathscr{C}^{m^\prime}$-norm for the parameter $x_0\in X$.
Note that the summation in \eqref{e:main} starts from $r=2q$
and \eqref{e:4.19} due
to \cite[Th 1.18]{ma-ma08}.
\section{Berezin-Toeplitz quantization on orbifolds} \label{pbs4}
After the pioneering work of Berezin, the Berezin-Toeplitz
quantization achieved generality for compact
K\"ahler manifolds and trivial bundle $E$
through the works \cite{BouGou81,BouSt76, BMS94}.
We refer to \cite{ma:ICMtalk,MM07,ma-ma08,MM11}
for more references and background.
The theory of Berezin-Toeplitz quantization on K\"ahler
and symplectic orbifolds was first established by
Ma and Marinescu \cite[Theorems 6.14, 6.16]{ma-ma08a}
by using as quantum spaces the kernel of the spin$^c$ Dirac operator.
Especially,
they showed that the set of Toeplitz operators forms an algebra.
The main tool was the asymptotic expansion of the Bergman kernel
associated with the spin$^c$ Dirac operator of Dai-Liu-Ma \cite{DLM04a}.
In this Section, we establish the corresponding theory
for the renormalized Bochner-Laplacian on symplectic orbifolds.
In \cite[\S 5.4]{MM07} one can find more explanations and references for
Sections \ref{pbs4.1} and \ref{pbs4.2}.
For related topics about orbifolds we refer to \cite{ALR}.
This Section is organized as follows.
In Section \ref{pbs4.1} we recall the basic definitions about orbifolds.
In Section \ref{pbs4.2} we explain the asymptotic expansion
of Bergman kernel on symplectic orbifolds, which we apply in
Section \ref{pbs4.3} to derive the Berezin-Toeplitz quantization on
symplectic orbifolds.
\subsection{Basic definitions on orbifolds} \label{pbs4.1}
We define at first a category $\mathcal{M}_s$ as follows~:
The objects of $\mathcal{M}_s$
are the class of pairs $(G,M)$ where $M$ is a connected smooth manifold
and $G$ is a finite group acting effectively on $M$
(i.e., if $g\in G$ such that $gx=x$ for any $x\in M$,
then $g$ is the unit element of $G$). If $(G,M)$ and
$(G',M')$ are two objects, then a morphism $\Phi: (G,M)\rightarrow
(G',M')$ is a family of open embeddings $\varphi: M\rightarrow M'$
satisfying~:
{i)} For each $\varphi \in \Phi $, there is an injective group
homomorphism
$\lambda_{\varphi}~: G\rightarrow G' $ that makes $\varphi$ be
$\lambda_{\varphi}$-equivariant.
{ii)} For $g\in G', \varphi \in \Phi $, we define
$g\varphi : M \rightarrow M'$
by $(g\varphi)(x) = g\varphi(x)$ for $x\in M$.
If $(g\varphi)(M) \cap \varphi(M) \neq \emptyset$,
then $g\in \lambda_{\varphi}(G)$.
{ iii)} For $\varphi \in \Phi $, we have $\Phi = \{g\varphi, g\in G'\}$.
\begin{defn}\label{pbt4.1} Let $X$ be a paracompact Hausdorff space.
A $m$-dimensional \emph{orbifold chart} on $X$ consists of a
connected open set $U$ of $X$,
an object $(G_U,\widetilde{U})$ of $\mathcal{M}_s$ with
$\dim\widetilde{U}=m$,
and a ramified covering $\tau_U:\widetilde{U}\to U$ which is
$G_U$-invariant and induces a homeomorphism
$U \simeq \widetilde{U}/G_U$. We denote the chart by
$(G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow}U$.
A $m$-dimensional \emph{orbifold atlas} $\mathcal{V}$ on $X$
consists of a family of $m$-dimensional orbifold charts
$\mathcal{V} (U)
=((G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow} U)$
satisfying the following conditions\,:
{i)} The open sets $U\subset X$ form a covering $\mathcal{U}$ with
the property:
\begin{equation}\label{pb4.1}
\text{For any $U, U'\in \mathcal{U}$ and $x\in U\cap U'$,
there is
$U''\in \mathcal{U}$ such that $x\in U''\subset U\cap U'$}.
\end{equation}
{ii)} for any $U, V\in \mathcal{U}, U\subset V$ there exists a morphism
$\varphi_{VU}:(G_U,\widetilde{U})\rightarrow (G_V,\widetilde{V})$,
which covers the inclusion $U\subset V$ and satisfies
$\varphi_{WU}=\varphi_{WV} \circ \varphi_{VU}$
for any $U,V,W\in \mathcal{U}$, with $U\subset V \subset W$.
\end{defn}
It is easy to see that there exists a unique maximal orbifold atlas
$\mathcal{V}_{max}$ containing $\mathcal{V}$;
$\mathcal{V}_{max}$ consists of all orbifold charts
$(G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow} U$,
which are locally isomorphic to charts from $\mathcal{V}$ in
the neighborhood
of each point of $U$. A maximal orbifold atlas $\mathcal{V}_{max}$
is called an \emph{orbifold structure} and the pair
$(X,\mathcal{V}_{max})$
is called an orbifold. As usual, once we have an orbifold atlas
$\mathcal{V}$ on $X$ we denote the orbifold by $(X,\mathcal{V})$,
since $\mathcal{V}$ determines uniquely $\mathcal{V}_{max}$\,.
Note that if $\mathcal{U}^\prime$ is a refinement of $\mathcal{U}$
satisfying \eqref{pb4.1}, then there is an orbifold
atlas $\mathcal{V}^\prime$ such that
$\mathcal{V} \cup \mathcal{V}^\prime $
is an orbifold atlas, hence
$\mathcal{V} \cup \mathcal{V}^\prime \subset \mathcal{V}_{max}$.
This shows that we may choose $\mathcal{U}$ arbitrarily fine.
Let $(X,\mathcal{V})$ be an orbifold. For each $x\in X$, we can choose a
small neighborhood $(G_x, \widetilde{U}_x)\to U_x$ such
that $x\in \widetilde{U}_x$ is a fixed point of $G_x$
(it follows from the definition that such a $G_x$ is unique up to
isomorphisms for each $x\in X$).
We denote by $|G_x|$ the cardinal of $G_x$.
If $|G_x|=1$, then $X$ has a smooth manifold structure
in the neighborhood of $x$, which is called a smooth point of $X$.
If $|G_x|>1$, then $X$ is not a smooth manifold in
the neighborhood of $x$,
which is called a singular point of $X$. We denote by
$X_{sing}= \{x\in X; |G_x|>1\}$ the singular set of $X$,
and $X_{reg}= \{x\in X; |G_x|=1\}$ the regular set of $X$.
It is useful to note that on an orbifold $(X,\mathcal{V})$ we can
construct partitions of unity. First, let us call a function on $X$ smooth,
if its lift to any chart of the orbifold atlas $\mathcal{V}$ is smooth
in the usual sense. Then the definition and construction of a smooth
partition of unity associated to a locally finite covering carries over
easily from the manifold case. The point is to construct
smooth $G_U$-invariant functions with compact support on
$(G_U,\widetilde{U})$.
In Definition \ref{pbt4.1} we can replace $\mathcal{M}_s$ by a category
of manifolds with an additional structure such as orientation,
Riemannian metric, almost-complex structure or complex structure.
We impose that the morphisms (and the groups) preserve the specified
structure. So we can define oriented, Riemannian,
almost-complex or complex orbifolds.
Let $(X,\mathcal{V})$ be an arbitrary orbifold. By the above definition,
a \emph{Riemannian metric} on $X$ is a Riemannian metric $g^{TX}$
on $X_{reg}$ such that the lift of $g^{TX}$ to any chart of the orbifold
atlas $\mathcal{V}$ can be extended to a smooth Riemannian metric.
Certainly, for any $(G_U, \wi{U})\in \mathcal{V}$, we can always
construct a $G_U$-invariant Riemannian metric on $\wi{U}$.
By a partition of unity argument, we see that there exist
Riemannian metrics on the orbifold $(X,\mathcal{V})$.
\begin{defn}\label{pbt4.4}
An orbifold vector bundle $E$ over an
orbifold $(X,\mathcal{V})$ is defined as follows\,: $E$ is an orbifold
and for $U\in \mathcal{U}$, $(G_U^{E}, \widetilde{p}_U:
\widetilde{E}_U \rightarrow \widetilde{U})$ is
a $G_U^{E}$-equivariant vector bundle, $(G_U^{E}, \widetilde{E}_U)$
(resp. $(G_U=G_U^{E}/K_U^{E}, \widetilde{U})$,
$K_U^{E}= \ke (G_U^{E}\rightarrow\mbox{{\rm Diffeo}} (\widetilde{U})))$
is the orbifold structure of $E$ (resp. $X$) and morphisms for
$(G_U^{E}, \widetilde{E}_U)$ are morphisms of equivariant vector bundles.
If $G_U^E$ acts effectively on $\widetilde{U}$ for $U\in \mathcal{U}$,
i.e., $K_U^{E} = \{ 1\}$, we call $E$ a proper orbifold vector bundle.
\end{defn}
Note that any structure on $X$ or $E$ is locally
$G_x$ or $G_{U_x}^{E}$-equivariant.
\begin{rem}\label{pbt4.5}
Let $E$ be an orbifold vector bundle on $(X,\mathcal{V})$.
For $U\in \mathcal{U}$,
let $\wi{E ^{\pr}_U}$ be the maximal $K_U^{E}$-invariant sub-bundle of
$\wi{E}_U$ on $\wi{U}$. Then $(G_U, \wi{E ^{\pr}_U})$ defines a
proper orbifold vector bundle on $(X, \mathcal{V})$, denoted by $E ^{\pr}$.
The (proper) orbifold tangent bundle $TX$ on an orbifold $X$ is defined by
$(G_U, T\widetilde{U} \rightarrow \widetilde{U})$, for $U\in \mathcal{U}$.
In the same vein we introduce the cotangent bundle $T^*X$.
We can form tensor products of bundles by taking the tensor
products of their local expressions in the charts of an orbifold atlas.
Note that a Riemannian metric on $X$ induces a section of
$T^*X\otimes T^*X$ over $X$
which is a positive definite bilinear form on $T_xX$ at each point $x\in X$.
\end{rem}
Let $E \rightarrow X$ be an orbifold vector bundle and
$k\in \NN\cup\{\infty\}$. A section $s: X\rightarrow E$ is called
$\mathscr{C}^k$
if for each $U\in \mathcal{U}$, $s|_{U}$ is covered by
a $G_U^{E}$-invariant $\mathscr{C}^k$
section $\widetilde{s}_U : \widetilde{U} \rightarrow \widetilde{E}_U$.
We denote by $\mathscr{C}^k(X,E)$
the space of $\mathscr{C}^k$ sections of $E$ on $X$.
If $X$ is oriented, we define the integral
$\int_{X} \alpha$ for a form $\alpha$
over $X$ (i.e., a section of $ \Lambda (T^*X)$ over $X$) as follows.
If $\supp(\alpha) \subset U\in \mathcal U$, set
\begin{equation}\label{pb4.5}
\int_{X} \alpha: = \frac{1}{|G_U|} \int_{\widetilde{U}}
\widetilde{\alpha}_U.
\end{equation}
It is easy to see that the definition is independent of the chart.
For general $\alpha$ we extend the definition by using a partition of unity.
If $X$ is an oriented Riemannian orbifold,
there exists a canonical volume element $dv_X$ on $X$, which
is a section of
$\Lambda^m(T^*X)$, $m=\dim X$. Hence, we can also integrate
functions on $X$.
Assume now that the Riemannian orbifold $(X,\mathcal V)$ is compact.
For $x,y\in X$, put
\begin{eqnarray}\begin{array}{l}
d(x,y) = \mbox{Inf}_\gamma \Big \{ \sum_i \int_{t_{i-1}}^{t_i}
|\frac{\partial }{\partial t}\widetilde{\gamma}_i(t)| dt \Big |
\gamma: [0,1] \to X, \gamma(0) =x, \gamma(1) = y,\\
\hspace*{15mm} \mbox{such that there exist }
t_0=0< t_1 < \cdots < t_k=1,
\gamma([t_{i-1}, t_i])\subset U_i, \\
\hspace*{15mm}U_i \in \mathcal{U}, \mbox{ and a } \mathscr{C}^{\infty}
\mbox{ map } \widetilde{\gamma}_i: [t_{i-1}, t_i] \to \widetilde{U}_i
\mbox{ that covers } \gamma|_{[t_{i-1}, t_i]} \Big \}.
\end{array}\nonumber
\end{eqnarray}
Then $(X, d)$ is a metric space.
Let us discuss briefly kernels and operators on orbifolds.
For any open set $U\subset X$ and orbifold chart
$(G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow} U$,
we will add a superscript $\, \wi{}\,$ to indicate the corresponding
objects on $\widetilde{U}$.
Assume that
$\wi{\mK}(\wi{x},\wi{x}^{\,\prime})\in \mathscr{C}^\infty(\wi{U}
\times \wi{U},
\pi_1^* \wi{E}\otimes \pi_2^* \wi{E}^*)$
verifies
\begin{align}\label{pb4.3}
(g,1)\wi{\mK}(g^{-1}\wi{x},\wi{x}^{\,\prime})
=(1,g^{-1})\wi{\mK}(\wi{x},g\wi{x}^{\,\prime})
\quad \text{ for any $g\in G_{U}$,}
\end{align}
where $(g_1,g_2)$ acts on
$\wi{E}_{\wi{x}}\times \wi{E}_{\wi{x}^{\,\prime}}^*$
by $(g_1,g_2)(\xi_1,\xi_2)= (g_1\xi_1,g_2 \xi_2)$.
We define the operator $\wi{\mK}: \mathscr{C}^\infty_0(\wi{U}, \wi{E})
\to \mathscr{C}^\infty(\wi{U}, \wi{E})$ by
\begin{equation}\label{pb4.4}
(\wi{\mK}\, \wi{s}) (\wi{x})= \int_{\wi{U}} \wi{\mK}(\wi{x},
\wi{x}^{\,\prime})
\wi{s}(\wi{x}^{\,\prime}) dv_{\wi{U}} (\wi{x}^{\,\prime})
\quad\text{for $\wi{s} \in \mathscr{C}^\infty_0(\wi{U}, \wi{E})$\,.}
\end{equation}
Any element $g\in G_U$ acts on $\mathscr{C}^\infty(\wi{U}, \wi{E})$ by:
$(g \cdot \wi{s})(\wi{x}):=g \cdot \wi{s}(g^{-1}\wi{x})$,
where $\wi{s} \in \mathscr{C}^\infty(\wi{U}, \wi{E})$.
We can then identify
an element $s \in \mathscr{C}^\infty({U}, {E})$
with an element $\wi{s} \in \mathscr{C}^\infty(\wi{U}, \wi{E})$
verifying $g\cdot \wi{s}=\wi{s}$ for any $g\in G_U$.
With this identification, we define the operator
$\mK: \mathscr{C}^\infty_0(U,{E})\to \mathscr{C}^\infty({U},{E})$ by
\begin{equation}\label{pb4.6}
({\mK} s)(x)=\frac{1}{|G_U|}\int_{\wi{U}} \wi{\mK}(\wi{x},
\wi{x}^{\,\prime})
\wi{s}(\wi{x}^{\,\prime}) dv_{\wi{U}} (\wi{x}^{\,\prime})
\quad\text{for $s \in \mathscr{C}^\infty_0({U},{E})$\,,}
\end{equation}
where $\wi{x}\in\tau^{-1}_U(x)$.
Then the smooth kernel $\mK(x,x^\prime)$ of the operator $\mK$
with respect to $dv_X$ is (cf. \cite[(5.18)]{DLM04a})
\begin{equation}\label{pb4.7}
\mK(x,x^\prime)= \sum_{g\in G_U} (g,1)\wi{\mK}(g^{-1}\wi{x},
\wi{x}^{\,\prime}).
\end{equation}
Let $\mK_1,\mK_2$ be two operators as above and assume that the
kernel of one of $\wi{\mK}_1, \wi{\mK}_2$ has compact support.
By \eqref{pb4.5}, \eqref{pb4.3} and \eqref{pb4.6},
the kernel of $\mK_1\circ \mK_2$ is given by
\begin{equation}\label{pb4.9}
(\mK_1\circ \mK_2)(x,x^\prime)= \sum_{g\in G_U}
(g,1)(\wi{\mK}_1\circ \wi{\mK}_2)(g^{-1}\wi{x},\wi{x}^{\,\prime}).
\end{equation}
\subsection{Bergman kernel on symplectic orbifolds} \label{pbs4.2}
Let $(X,\omega)$ be a compact symplectic orbifold of dimension $2n$.
Assume that there exists a proper orbifold Hermitian line bundle $(L,h^L)$
on $X$ with a Hermitian connection
$\nabla^L : \mathscr{C}^\infty(X,L)\to
\mathscr{C}^\infty(X,T^*X\otimes L)$
satisfying the prequantization condition:
\begin{equation}\label{0.-1}
\frac{i}{2\pi}R^L=\omega.
\end{equation}
where $R^L=(\nabla^L)^2$ is the curvature of $\nabla^L$.
Let $(E, h^E)$ be a proper orbifold Hermitian vector bundle on $X$
equipped with a Hermitian connection $\nabla^E$ and $R^E$ be
the curvature of $\nabla^E$.
Let $g^{TX}$ be a Riemannian metric on $X$. Let $\Delta_p$ be
the renormalized Bochner-Laplacian acting on
$\mathscr{C}^\infty(X,L^p\otimes E)$ by \eqref{e:Delta_p}.
With the same proof as in \cite[Corollary 1.2]{ma-ma02},
we can establish the spectral gap property.
\begin{thm}\label{t:gap}
Let $(X,\om)$ be a compact symplectic orbifold, $(L,\nabla^{L}, h^L)$
be a prequantum Hermitian proper orbifold line bundle
on $(X,\om)$ and $(E,\nabla^{E},h^E)$ be an arbitrary Hermitian
proper orbifold vector bundle on $X$. There exists a constant $C_L>0$
such that for any $p$
\begin{equation}\label{pb4.17}
\sigma(\Delta_p)\subset [-C_L,C_L]\cup [2p\mu_0-C_L,+\infty),
\end{equation}
where $\mu_0>0$ is given by \eqref{mu}.
\end{thm}
From now on, we assume $p>C_L(2\mu_0)^{-1}$. We consider
the subspace $\mathcal H_p\subset L^2(X,L^p\otimes E)$
spanned by the eigensections of $\Delta_p$ corresponding to eigenvalues
in $[-C_L,C_L]$. We define the generalized Bergman kernel
\begin{align}\label{e:5.10}
P_p(\cdot,\cdot)\in \mathscr{C}^\infty(X\times X,
\pi_1^*(L^p\otimes E)\otimes \pi_2^*((L^p\otimes E)^*))
\end{align}
as the smooth kernel with respect to the
Riemannian volume form $dv_X(x')$ of the orthogonal projection
(Bergman projection) $P_{\mathcal H_p}$ in $L^2(X,L^p\otimes E)$
onto $\mathcal H_p$.
Consider an open set $U\subset X$ and orbifold chart
$(G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow} U$. Recall that
we add a superscript $\, \wi{}\,$ to indicate the corresponding
objects on $\widetilde{U}$.
The Riemannian metric $g^{TX}$ can be lifted to
a $G_U$-invariant Riemannian metric $g^{\wi{U}}$ on $\wi{U}$.
We denote by $B^{\wi{U}}(\wi{x},\varepsilon)$ and
$B^{T_{\wi{x}}\wi{U}}(0,\varepsilon)$ the open balls in
$\wi{U}$ and $T_{\wi{x}}\wi{U}$ with center $\wi x$ and $0$
and radius $\varepsilon$, respectively. We will always assume that
$\tau_U$ extends to
$(G_U,\widetilde{V})\stackrel{\tau_V}{\longrightarrow} V$
with $U\subset\subset V$ and $\wi{U}\subset\subset \wi{V}$.
Let $\partial U=\ov{U}\setminus U$ and $\partial \wi{U}
=\ov{\wi{U}}\setminus \wi{U}$. Fix $a^X>0$ such that
for every open set $U\subset X$ and orbifold chart
$(G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow} U$,
for every $\varepsilon\leq a^X$ and for every $\wi{x}\in \wi{U}$
such that $d(\wi{x},\partial \wi{U})\leq \varepsilon$, the exponential
map $T_{\wi{x}}\wi{U}\ni Z\mapsto \exp^X_{\wi{x}}(Z)\in \wi{U}$
is a diffeomorphism from $B^{T_{\wi{x}}\wi{U}}(0,\varepsilon)$ onto
$B^{\wi{U}}(\wi{x},\varepsilon)$. Throughout in what follows,
$\varepsilon$ runs in the fixed interval $(0,a^X/4)$.
Let $f : \mathbb R \to [0,1]$ be a smooth even function such that
$f(v)=1$ for $|v| \leqslant \varepsilon/2$,
and $f(v) = 0$ for $|v| \geqslant \varepsilon$. Set
\begin{equation} \label{0c3}
F(a)= \Big(\int_{-\infty}^{+\infty}f(v) dv\Big)^{-1}
\int_{-\infty}^{+\infty} e ^{i v a}\, f(v) dv.
\end{equation}
Then $F(a)$ is an even function and lies in the Schwartz space
$\mathcal{S} (\mathbb R)$ and $F(0)=1$.
Let $\wi{F}$ be the holomorphic function on
$\mathbb C$ such that $\wi{F}(a ^2) =F(a)$.
The restriction of $\wi{F}$
to $\mathbb R$ lies in the Schwartz space $\mS (\mathbb R)$.
Then there exists
$\{c_j\}_{j=1}^{\infty}$ such that for any $k\in \mathbb N$, the function
\begin{equation} \label{0c5}
\wi{F}_k(a)= \wi{F}(a) - \sum_{j=1}^k c_j a ^j \wi{F}(a), \quad c_{k+1}
=\frac{1}{(k+1)!}\wi{F}_k^{(k+1)}(0),
\end{equation}
verifies
\[
\wi{F}_k^{(i)}(0)= 0 \quad \mbox{\rm for any} \ 0< i\leqslant k.
\]
Using the same arguments as in \cite{ma-ma08}, one can show the
analogue of \cite[Proposition 1.2]{ma-ma08}:
\begin{prop}\label{0t3.0}
For any $k,m\in \mathbb N$, there exists $C_{k,m}>0$ such that for
$p\geqslant1$
\begin{align}\label{pb4.21}
\left|\wi{F}_k\big(\tfrac{1}{\sqrt{p}}\Delta_{p}\big)(x,x')
- P_{p}(x,x')\right|_{\mathscr{C}^m(X\times X)}
\leqslant C_{k,m} p^{-\frac{k}{2} +2(2m+2n+1)}.
\end{align}
Here the $\mathscr{C}^m$ norm is induced by $\nabla^L,\nabla^E$ and
$h^L,h^E,g^{TX}$.
\end{prop}
Using \eqref{0c3}, \eqref{0c5} and the property of
the finite propagation speed of solutions of hyperbolic equations
\cite[Appendix D.2]{MM07}
(which still holds on orbifolds as pointed out in \cite{Ma05})
it is clear that for $x,x'\in X$,
$\wi{F}_k\big(\frac{1}{\sqrt{p}}\Delta_{p}\big)(x,\cdot)$
only depends on
the restriction of $\Delta_{p}$ to $B^X(x,\varepsilon p^{-\frac{1}{4}})$,
and $\wi{F}_k\big(\frac{1}{\sqrt{p}}\Delta_{p}\big)(x,x')= 0$,
if $d(x, x') \geqslant\varepsilon p^{-\frac{1}{4}}$.
Consider an open set $U\subset X$ with an orbifold chart
$(G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow} U$.
Let $$U_1=\{ x\in U, d(x,\partial U)<2\varepsilon\}.$$
For any $x_0\in U_1$,
the exponential map $\exp_{\widetilde x_0}$ is a diffeomorphism from
$B^{T_{\widetilde x_0}\widetilde{U_1}}(0,2\varepsilon)$ onto
$B^{\widetilde{U_1}}(\widetilde x_0,2\varepsilon)$ which is
$G_{x_0}$-equivariant. Thus we can extend everything from
$B^{T_{\widetilde x_0}\widetilde{U_1}}(0,2\varepsilon)$ to
$\widetilde{X_0}:=T_{\widetilde x_0}\widetilde{U_1}$ as in
Section~\ref{expansions} (cf. \cite[\S 1.2]{ma-ma08}) which
is automatically $G_{x_0}$-equivariant. Let $P_{\widetilde{X_0},p}$
be the spectral projection of the renormalized Bochner-Laplacian
$\Delta_p^{\widetilde{X_0}}$ on
$\mathscr{C}^\infty(\widetilde{X_0}, L_)^p\otimes E_0)$,
corresponding to the interval $[-C_{L_0},C_{L_0}]$, and
$P_{\widetilde{X_0},p}(\wi{x},\wi{y})$ be the Schwartz kernel of
$P_{\widetilde{X_0},p}$ with respect to the volume form
$dv_{\widetilde{X_0}}$. Let $P_{X_0,p}(x,y)$ be the corresponding
object on $G_{x_0}\setminus T_{\widetilde x_0}\widetilde{U_1}$, then,
by \eqref{pb4.7}, we have
\begin{equation}\label{pb4.7a}
P_{X_0,p}(x,y)= \sum_{g\in G_{x_0}} (g,1)
P_{\widetilde{X_0},p}(g^{-1}\wi{x},\wi{y}^{\,\prime}).
\end{equation}
By Proposition \ref{0t3.0}, we get the analogue of
\cite[Prop. 1.3]{ma-ma08}: for any $\ell, m\in\mathbb N$,
there exists $C_{\ell,m}>0$ such that, for any $x,y \in B(x_0,\varepsilon)$
and $p\in \mathbb N^{*}$
\begin{equation}\label{pb4.7b}
|P_{p}(x,y)-P_{X_0,p}(x,y)|\leq C_{\ell,m}p^{-\ell}.
\end{equation}
\subsection{Berezin-Toeplitz quantization on symplectic orbifolds}
\label{pbs4.3}
We apply now the results of Section \ref{pbs4.2} to establish
the Berezin-Toeplitz quantization on symplectic orbifolds
by using as quantum spaces the spaces $\mH_p$.
We use the notations and assumptions of that Section.
We will closely follow and slightly modify the arguments
of \cite[\S 6.3, 6.4]{ma-ma08a}.
Thus we have the following definition.
\begin{defn}\label{toet6.0}
A \textbf{\em Toeplitz operator}
is a sequence $\{T_p\}_{p\in \mathbb N}$ of bounded linear operators
\begin{equation}\label{toe6.1}
T_{p}:L^2(X, L^p\otimes E)\longrightarrow L^2(X, L^p\otimes E)\,,
\end{equation}
satisfying the following conditions:
\begin{description}
\item[(i)] For any $p\in \mathbb N$, we have
\begin{align}\label{e:5.17}
T_p=P_{\mathcal H_p}T_pP_{\mathcal H_p}.
\end{align}
\item[(ii)] There exists a sequence
$g_l\in \mathscr{C}^\infty(X,\operatorname{End}(E))$ such that
\begin{align}\label{e:5.18}
T_p=P_{\mathcal H_p}\Big(\sum_{l=0}^\infty p^{-l}g_l\Big)
P_{\mathcal H_p}+\mathcal O(p^{-\infty}),
\end{align}
i.e., for any $k\geq 0$ there exists $C_k>0$ such that
\begin{align}\label{e:5.19}
\Big\|T_p-P_{\mathcal H_p}\Big(\sum_{l=0}^k p^{-l}g_l\Big)
P_{\mathcal H_p}\Big\|\leq C_kp^{-k-1}.
\end{align}
\end{description}
\end{defn}
For any section $f\in \mathscr{C}^{\infty}(X,\End(E))$,
the \textbf{\em Berezin-Toeplitz quantization} of $f$ is defined by
\begin{equation}\label{toe6.3}
T_{f,\,p}:L^2(X,L^p\otimes E)\longrightarrow L^2(X,L^p\otimes E)\,,
\quad T_{f,\,p}=P_p\,f\,P_p\,.
\end{equation}
The Schwartz kernel of $T_{f,\,p}$ is given by
\begin{equation} \label{toe2.5}
T_{f,\,p}(x,x')=\int_XP_p(x,x'')f(x'')P_p(x'',x')\,dv_X(x'')\,.
\end{equation}
\begin{lemma} \label{toet6.1}
For any $\varepsilon_0>0$ and any $l,m\in\NN$ there exists
$C_{l,m,{\varepsilon}}>0$ such that
\begin{equation} \label{toe6.4}
\big|T_{f,\,p}(x,x')\big|_{\mathscr{C}^m(X\times X)}
\leqslant C_{l,m,{\varepsilon}}p^{-l}
\end{equation}
for all $p\geqslant 1$ and all $(x,x')\in X\times X$
with $d(x,x')>\varepsilon_0$.
\end{lemma}
\begin{proof}
By using Proposition \ref{0t3.0}, the proof is exactly the same as the proof
of \cite[Lemma 4.2]{ma-ma08a}.
\end{proof}
Next we obtain the asymptotic expansion of the kernel
$T_{f,\,p}(x,x')$ in a neighborhood of the diagonal. We will
use \cite[Condition 6.7]{ma-ma08a}.
Recall that the (proper) orbifold tangent bundle $TX$ on an orbifold $X$
is defined by a family of $G_U$-equivariant vector bundles
$(G_U, T\widetilde{U} \rightarrow \widetilde{U})$, for
$U\in \mathcal{U}$.
Consider the fiberwise product
$TX\times_X TX=\{(Z,Z^\prime)\in
T_{x_0}X\times T_{x_0}X : x_0\in X\}$.
Let $\pi : TX\times_X TX\to X$ be the natural projection given by
$\pi(Z,Z^\prime)=x_0$. We say that
$Q_{r,\,x_0}\in \End( E)_{x_0}[\wi{Z},\wi{Z}^{\prime}]$,
if for any $U\in \mathcal U$, it induces a smooth section
$Q_{r,\,x_0}\in \End( E)_{x_0}[\wi{Z},\wi{Z}^{\prime}]$
of the vector bundle $\pi^*(\operatorname{End}(E))$ on
$T\wi{U}\times_{\wi{U}} T\wi{U}$ defined for all $\wi{x}_0\in \wi{U}$
and $Z,Z^\prime\in T_{\wi{x}_0}\wi{U}$ and polynomial in
$Z,Z^\prime\in T_{\wi{x}_0}\wi{U}$.
Let $\{\Xi_p\}_{p\in\NN}$ be a sequence of linear operators
$\Xi_p: L^2(X,L^p\otimes E)\longrightarrow L^2(X,L^p\otimes E)$
with smooth kernel $\Xi_p(x,y)$ with respect to $dv_X(y)$.
\begin{condition}\label{coe2.71}
Let $k\in\NN$. Assume that there exists a family
$\{Q_{r,\,x_0}\}_{0\leqslant r\leqslant k,\,x_0\in X}$, satisfying
the conditions:
\begin{itemize}
\item[(a)] $Q_{r,\,x_0}\in \End( E)_{x_0}[\wi{Z},\wi{Z}^{\prime}]$\,,
$\wi{Z},\wi{Z}^\prime\in T_{\wi{x}_0}X$,
\item[(b)] $\{Q_{r,\,x_0}\}_{x_0\in X}$ is smooth with respect
to the parameter $x_0\in X$,
\end{itemize}
and, for every open set $U\in\mathcal{U}$ and every orbifold chart
$(G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow} U$,
a sequence of kernels
\[
\{\wi{\Xi}_{p, U}(\wi{x},\wi{x}^{\,\prime})\in
\mathscr{C}^\infty(\wi{U} \times \wi{U},
\pi_1^* (\wi{L}^p\otimes \wi{E})\otimes
\pi_2^* (\wi{L}^p\otimes\wi{E})^*)\}_{p\in\NN}
\]
such that, for every $\varepsilon''>0$ and every
$\wi{x},\wi{x}^{\,\prime}\in \wi{U}$,
\begin{equation} \label{toe6.5}
\begin{split}
& (g,1)\wi{\Xi}_{p, U}(g^{-1}\wi{x},\wi{x}^{\,\prime})=
(1,g^{-1})\wi{\Xi}_{p, U}(\wi{x},g\wi{x}^{\,\prime})
\quad \text{for any } \, \, g\in G_{U}\; \text{(cf. \eqref{pb4.3})}, \\
&\wi{\Xi}_{p, U}(\wi{x},\wi{x}^{\,\prime})= \cO(p^{-\infty}) \quad
\quad\text{for}\, \, d(x,x^\prime)>\varepsilon'',\\
&\Xi_{p}(x,x^\prime)
= \sum_{g\in G_{U}} (g,1)\wi{\Xi}_{p, U}(g^{-1}\wi{x},\wi{x}^{\,\prime})
+ \cO(p^{-\infty}),
\end{split}\end{equation}
and moreover, for every relatively compact open subset
$\wi{V}\subset \wi{U}$, the following relation is valid for any
$\wi{x}_0\in \wi{V}$:
\begin{equation} \label{toe6.6}
p^{-n}\, \wi{\Xi}_{p,U,\wi{x}_0}(\wi{Z},\wi{Z}^\prime)
\cong \sum_{r=0}^k (Q_{r,\,\wi{x}_0} \cP_{\wi{x}_0})
(\sqrt{p}\wi{Z},\sqrt{p}\wi{Z}^{\prime})p^{-\frac{r}{2}}
+\mO(p^{-\frac{k+1}{2}})\,.
\end{equation}
which means that there exist $\varepsilon^\prime>0$ and $C_0>0$
with the following property:
for any $m\in \mathbb N$, there exist $C>0$ and $M>0$ such that for
any $\wi{x}_0\in \wi{V}$, $p\geq 1$ and $\wi{Z},\wi{Z}^\prime
\in T_{\wi{x}_0}\wi{U}$, $|\wi{Z}|, |\wi{Z}^\prime|<\varepsilon^\prime$,
we have with $\kappa$ in \eqref{e:kappa},
\begin{multline}\label{e:5.25}
\Bigg|p^{-n}\, \wi{\Xi}_{p,U,\wi{x}_0}(\wi{Z},\wi{Z}^\prime)
\kappa^{\frac 12}(\wi{Z})\kappa^{\frac 12}(\wi{Z}^\prime)
-\sum_{r=0}^k(Q_{r,\,\wi{x}_0} \cP_{\wi{x}_0})
(\sqrt{p}\wi{Z},\sqrt{p}\wi{Z}^{\prime})p^{-\frac{r}{2}}
\Bigg|_{\mathscr{C}^{m}(\wi{V}\times \wi{V})}\\
\leq Cp^{-\frac{k+1}{2}}(1+\sqrt{p}|\wi{Z}|+\sqrt{p}|\wi{Z}^\prime|)^M
\exp(-\sqrt{C_0p}|\wi{Z}-\wi{Z}^\prime|)+\mathcal O(p^{-\infty}).
\end{multline}
\end{condition}
\begin{notation}\label{noe2.71}
If the sequence $\{\Xi_p : L^2(X,L^p\otimes E)\longrightarrow
L^2(X,L^p\otimes E)\}_{p\in\NN}$ satisfies Condition \ref{coe2.71},
we write
\begin{equation} \label{toe6.7}
p^{-n}\, \Xi_{p,\,x_0}(Z,Z^\prime)\cong \sum_{r=0}^k
(Q_{r,\,x_0} \cP_{x_0})(\sqrt{p}Z,\sqrt{p}Z^{\prime})p^{-\frac{r}{2}}
+\mO(p^{-\frac{k+1}{2}})\,.
\end{equation}
\end{notation}
As in \cite[Lemma 6.9]{ma-ma08a}, one can show that the smooth family
$Q_{r,\,x_0}\in \End( E)_{x_0}[\wi{Z},\wi{Z}^{\prime}]$
in Condition \ref{coe2.71} is uniquely determined by $\Xi_p$.
\begin{thm} \label{toet2.2}
There exist polynomials
$J_{r,\,x_0}\in \End(E)_{x_0}[\wi{Z},\wi{Z}^{\prime}]$
such that, for any $k\in \NN$, $Z,Z^\prime \in T_{x_0}X$,
$\abs{Z},\abs{Z^{\prime}}< \varepsilon$,
we have
\begin{equation} \label{toe2.9}
p^{-n} P_{p,\,x_0}(Z,Z^\prime) \cong \sum_{r=0}^{k-1}
(J_{r,\,x_0} \cP_{x_0})(\sqrt{p}Z,\sqrt{p}Z^{\prime})p^{-\frac{r}{2}}
+\mO(p^{-\frac{k}{2}})\,,
\end{equation}
in the sense of Notation \ref{noe2.71}.
\end{thm}
\begin{proof}
By Theorem~\ref{t:main} for $P_{\widetilde{X_0},p}(\wi{x},\wi{y})$,
\eqref{pb4.7a} and \eqref{pb4.7b}, we get Theorem~\ref{toet2.2}
as the analogue of \cite[Lemmas 4.5, 6.10]{ma-ma08a}.
\end{proof}
From \eqref{toe2.5} and \eqref{toe2.9}, we deduce an analogue
of \cite[Lemmas 4.6, 4.7 and 6.10]{ma-ma08a}.
\begin{lemma} \label{toet2.3}
Let $f\in \mathscr{C}^\infty(X,\End(E))$.
There exists a family $\{Q_{r,\,x_0}(f)\}_{r\in\NN,\,x_0\in X}$ such that
\begin{itemize}
\item[(a)] $Q_{r,\,x_0}(f)\in\End(E)_{x_0}[Z,Z^{\prime}]$
are polynomials with the same parity as $r$,
\item[(b)] $\{Q_{r,\,x_0}(f)\}_{r\in\NN,\,x_0\in X}$ is smooth with
respect to $x_0\in X$,
\item[(c)] for every $k\in \NN$, $x_0\in X$,
$Z,Z^\prime \in T_{x_0}X$, $\abs{Z},\abs{Z^{\prime}}<\varepsilon/2$
we have
\begin{equation} \label{toe2.13}
p^{-n}T_{f,\,p,\,x_0}(Z,Z^{\prime})
\cong \sum^k_{r=0}(Q_{r,\,x_0}(f)\cP_{x_0})
(\sqrt{p}Z,\sqrt{p}Z^{\prime})
p^{-\frac{r}{2}} + \mO(p^{-\frac{k+1}{2}})\,,
\end{equation}
in the sense of Notation \ref{noe2.71}.
\end{itemize}
Here $Q_{r,\,x_0}(f)$ are expressed by
\begin{equation} \label{toe2.14}
Q_{r,\,x_0}(f) = \sum_{r_1+r_2+|\alpha|=r}
\cK\Big[J_{r_1,\,x_0}\;,\;
\frac{\partial ^\alpha f_{\,x_0}}{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !} J_{r_2,\,x_0}\Big]\,.
\end{equation}
Especially,
\begin{align} \label{toe2.15}
Q_{0,\,x_0}(f)= f(x_0) .
\end{align}
\begin{equation} \label{toe2.20}
Q_{1,\,x_0}(f)= f(x_0) J_{1,\,x_0} + \cK\Big[J_{0,\,x_0},
\frac{\partial f_{x_0}}{\partial Z_j}(0) Z_j J_{0,\,x_0}\Big].
\end{equation}
\end{lemma}
Here, for any polynomials $F,G\in \mathbb C[Z,Z^\prime]$,
the polynomial $\mathcal K[F,G]\in \mathbb C[Z,Z^\prime]$
is defined by the relation
\[
((F\cP_{x_0})\circ (G\cP_{x_0}))(Z,Z^\prime)=(\mathcal K[F,G]
\cP_{x_0})(Z,Z^\prime),
\]
where $((F\cP_{x_0})\circ (G\cP_{x_0}))(Z,Z^\prime)$ is the kernel
of the composition $(F\cP_{x_0})\circ (G\cP_{x_0})$ of the operators
$F\cP_{x_0}$ and $G\cP_{x_0}$ in $L^2(T_{x_0}X)$ with kernels
$(F\cP_{x_0})(Z,Z^\prime)$ and $(G\cP_{x_0})(Z,Z^\prime)$, respectively.
Now we can proceed by a word for word repetition of the corresponding
arguments in \cite{ma-ma08a}. So we just give statements of the main
results.
First, the following analogue of \cite[Theorem 6.11]{ma-ma08a} provides
a useful criterion for a given family to be a Toeplitz operator.
\begin{thm}\label{toet6.4}
Let $\{T_p:L^2(X,L^p\otimes E)\longrightarrow L^2(X,L^p\otimes E)\}$
be a family of bounded linear operators which satisfies
the following three conditions:
\begin{itemize}
\item[(i)] For any $p\in \NN$,
$P_{\mathcal H_p}\,T_p\,P_{\mathcal H_p}=T_p$\,.
\item[(ii)] For any $\varepsilon_0>0$ and any $l\in\NN$,
there exists $C_{l,\varepsilon_0}>0$ such that
for all $p\geqslant 1$ and all $(x,x')\in X\times X$
with $d(x,x')>\varepsilon_0$,
\begin{equation} \label{toe6.21}
|T_{p}(x,x')|\leqslant C_{l,{\varepsilon_0}}p^{-l}.
\end{equation}
\item[(iii)] There exists a family of polynomials
$\{\mQ_{r,\,x_0}\in\End(E)_{x_0}[Z,Z^{\prime}]\}_{x_0\in X}$
such that\,{\rm:}
\begin{itemize}
\item[(a)] each $\mQ_{r,\,x_0}$ has the same parity as $r$,
\item[(b)] the family is smooth in $x_0\in X$ and
\item[(c)] for every $k\in\NN$, we have
\begin{equation} \label{toe6.22}
p^{-n}T_{p,\,x_0}(Z,Z^{\prime})\cong
\sum^k_{r=0}(\mQ_{r,\,x_0}\cP_{x_0})
(\sqrt{p}Z,\sqrt{p}Z^{\prime})p^{-\frac{r}{2}} + \mO(p^{-\frac{k+1}{2}})
\end{equation}
in the sense of \eqref{toe6.7}.
\end{itemize}
\end{itemize}
Then $\{T_p\}$ is a Toeplitz operator.
\end{thm}
Finally, we show that the set of Toeplitz operators on a compact orbifold is
closed under the composition of operators, so forms an algebra
(an analogue of \cite[Theorems 6.13 and 6.16]{ma-ma08a}).
\begin{thm}\label{toet6.7}
Let $(X,\om)$ be a compact symplectic orbifold and $(L,\nabla^{L},h^L)$
be a Hermitian proper orbifold line bundle satisfying
the prequantization condition \eqref{0.-1}.
Let $(E,\nabla^E ,h^E)$ be an arbitrary Hermitian proper orbifold
vector bundle on $X$.
Given $f,g\in \mathscr{C}^\infty(X,\End(E))$, the product of
the Toeplitz operators
$T_{f,\,p}$ and $T_{g,\,p}$ is a Toeplitz operator,
more precisely, it admits an asymptotic expansion
\begin{align}\label{e:5.33}
T_{f,\,p}T_{g,\,p}=\sum_{r=0}^\infty p^{-r}T_{C_r(f,g),\,p}
+ \mathcal O(p^{-\infty}),
\end{align}
where $C_r(f,g)\in \mathscr{C}^\infty(X,\End(E))$
and $C_r$ are bidifferential operators defined locally
on each covering $\wi{U}$ of an orbifold chart
$(G_U,\widetilde{U})\stackrel{\tau_U}{\longrightarrow}U$.
In particular $C_0(f,g)=fg$.
If $f,g\in \mathscr{C}^\infty(X)$, then
\begin{align}\label{e:5.34}
[T_{f,p}, T_{g,p}]=\frac{\sqrt{-1}}{p}T_{\{f,g\},p}+\mathcal O(p^{-2}).
\end{align}
For any $f\in \mathscr{C}^\infty(X,\End(E))$, the norm of $T_{f,p}$
satisfies
\begin{align}\label{e:5.35}
\lim_{p\to\infty}\|T_{f,p}\|=\|f\|_{\infty}
:=\sup_{\substack{x\in X\\0\neq u\in E_x}}|f(x)(u)|_{h^E}/|u|_{h^E}.
\end{align}
\end{thm}
\begin{rem}\label{toet4.31} As mentioned in
\cite[Remark 6.14]{ma-ma08a}, by
\cite[Theorems 6.13, 6.16]{ma-ma08a},
on every compact symplectic orbifold $X$ admitting a prequantum
line bundle $(L, \nabla^{L}, h^L)$, one can define in
a canonical way an associative star-product $f*g
=\sum_{l=0}^\infty \hbar^{l}C_l(f,g)\in \mathscr{C}^\infty(X)[[\hbar]]$
for every $f,g\in \mathscr{C}^\infty(X)$,
called the \textbf{\emph{Berezin-Toeplitz star-product}} by using
the kernel of the spin$^c$ Dirac operator. Moreover, $C_l(f,g)$
are bidifferential operators defined locally as in the smooth case.
Theorem \ref{toet6.7} shows that one can also use the eigenspaces
of small eigenvalues of the renormalized Bochner-Laplacian.
\end{rem}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,030 |
\section{Introduction}
The problem of finding eigenvalues and eigenvectors of a given (finite-dimensional) matrix $A$ is very old and extremely relevant. The number of applications in which this is needed is countless. Just to cite a single one, the most relevant for us, eigenvalues of a quantum mechanical Hamiltonian $H$ are the energies allowed for the system ${\cal S}$ one is investigating, \cite{messiah}. In ordinary quantum mechanics, $H$ is taken to be Hermitian\footnote{All along this paper Hermitian and self-adjoint will be used as synonimous.}: $H=H^\dagger$, \cite{rs}. Here $H^\dagger$ is the (Dirac)-adjoint of $H$, \cite{baginbook}: in practice, if $H$ is an $n\times n$ matrix, $H^\dagger$ is simply the transpose and complex-conjugate\footnote{More in general, $H^\dagger$ is the operator satisfying the equality $\left<f,Hg\right>=\left<H^\dagger f,g\right>$, for all $f,g\in{\cal H}$, the Hilbert space where ${\cal S}$ lives, and $\left<.,.\right>$ is its scalar product.} of $H$. The Hamiltonian is not the only operator of ${\cal S}$ which is usually assumed to be Hermitian. Other operators usually can be associated to ${\cal S}$ with the same property: these are the so-called {\em observables} of ${\cal S}$. Clearly, the eigenvalues of all these operators are also necessarily real. Few decades ago, with the seminal paper \cite{ben1}, it became clear to the community of physicists (the mathematicians were already aware of this!) that reality of eigenvalues is implied by Hermiticity, but not vice-versa: non-Hermitian operators exist, both in finite and in infinite dimensional Hilbert spaces, whose eigenvalues are all real, see \cite{baginbagbook} and references therein for several examples. Moreover, it is now clear that some of the eigenvalues of a {\em physically relevant} (non Hermitian) Hamiltonian could easily be complex. This is what happens, for instance, when $PT$-symmetry is broken, \cite{benbook}, i.e. when the parameters of the Hamiltonian change in a proper way, see \cite{jli,guo} and references therein, or when some effective Hamiltonian is used to describe gain and loss effects. With this in mind it should be clear why we are so interested in finding possibly complex eigenvalues, and their related eigenvectors, of a given matrix, describing some physical characteristic of ${\cal S}$. Needless to say, many softwares exist in the market since many years which compute eigenvalues and eigenvectors of a given matrix. Some of them works mainly numerically (like Matlab). Others, like Mathematica, produce often analytical results. However other techniques may be relevant when the dimension of the matrices become larger and larger, as, for instances, in the case of the matrices involved in ranking web pages. In this case, power method looks more efficient, other than being mathematically quite interesting. We find also rather elegant the dynamical approach, considered in \cite{tang}, which associates a suitable differential equation to a matrix whose eigenvalues we need to compute. A review of strategies can be found in \cite{rev}.
The aim of this paper is to merge these two aspects, complexity of eigenvalues and mathematical strategies designed especially for large matrices, to propose new algorithms, based on a solid mathematical ground, to compute eigenvalues of non Hermitian matrices. In Section \ref{sect2} we extend the dynamical approach analyzed in \cite{tang}. Section \ref{sec:fixedpoint} contains our {\em modified power method}. Applications are discussed in Section \ref{sect4}. In particular, we propose an application in the realm of quantum mechanics to the finite-dimensional Swanson model, \cite{bag2018}. Conclusions are given in Section \ref{sect5}.
\section{Dynamical approach: introductory results for $A=A^\dagger$}\label{sect2}
In the first part of this section, for reader's convenience, we briefly review the approach proposed in \cite{tang}, while in the second part we will propose a possible extention of this approach, useful for our particular pourpouses.
Let $x\in{\cal H}=\mathbb{C}^n$ be an $n$-dimensional vector, and $A=A^\dagger$ an Hermitian matrix on ${\cal H}$. Of course, since $A$ is Hermitian, its eigenvalues are necessarily real\footnote{In \cite{tang} the Hilbert space was taken to be $\mathbb{R}^n$. We prefer to adopt this more general choice here, also in view of our interest in quantum mechanics.}. We want to compute eigenvalues and eigenvectors of $A$ (or, at least, some of them).
The approach we discuss here is {\em dynamical}: we introduce a differential equation on $\mathbb{R}^n$, and we show that the solution of this equation converges to the eigenstate of $A$ corresponding to its highest eigenvalue, the so-called {\em leading eigenvector}. We start considering a vector $x(t)\in\mathbb{R}^n\subset {\cal H}$, depending on a continuous parameter $t\geq0$, and we assume $x(t)$ satisfies the following equation:
\begin{equation} \frac{dx(t)}{dt}=-x(t)+f(x(t))=\|x(t)\|^2Ax(t)-\left<x(t),Ax(t)\right>x(t),
\label{21}\end{equation} where we have defined $f(x)=\left(\|x\|^2A+(1-\left<x,Ax\right>)\right)x$. It is clear that $f$ is a non-linear function from ${\cal H}$ into ${\cal H}$.
\begin{defn}\label{def1}
A vector $\xi(t)\neq0$ is an equilibrium point of (\ref{21}) if $f(\xi(t))=\xi(t)$, for all $t\geq0$.
\end{defn}
It is clear that if $\xi(t)$ is an equilibrium point of (\ref{21}), then $\xi(t)=\xi(0)=:\xi$ is an eigenstate of $A$ with eigenvalue $\lambda_\xi=\frac{\left<\xi,A\xi\right>}{\|\xi\|^2}$, and vice-versa. Notice that $\lambda_\xi$ is well defined, since $\xi\neq0$, and therefore $\|\xi\|>0$. Moreover, since $A=A^\dagger$, $\lambda_\xi\in\mathbb{R}$.
Following \cite{tang} we look for a solution of (\ref{21}) in terms of the (unknown) eigenvectors of $A$: $Ae_j=\lambda_je_j$, $j=1,2,\ldots,n$: $x(t)=\sum_{i=1}^{n}c_i(t)e_i$. The equation for $c_i(t)$ can be deduced using the ortogonality of the $e_i$'s and the equality $\|x(t)\|^2=\|x(0)\|^2$, which can be proved easily since
\begin{equation}
\frac{d\|x(t)\|^2}{dt}=\left<\dot x(t),x(t)\right>+\left<x(t),\dot x(t)\right>=0,
\label{add5}\end{equation}
after inserting twice (\ref{21}). We get
\begin{equation}
\dot c_i(t)=F_i(t)c_i(t), \qquad F_i(t)=\|x(0)\|^2\lambda_i-D(t),
\label{22}\end{equation}
where $D(t)=\sum_{k=1}^{n}\lambda_k|c_k(t)|^2$. Notice that $F_i(t)$ is a real function of time. Rewriting the differential equation for $c_i(t)$ in its integral form,
$
c_i(t)=e^{\int_{0}^{t}F_i(s)\,ds}c_i(0),
$
we see that, if $c_i(0)\in\mathbb{R}$, then $c_i(t)\in\mathbb{R}$ for all $t\geq0$. Working under this natural assumption we rewrite $D(t)=\sum_{k=1}^{n}\lambda_kc_k(t)^2$. The explicit form of the $c_i(t)$ can now be deduced and we get
\begin{equation}
c_i(t)=\frac{c_i(0)\|x(0)\|}{\sqrt{\sum_{k=1}^{n}c_k(0)^2e^{2\|x(0)\|^2(\lambda_k-\lambda_j)t}}},
\label{23}\end{equation}
so that the general solution of (\ref{21}) is
\begin{equation} x(t)=\sum_{i=1}^{n}\frac{c_i(0)\|x(0)\|}{\sqrt{\sum_{k=1}^{n}c_k(0)^2e^{2\|x(0)\|^2(\lambda_k-\lambda_j)t}}}\,e_i.
\label{24}
\end{equation}
Suppose now that the eigenvalues of $A$ are non degenerate. Hence we have $n$ different eigenvalues which we label as a decreasing sequence: $\lambda_1>\lambda_2>\cdots>\lambda_n$. Calling $B_{k,j}(t)=e^{2\|x(0)\|^2(\lambda_k-\lambda_j)t}$ we see the following: if $k>j$, $\lambda_k<\lambda_j$ and therefore $B_{k,j}(t)\rightarrow0$ when $t\rightarrow\infty$. If $k=j$, $\lambda_k=\lambda_j$ and therefore $B_{k,j}(t)=1$ for all $t$. Moreover, if $k<j$, $\lambda_k>\lambda_j$ and therefore $B_{k,j}(t)\rightarrow\infty$ when $t\rightarrow\infty$. With this in mind, and assuming for concreteness that $c_1(0)>0$, it follows that
$$
x(t)\rightarrow \|x(0)\|e_1,
$$
when $t\rightarrow\infty$: the solution of the differential equation (\ref{21}) converges to the (non-normalized, in general) eigenvector of $A$ corresponding to its highest eigenvalue. This is true, of course, if $c_1(0)$ in
(\ref{23}) and (\ref{24}) is different from zero. If we rather take, as initial vector $x(0)$, a vector with $c_1(0)=0$ and $c_2(0)\neq0$, $x(0)=\sum_{i=2}^{n}c_i(0)e_i$, we can check that
$$
x(t)\rightarrow \|x(0)\|e_2,
$$
at least if $c_2(0)>0$. Incidentally, this result also shows that, if we consider a vector $x(0)$ which is orthogonal to $e_1$, then it remains orthogonal to $e_1$ even in the limit $t\rightarrow\infty$. These steps can be repeated and all the eigenvectors of $A$ can be found, in principle
The situation generalizes easily when some eigenvalue is degenerate. Suppose, just to be concrete, that $\lambda_1=\lambda_2>\lambda_3=\lambda_4>\lambda_5>\cdots>\lambda_n$: two eigenvalues are degenerate, and both have degeneracy two. In this case, repeating the previous analysis, we see that
$$
x(t)\rightarrow \|x(0)\|\frac{c_1(0)e_1+c_2(0)e_2}{\sqrt{c_1(0)^2+c_2(0)^2}},
$$
at least if both $c_1(0)$ and $c_2(0)$ are not zero. Notice that this vector belongs to the eigenspace of the highest eigenvalue of $A$, ${\cal E}_{max}$, as before, even if this eigenspace is now two-dimensional. The two leading eigenvectors can now be fixed by choosing any two orthonormal vectors in ${\cal E}_{max}$.
\subsection{Extension to $A\neq A^\dagger$}\label{sect3}
Our next task is to generalize the above results to the case in which $A\neq A^\dagger$. As we have discussed in the Introduction, this is relevant in connection with pseudo hermitian quantum mechanics or for similar extensions of ordinary quantum mechanics, where the Hamiltonian of the physical system is not required to be Hermitian but still, most of the times, has a real spectrum, \cite{baginbook,benbook}. Also, it can be quite useful in other relevant situations, in which {\em physical} Hamiltonians turn out to have complex eigenvalues, like often happen in quantum optics or in gain-loss systems, see \cite{op1,op2,op3}. In this situation the first difference, with respect with what is discussed in Section \ref{sect2}, is that the eigenvectors of $A$, $\{\varphi_i\}$, are not, in general, mutually orthogonal: $\left<\varphi_i,\varphi_j\right>\neq\delta_{i,j}$. Nevertheless it is well known, \cite{bag2016,chri}, that a biorthogonal basis of ${\cal H}$ exists, $\{\Psi_i\}$, $\left<\varphi_i,\Psi_j\right>=\delta_{i,j}$, such that
\begin{eqnarray}
A\varphi_i=\lambda_i\varphi_i, \qquad \mbox{and}\qquad A^\dagger\Psi_i=\overline{\lambda_i}\Psi_i,\label{biorto}
\end{eqnarray}
for all $i=1,2,\ldots,n$. Let now introduce two sets of unknown functions $c:=\{c_i(t), \,i=1,2,\ldots,n\}$ and $d:=\{d_i(t), \,i=1,2,\ldots,n\}$, and two related vectors:
\begin{equation}
x_\varphi(t)=\sum_{i=1}^{n}c_i(t)\varphi_i, \qquad x_\Psi(t)=\sum_{i=1}^{n}d_i(t)\Psi_i,
\label{31}\end{equation}
and let us assume that these functions satisfy two (apparently) different differential equations:
\begin{equation}
\left\{
\begin{array}{ll}
\frac{dx_\varphi(t)}{dt}=\left<x_\Psi(t),x_\varphi(t)\right>Ax_\varphi(t)-\left<x_\Psi(t),Ax_\varphi(t)\right>x_\varphi(t)\\
\frac{dx_\Psi(t)}{dt}=\left<x_\varphi(t),x_\Psi(t)\right>A^\dagger x_\Psi(t)-\left<x_\varphi(t),A^\dagger, x_\Psi(t)\right>x_\Psi(t).
\end{array}
\right.
\label{32}\end{equation}
which look quite similar to two copies of equation \eqref{21}.
This {\em doubling} is a typical effect of going from Hermitian to non-Hermitian operators, \cite{baginbook}: when $A=A^\dagger$, then $\varphi_i=\Psi_i$, $\lambda_i=\overline{\lambda_i}$, for all $i$, and the eigenvectors form an orthonormal basis for ${\cal H}$. Stated differently, moving from an Hermitian $A$ to a non-Hermitian one, often produces this kind of features: two Hamiltonians (one the adjoint of the other), two sets of eigenvalues, two families of eigenvectors, and so on, see \cite{baginbagbook}.
The system in (\ref{32}) is closed and nonlinear. As we will show in a moment, it is possible to find its explicit solution, extending what we have shown for equation (\ref{21}).
It is interesting to notice that, similarly to what we have found in (\ref{add5}), i.e. that $\|x(t)\|=\|x(0)\|$, the two scalar products appearing in equations (\ref{32}) turn out to be time-independent:
\begin{equation}
\left<x_\Psi(t),x_\varphi(t)\right>=\left<x_\Psi(0),x_\varphi(0)\right>=\sum_{k=1}^n \,\overline{d_k(0)}\, c_k(0)=\left<d,c\right>,
\label{34}\end{equation}
where $\left<d,c\right>$ is the usual scalar product of $d=\{d_k(0)\}$ and $c=\{c_k(0)\}$ in $\mathbb{C}^n$. The proof is a replica of that for $\|x(t)\|$:
$$
\frac{d}{dt}\left<x_\Psi(t),x_\varphi(t)\right>=\left<\dot x_\Psi(t),x_\varphi(t)\right>+\left<x_\Psi(t),\dot x_\varphi(t)\right>=0,
$$
after inserting (\ref{32}). Hence equations in (\ref{32}) can be rewritten as
\begin{equation}
\left\{
\begin{array}{ll}
\frac{dx_\varphi(t)}{dt}=\left<x_\Psi(0),x_\varphi(0)\right>Ax_\varphi(t)-\left<x_\Psi(t),Ax_\varphi(t)\right>x_\varphi(t)\\
\frac{dx_\Psi(t)}{dt}=\left<x_\varphi(0),x_\Psi(0)\right>A^\dagger x_\Psi(t)-\left<x_\varphi(t),A^\dagger, x_\Psi(t)\right>x_\Psi(t).
\end{array}
\right.
\label{32bis}\end{equation}
We introduce the following definition:
\begin{defn}\label{def2}
Two vectors $(\xi_\varphi(t),\xi_\Psi(t))\neq(0,0)$, $\xi_\varphi(t)=\sum_{i=1}^{n}c_i(t)\varphi_i$ and $\xi_\psi(t)=\sum_{i=1}^{n}d_i(t)\Psi_i$, form an equilibrium pair of (\ref{32bis}) if the right-hand sides of (\ref{32bis}) are both zero.
\end{defn}
Next we prove this lemma:
\begin{lemma}\label{lemma3}
If $(\xi_\varphi(t),\xi_\Psi(t))$ is an equilibrium pair of (\ref{32bis}) then $(\xi_\varphi(t),\xi_\Psi(t))=(\xi_\varphi(0),\xi_\Psi(0))$, and, if $\left<\xi_\Psi(0),\xi_\varphi(0)\right>\neq0$,
\begin{equation}
A\xi_\varphi(0)=\frac{\left<\xi_\Psi(0),A\xi_\varphi(0)\right>}{\left<\xi_\Psi(0),\xi_\varphi(0)\right>}\,\xi_\varphi(0), \qquad A^\dagger\xi_\Psi(0)=\frac{\left<\xi_\varphi(0),A^\dagger\xi_\Psi(0)\right>}{\left<\xi_\varphi(0),\xi_\Psi(0)\right>}\,\xi_\Psi(0).
\label{33}\end{equation}
Viceversa, if $(\xi_\varphi(t),\xi_\Psi(t))=(\xi_\varphi(0),\xi_\Psi(0))$, with $\left<\xi_\Psi(0),\xi_\varphi(0)\right>\neq0$, and if $(\xi_\varphi(t),\xi_\Psi(t))$ satisfy (\ref{33}), then $(\xi_\varphi(t),\xi_\Psi(t))$ is an equilibrium pair of (\ref{32bis}).
\end{lemma}
\begin{proof}
Let us assume first that $(\xi_\varphi(t),\xi_\Psi(t))$ is an equilibrium pair of (\ref{32bis}). Hence the right-hand sides of (\ref{32bis}) are both zero, which implies that $\dot \xi_\varphi(t)=\dot \xi_\Psi(t)=0$. Then
$(\xi_\varphi(t),\xi_\Psi(t))=(\xi_\varphi(0),\xi_\Psi(0))$. Now, since for instance $\left<\xi_\Psi(0),\xi_\varphi(0)\right>A\xi_\varphi(t)-\left<\xi_\Psi(t),A\xi_\varphi(t)\right>\xi_\varphi(t)=0$, the first equality in (\ref{33}) easily follows recalling also that $\left<\xi_\Psi(0),\xi_\varphi(0)\right>\neq0$.
Viceversa, if $(\xi_\varphi(t),\xi_\Psi(t))=(\xi_\varphi(0),\xi_\Psi(0))$, and if, for instance, $A\xi_\varphi(0)=\frac{\left<\xi_\Psi(0),A\xi_\varphi(0)\right>}{\left<\xi_\Psi(0),\xi_\varphi(0)\right>}\,\xi_\varphi(0)$, then $\left<\xi_\Psi(0),\xi_\varphi(0)\right>A\xi_\varphi(0)-\left<\xi_\Psi(0),A\xi_\varphi(0)\right>\xi_\varphi(0)=0$. Therefore $$\left<\xi_\Psi(0),\xi_\varphi(0)\right>A\xi_\varphi(t)-\left<\xi_\Psi(t),A\xi_\varphi(t)\right>\xi_\varphi(t)=0$$ as well, as we had to prove
\end{proof}
From formulas in (\ref{33}) we conclude that an equilibrium pair of (\ref{32}) is a set of time-independent eigenvectors of $A$ and $A^\dagger$ respectively, with eigenvalues given in terms of $\xi_\varphi(t)$ and $\xi_\Psi(t)$, both computed at $t=0$ (or at any $t>0$, being these vectors constant in time).
Using now (\ref{34}), together with the expansions in (\ref{31}), and the biorthogonality of the sets $\{\varphi_i\}$ and $\{\Psi_i\}$, the equations in (\ref{32bis}) can be rewritten as follows:
\begin{equation}
\left\{
\begin{array}{ll}
\dot c_k(t)=\left(\left<d,c\right>\lambda_k-\Lambda(t)\right)c_k(t),\\
\dot d_k(t)=\left(\left<c,d\right>\overline{\lambda_k}-\overline{\Lambda(t)}\right) d_k(t),\\
\end{array}
\right.
\label{35}\end{equation}
where we have defined
\begin{equation}
\Lambda(t)=\sum_{k=1}^{n}\lambda_k\,\overline{d_k(t)}\,c_k(t).
\label{36}\end{equation}
It is convenient now to introduce two auxiliary functions $p_k(t)=e^{-\left<d,c\right>\lambda_kt}c_k(t)$ and $q_k(t)=e^{-\left<c,d\right>\overline{\lambda_k}\,t}d_k(t)$. Hence we can rewrite (\ref{35}) as follows:
\begin{equation}
\left\{
\begin{array}{ll}
\dot p_k(t)=-\Lambda(t) p_k(t),\\
\dot q_k(t)=-\overline{\Lambda(t)} d_k(t).\\
\end{array}
\right.
\label{37}\end{equation}
where now we rewrite $\Lambda(t)$ as follows:
$$
\Lambda(t)=\sum_{k=1}^{n}\lambda_k\,e^{2\left<d,c\right>\lambda_kt}\,\overline{q_k(t)}\,p_k(t).
$$
Next, let us consider the following initial conditions for $c_j(t)$ and $d_j(t)$: $d_j(0)=\overline{c_j(0)}$, for all $j=1,2,\ldots,n$. Notice that this choice automatically implies that $c$ and $d$ are not orthogonal, as required in Lemma \ref{lemma3}. Indeed we have $\left<d,c\right>=\sum_{k=1}^{n}\,(c_k(0))^2>0$ if we restrict, as we will do here, to real and not all zero $c_k(0)$. Hence $q_j(0)=\overline{p_j(0)}$, and the solutions for $p_k(t)$ and $q_k(t)$ are easily related: $q_k(t)=\overline{p_k(t)}$. The system in (\ref{37}) simplifies, giving a single equation
\begin{equation}
\dot p_k(t)=-\Lambda(t) p_k(t),
\label{38}\end{equation}
with
$$
\Lambda(t)=\sum_{k=1}^{n}\lambda_k\,e^{2\left<d,c\right>\lambda_kt}\,(p_k(t))^2,
$$
and we also get $\left<d,c\right>=\sum_{k=1}^{n}\,(c_k(0))^2=\sum_{k=1}^{n}\,(p_k(0))^2$ which is strictly positive under our assumptions. From (\ref{38}) we deduce that for all $k$ and $l$ the ratio $\frac{p_k(t)}{p_l(t)}$ is independent of time:
$$
\frac{p_k(t)}{p_l(t)}=\frac{p_k(0)}{p_l(0)},
$$
at least if $p_l(0)=c_l(0)\neq 0$.
Using this property we obtain the following differential equation:
$$
\frac{d}{dt}\frac{1}{p_k^2(t)}=\frac{2}{p_k^2(0)}G(t),
$$
where $G(t)=\sum_{k=1}^{n}\lambda_k\,e^{2\left<d,c\right>\lambda_kt}\,( p_k(0))^2$ so that, after few simple computations and going back to $c_k(t)$, we obtain
$$
c_k(t)=\frac{|c_k(0)|\sqrt{\left<d,c\right>}}{\sqrt{\sum_{l=1}^{n}c_l(0)^2e^{2\left<d,c\right>(\lambda_l-\lambda_k)t}}}.
$$
Finally, from (\ref{31}), we get
\begin{equation} x_\varphi(t)=\sum_{k=1}^{n}\frac{c_k(0)\sqrt{\chi_c(0)}}{\sqrt{\sum_{l=1}^{n}c_l(0)^2e^{2\chi_c(0)(\lambda_l-\lambda_k)t}}}\,\varphi_k.
\label{39}
\end{equation}
Similarly, $x_\Psi(t)=\sum_{k=1}^{n}\overline{c_k(t)}\,\Psi_k$
Now the next steps are almost identical to those for the Hermitian case: for instance, suppose that the eigenvalues of $A$ are non degenerate and that $\Re(\lambda_1)>\Re(\lambda_2)>\cdots>\Re(\lambda_n)$. Here $\Re(z)$ is the real part of $z$. Then, if $c_1(0)>0$, it follows that
$$
x_\varphi(t)\rightarrow \,\sqrt{\chi_c(0)}\varphi_1, \qquad x_\Psi(t)\rightarrow \,\sqrt{\chi_c(0)}\Psi_1,
$$
when $t\rightarrow\infty$: the solutions of the differential equations in (\ref{32}) converge to the (non-normalized, in general) eigenvectors of $A$ and $A^\dagger$ corresponding to the eigenvalue with the highest real part, called again {\em the dominant eigenvectors}.
Of course, in order to determine the eigenvectors $\varphi_i$ of $A$ and $\Psi_i$ of $A^\dagger$ related to the other $\lambda_i$, $i>1$, it would be enough to start with initial conditions for \eqref{32} which are orthogonal respectively to the eigenvectors $\Psi_1$ and $\varphi_1$, that is
$\left<x_{\varphi}(0),\Psi_1\right>=\left<x_{\Psi}(0),\varphi_1\right>=0$ or, equivalently, taking $c_1(0)=d_1(0)=0$. Hence, supposing that for instance $c_2(0)\neq0,d_2(0)\neq0$, we would have
$$
x_\varphi(t)\rightarrow \,\sqrt{\chi_c(0)}\,\varphi_2, \qquad x_\Psi(t)\rightarrow \,\sqrt{\chi_c(0)}\,\Psi_2.
$$
Similarly, by requiring that $c_1(0)=c_2(0),\ldots=c_k(0)=d_1(0)=d_2(0)\ldots=d_k(0)=0,$ we shall have $
x_\varphi(t)\rightarrow \,\sqrt{\chi_c(0)}\,\varphi_{k+1}$ and $x_\Psi(t)\rightarrow \,\sqrt{\chi_c(0)}\,\Psi_{k+1}$. We conclude that all the eigenvectors ca be found, in principle.
\vspace{2mm}
{\bf Remark:--}.
It is much simpler to recover the eigenvalue with the smallest real part. In fact, it is enough to consider the system of equation \eqref{32bis}, and consequently \eqref{35}, with $A$ and $A^\dagger$ replaced respectively by $-A$ and $-A^\dagger$. In this case the eigenvalues follow the order $-\Re(\lambda_n)>-\Re(\lambda_{n-1})>\ldots>-\Re(\lambda_1)$; hence the numerical solution in \eqref{39} will converge in the infinite time limit:
$$
x_\varphi(t)\rightarrow \,\sqrt{\chi_c(0)}\,\varphi_n, \qquad x_\Psi(t)\rightarrow \,\sqrt{\chi_c(0)}\,\Psi_n,
$$
corresponding to the lowest (in real part) eigenvalues $\lambda_n$ of $A$ and $A^\dagger$ (highest of $-A$ and $-A^\dagger$).
\vspace{2mm}
\section{A fixed point strategy}\label{sec:fixedpoint}
In this section we consider a different approach for finding the eigensystem of a given matrix $A$, not necessarily self-adjoint, based on an iterative procedure. The proof of the convergence of the procedure is given introducing a suitable contraction map and proving that its unique fixed point is indeed one of the eigenvectors of $A$. Our strategy is based on some results discussed in \cite{rev,stak,schae}, which we extend here to cover the case of possibly non real eigenvalues of $A$.
The main idea of our fixed point approach is based on the following simple considerations: assume the $N\times N$ matrix $A$ admits $N$, possibly not all different, (complex) eigenvalues $\lambda_j$. In particular, we assume that
\begin{equation}
|\lambda_1|>|\lambda_2|\geq |\lambda_3|\geq \cdots \geq |\lambda_N|.
\label{40}\end{equation}
This means, in particular, that $\lambda_1$ has multiplicity one. Using the same notation as in the previous section, this is called the { dominant eigenvalue}, and the related eigenvalue $u_1$ is the { dominant eigenstate}. The other eigenvalues can be degenerate, but still we can construct suitable linear combinations such that the set of eigenvectors of $A$, ${\cal F}_u=\{u_j\}$, $Au_j=\lambda_ju_j$, $j=1,2,\ldots,N$, is a basis for ${\cal H}=\mathbb{C}^N$. Of course, ${\cal F}_u$ is not, in general, an o.n. basis. However, as in Section \ref{sect3}, ${\cal F}_u$ admits a unique biorthogonal set ${\cal F}_v=\{v_j\}$, $\left<v_k,u_j\right>=\delta_{k,j}$, which is automatically a set of eigenstates of $A^\dagger$: $A^\dagger v_j=\overline{\lambda_j}\,v_j$, for all $j$.
Let now $x_0\in{\cal H}$, $\|x_0\|=1$, be such that $\left<v_1,x_0\right>\neq0$. Then $x_0=\sum_{i=1}^{N}\alpha_iu_i$, $\alpha_i=\left<v_i,x_0\right>$. In particular, $\alpha_1\neq0$. Now, it is clear that
\begin{equation}
A^kx_0=\lambda_1^k\left(\alpha_1u_1+\sum_{i=2}^{N}\alpha_i\left(\frac{\lambda_i}{\lambda_1}\right)^ku_i\right),
\label{main40}
\end{equation}
for all $k=0,1,2,3,\ldots$. Since $\lambda_1$ is the dominant eigenvalue, $\left|\frac{\lambda_i}{\lambda_1}\right|<1$ for all $i=2,3,4,\ldots$, and the $k$-th power of this ratio goes to zero when $k\rightarrow\infty$. Of course, the larger the difference between $|\lambda_1|$ and $|\lambda_2|$, the faster the convergence of the sum $\sum_{i=2}^{N}\alpha_i\left(\frac{\lambda_i}{\lambda_1}\right)^ku_i$. If we further consider $x_k=\frac{A^kx_0}{\|A^kx_0\|}$, we deduce that, a part from corrections of orders $O\left(\left|\frac{\lambda_2}{\lambda_1}\right|^k\right)$,
$$
x_k\simeq \left(\frac{\lambda_1}{|\lambda_1|}\right)^k\frac{\alpha_1}{|\alpha_1|} \,u_1.
$$
Of course, this sequence converges if $\lambda_1>0$ . If $\lambda_1<0$, $\{x_k\}$ oscillates between two opposite vectors, both proportional to $u_1$, the dominant eigenvector. Of course, the sequence $\{x_k\}$ {\em oscillates even more} if $\lambda_1$ is complex. It may be worth noticing that, in this procedure it is essential that $\alpha_1\neq0$. If we call $x$ the limit of $\left(\frac{|\lambda_1|}{\lambda_1}\right)^k\frac{|\alpha_1|}{\alpha_1}x_k$, $\lambda_1$ can be found as usual: $\lambda_1=\frac{\left<x,Ax\right>}{\left<x,x\right>}$.
{In complete analogy, writing $y_0=\sum_{i=1}^{N}\beta_iv_i$, $\beta_i=\left<u_i,y_0\right>$ the initial vector with $\beta_1\neq0$, we retrieve
\begin{eqnarray}
y_k=\frac{\left(A^\dag\right)^ky_0}{\|\left(A^\dag\right)^ky_0\|}\simeq \left(\frac{\bar\lambda_1}{|\bar\lambda_1|}\right)^k\frac{\beta_1}{|\beta_1|} \,v_1,\label{secv1}
\end{eqnarray} which converge to an eigenvector proportional to $v_1$.}
\vspace{2mm}
{\bf Remarks:--} (1) Numerical implementation of this strategy clearly shows the effect of the sign of $\lambda_1$ which is positive when the sequence $\{x_k\}$ converges, while is negative when $\{x_k\}$ oscillates, for large $k$, between two opposite vectors, $x$ and $-x$.
(2) The same procedure can be easily extended to the case of a $d$-degenerate $\lambda_1$, $d>1$. In this case the sequence converges to an element of the $d$-dimensional subspace corresponding to the eigenvalue $\lambda_1$.
\vspace{2mm}
{The above strategy, which will be made rigorous soon, can be extended to find more eigenvectors others than the dominant ones, $u_1$ and $v_1$, by making use of the biorhonormal sets ${\cal F}_u$ and ${\cal F}_v$. In fact, once $u_1,v_1$ have been deduced, we consider a new {\em trial vector}, which we call again $x_0$, which is orthogonal to $v_1$: $\left<x_0,v_1\right>=0$. This implies that $x_0=\sum_{i=2}^{N}\alpha_i u_i$, $\alpha_i=\left<v_i,x_0\right>$. Hence, repeating the same steps as before, we deduce that
$$
x_k=\frac{A^kx_0}{\|A^kx_0\|}\simeq \left(\frac{\lambda_2}{|\lambda_2|}\right)^k\frac{\alpha_2}{|\alpha_2|} \,u_2,
$$
and all our previous considerations can be repeated. In particular, again we conclude that $x_k$ converges up to a phase to the second dominant eigenvector $u_2$.
The procedure can be continued by finding $v_2$ as we did for $v_1$ in \eqref{secv1}, and iterated more for all the eigenvalues and eigenvectors.
}
\subsection{The contraction}
Let us now define a map $T=\frac{1}{\lambda_1}\,A$, and let us fix a (normalized) $x_0\in{\cal H}$. We define the following set:
$$
{\cal C}_{x_0}=\left\{f\in{\cal H};\quad \left<v_1,f\right>=\left<v_1,x_0\right> \right\}.
$$
Of course, since both $\lambda_1$ and $v_1$ are unknown, when we start our procedure, both $T$ and ${\cal C}_{x_0}$ cannot be explicitly identified. However, as we will show in the rest of this section, they are useful tools to prove the convergence of the power method also in presence of complex eigenvalues.
From its definition we see that all the vectors in ${\cal C}_{x_0}$ have the same projections on $v_1$. It is clear that $T$ maps ${\cal C}_{x_0}$ into ${\cal C}_{x_0}$. In fact, let $f\in{\cal C}_{x_0}$. Then, expanding $f=\sum_{j=1}^{N}\left<v_j,f\right>u_j$, we have
$$
Tf=\frac{1}{\lambda_1}\sum_{j=1}^{N}\left<v_j,f\right>Au_j=\frac{1}{\lambda_1}\sum_{j=1}^{N}\left<v_j,f\right>\lambda_ju_j=\left<v_1,f\right>u_1+\frac{1}{\lambda_1}\sum_{j=2}^{N}\left<v_j,f\right>\lambda_ju_j,
$$
so that $\left<v_1,Tf\right>=\left<v_1,f\right>=\left<v_1,x_0\right>$.
Because of the fact that $A\neq A^\dagger$, the approach used in \cite{stak} does not work. This is because, in general, ${\cal F}_u$ is not an o.n. basis. Hence, if $f=\sum_{j=1}^Nf_ju_j$, $\|f\|^2\neq \sum_{j=1}^{N}|f_j|^2$. For this reason, we introduce a new norm $\|.\|_v$ in ${\cal H}$, which is more convenient for us. Notice however that, due to the fact that ${\cal H}$ is finite-dimensional, $\|.\|_v$ is equivalent to the standard norm $\|.\|$ of ${\cal H}$, $\|f\|=\sqrt{\left<f,f\right>}$. We put
\begin{equation}
\|f\|_v=\sup_{j}|\left<v_j,f\right>|.
\label{41}\end{equation}
The fact that this is a norm is clear. In particular, $\|f\|_v=0$ if and only if $\left<v_j,f\right>=0$ for all $j$, which implies that $f=0$, due to the fact that ${\cal F}_v$, being a basis, is complete in ${\cal H}$. Since $\|.\|_v$ and $\|.\|$ are equivalent, and since ${\cal C}_{x_0}$ is a closed subspace of a complete set, ${\cal C}_{x_0}$ is also complete. Our main result is contained in the following proposition:
\begin{prop}\label{prop::contraction}
$T$ is a contraction on ${\cal C}_{x_0}$. Hence it admits an unique fixed point $y_f\in{\cal C}_{x_0}$, $y_f=\left<v_1,x_0\right>u_1$.
\end{prop}
\begin{proof}
Let $x,y\in{\cal C}_{x_0}$: $x=\sum_{j=1}^{N}\alpha_ju_j$, $y=\sum_{j=1}^{N}\beta_ju_j$, $\alpha_j=\left<v_j,x\right>$ and $\beta_j=\left<v_j,y\right>$, with $\alpha_1=\beta_1$. Hence
$$
x-y=\sum_{j=2}^{N}(\alpha_j-\beta_j)u_j, \qquad Tx-Ty=\sum_{j=2}^{N}(\alpha_j-\beta_j)\left(\frac{\lambda_j}{\lambda_1}\right)u_j.
$$
Now, using (\ref{41}),
$$
\|x-y\|_v=\sup_{j}|\alpha_j-\beta_j|, \qquad \|Tx-Ty\|_v=\sup_{j}|\alpha_j-\beta_j|\left|\frac{\lambda_j}{\lambda_1}\right|\leq \left|\frac{\lambda_2}{\lambda_1}\right|\sup_{j}|\alpha_j-\beta_j|,
$$
so that, calling $\rho=\left|\frac{\lambda_2}{\lambda_1}\right|$, $\|Tx-Ty\|_v\leq \rho \|x-y\|_v$. Notice that (\ref{40}) implies that $\rho<1$. Hence $T$ is a contraction.
The fact that $y_f=\left<v_1,x_0\right>u_1$ is a fixed point follows from a direct computation:
$$
Ty_f=\frac{1}{\lambda_1}\left<v_1,x_0\right>Au_1=\frac{1}{\lambda_1}\left<v_1,x_0\right>\lambda_1u_1=\left<v_1,x_0\right>u_1=y_f.
$$
\end{proof}
\vspace{2mm}
{\bf Remarks:--} (1) It is clear that other fixed points of $T$ also exist: $\gamma u_1$, for all complex $\gamma$. This is because $T$ is a linear map. However, in ${\cal C}_{x_0}$, normalization of the fixed point cannot be changed, because of the condition $\left<v_1,f\right>=\left<v_1,x_0\right>$. This makes the fixed point of $T$ in ${\cal C}_{x_0}$ unique.
(2) It is interesting to extend this result to infinite-dimensional Hilbert spaces. This can be useful for possible applications to quantum mechanical systems living in ${\cal L}^2(\mathbb{R})$, the space of the square-integrable functions on $\mathbb{R}$, rather than only to elements of $\mathbb{C}^n$. But, as always when going from finite to infinite-dimensional vector spaces, mathematics is much more complicated. This is work in progress.
\vspace{2mm}
Once the fixed point $y_f$ is found, the related (dominant) eigenvalue is easily deduced:
\begin{equation}
\frac{\left<y_f,Ay_f\right>}{\left<y_f,y_f\right>}=\frac{\left<u_1,Au_1\right>}{\left<u_1,u_1\right>}=\lambda_1,
\label{42}
\end{equation} as expected.
This fixed point strategy can be slightly modified to deduce more eigenvalues and eigenvectors other than the dominant ones. We will now briefly sketch what is known as the {\em shifted inverse power} method, modified to take into account the fact that, in our particular case, $A\neq A^\dagger$.
Let $q\in\mathbb{C}$ be a fixed complex number and suppose $q\neq\lambda_j$, for all $j$. This implies that $B_q=(A-q1 \!\! 1)^{-1}$ exists. Also, under our assumptions, $u_j$ is an eigenstate of $B_q$ with eigenvalue $\mu_j=(\lambda_j-q)^{-1}$: $B_qu_j=\mu_ju_j$. Let us call $j_0$ the value of the integer such that $|\mu_{j_0}|>|\mu_j|$, for all $j\neq j_0$. We now introduce the map $D=\frac{B_q}{\mu_{j_0}}$, and repeat for $D$ what we have done for $T$ before. In particular, it is possible to prove that $D$ has an unique fixed point $Y_f=\left<v_{j_0},x_0\right>u_{j_0}$ in the space ${\cal C}_{x_0}^{j_0}=\left\{f\in{\cal H};\quad \left<v_{j_0},f\right>=\left<v_{j_0},x_0\right> \right\}$: $DY_f=Y_f$. Then, since
$$
\frac{\left<Y_f,DY_f\right>}{\left<Y_f,Y_f\right>}=1,
$$
we deduce that
$$
\frac{\left<Y_f,B_q^{-1}Y_f\right>}{\left<Y_f,Y_f\right>}=\mu_{j_0}=\frac{1}{\lambda_{j_0}-q},
$$
which gives back the value of $\lambda_{j_0}$ after few simple computations.
\subsection{The Schwartz Quotient}\label{sec::schwartz}
Proposition \ref{prop::contraction} in the previous section states that $T$ is a contraction on ${\cal C}_{x_0}$ and, as such, because of the properties of ${\cal C}_{x_0}$, it admits a unique fixed point. It is worth stressing (once more!) that $T$ is defined by working as if the dominant eigenvalues $\lambda_1$ was known, which is not the case, of course: $\lambda_1$ will be computed only at the end of our numerical implementation. To avoid this apparent paradox, it can be useful to define a new (iterative)-map which we shall prove converges to $T$.
First of all we define the so called \textit{ Schwartz quotients},
\begin{equation}
\gamma_m=\frac{\left< \left(A^\dag\right)^{m+1} x_{\psi}, A^m x_{\varphi}\right>}{\left< \left(A^\dag\right)^{m} x_{\psi}, A^m x_{\varphi},\right>},\label{squot}
\end{equation}
where $x_\varphi=\sum_{j=1}^{N}\alpha_j u_j$ and $x_\psi=\sum_{j=1}^{N}\beta_jv_j$ are two initial random vectors with
$\alpha_j=\left<v_j,x_\varphi\right>$ and $\beta_j=\left<u_j,x_\psi\right>$.
It is clear that $\gamma_m$ is an approximation of the dominant eigenvalue $\lambda_1$ and it is expected to converge to it for $m\rightarrow\infty$. In fact, making use of the bi-orthogonality conditions of the eigenvectors of $A$ and $A^\dag$, we have that
\begin{eqnarray*}
\gamma_m=\lambda_1
\frac{\alpha_1\bar{\beta}_1+\sum_{j=2}^N\alpha_1\bar{\beta}_j\left(\frac{\lambda_j}{\lambda_1}\right)^{2m+1}}{
\alpha_1\bar{\beta}_1+\sum_{j=2}^N\alpha_1\bar{\beta}_j\left(\frac{\lambda_j}{\lambda_1}\right)^{2m}}=&&\\
\lambda_1\left(\frac{\alpha_1\bar{\beta}_1}{
\alpha_1\bar{\beta}_1+\sum_{j=2}^N\alpha_j\bar{\beta}_j\left(\frac{\lambda_j}{\lambda_1}\right)^{2m}}+
\frac{\sum_{j=2}^N\alpha_j\bar{\beta}_j\left(\frac{\lambda_j}{\lambda_1}\right)^{2m+1}}{
\alpha_1\bar{\beta}_1+\sum_{j=2}^N\alpha_j\bar{\beta}_j\left(\frac{\lambda_j}{\lambda_1}\right)^{2m}}\right)=&&\\
\lambda_1\left(\frac{1}{1+O\left(\frac{\lambda_2}{\lambda_1}\right)^{2m}}\right)+ O\left(\lambda_2\alpha_2\bar{\beta}_2\left(\frac{\lambda_2}{\lambda_1}\right)^{2m}\right).&&
\end{eqnarray*}
Since $\lambda_1$ is the dominant eigenvalue, then $\left|\frac{\lambda_2}{\lambda_1}\right|<1$, and the $m$-th power of the ratio $\frac{\lambda_2}{\lambda_1}$ goes to zero when $m\rightarrow\infty$. Hence $\gamma_m\rightarrow\lambda_1$ for $m\rightarrow\infty$.
Then we define the map $\mathcal{S}_m=\frac{A}{\gamma_m} $ on
$$
{\cal C}_{x_\varphi}=\left\{f\in{\cal H};\quad \left<v_1,f\right>=\left<v_1,x_\varphi\right> \right\}.
$$
and we prove that $\mathcal{S}_m$ can be used to define a sequence of vectors converging to the fixed point of $T$. This is not surprising since, recalling that $\gamma_m\rightarrow\lambda_1$, it is reasonable that $S_m\frac{A}{\gamma_m}\rightarrow\frac{A}{\lambda_1}$ when $m\rightarrow\infty$. To be more rigorous, we observe that
$$
T^mx_\varphi=\alpha_1u_1+\sum_{j=2}^{N}\alpha_j\left(\frac{\lambda_j}{\lambda_1}\right)^{m}u_j,
$$
and
$$
S_mx_{m-1}=
\sum_{j=1}^{N}\alpha_j\left(\frac{\lambda_j^m}{\prod_{k=1}^m\gamma_k}\right)u_j,\\
$$
where $x_1=S_1x_\varphi,\ldots,x_m=S_mx_{m-1}$.
Hence
\begin{eqnarray*}
T^mx_\varphi-S_mx_{m-1}=\sum_{j=1}^{N}\alpha_j\left[\left(\frac{\lambda_j}{\lambda_1}\right)^m-\frac{\lambda_j^m}{\prod_{k=1}^m\gamma_k}\right]u_j.
\end{eqnarray*}
and
\begin{eqnarray*}
\|T^mx_\varphi-S_mx_{m-1}\|_v=\sup_j\left|\alpha_j\left[\left(\frac{\lambda_j}{\lambda_1}\right)^m-\frac{\lambda_j^m}{\prod_{k=1}^m\gamma_k}\right]\right|.
\end{eqnarray*}
Now, as $\gamma_m\approx\lambda_1+ O\left(\frac{\lambda_2}{\lambda_1}\right)^{2m}$,
\begin{eqnarray*}
\|T^mx_\varphi-S_mx_{m-1}\|_v=\sup_j\left|\alpha_j\left[\left(\frac{\lambda_j}{\lambda_1}\right)^m-\frac{\lambda_j^m}{\lambda_1^m+O\left(\left(\frac{\lambda_2}{\lambda_1}\right)^2\right)\lambda_1^{m-1}}\right]\right|=\\
\sup_j\left|\alpha_j\left[\left(\frac{\lambda_j}{\lambda_1}\right)^m-\frac{\lambda_j^m}{\lambda_1^m\left(1+O\left(\left(\frac{\lambda_2}{\lambda_1}\right)^2\right)\frac{1}{\lambda_1}\right)}\right]\right|\leq\\
\left|\left(\frac{\lambda_2}{\lambda
_1}\right)^m\right|\sup_j\left|\alpha_j\left(1-\frac{1}{1+O\left(\left(\frac{\lambda_2}{\lambda_1}\right)^2\right)\frac{1}{\lambda_1}}\right)\right|,
\end{eqnarray*}
so that
$$
\|T^mx_\varphi-S_mx_{m-1}\|_v\rightarrow0,
$$
for $m\rightarrow\infty$. Now, since $\lim_{m,\infty}T^mx_\varphi=u_1$, we can conclude that
also the sequence $S_mx_{m-1}\rightarrow u_1$ in the same limit.
\section{Numerical results}\label{sect4}
In this section we present some numerical applications of the strategies proposed in Sections \ref{sect3} and \ref{sec:fixedpoint}.
The dynamical approach analyzed in Section \ref{sect3} requires the numerical solution of the system of ODEs \eqref{32bis}. In order to solve it, we have used
a multi step variable order Adams-Bashforth-Moulton
time discretization method which usually requires the solutions at several preceding time
points to compute the current solution, \cite{sg}. Once the solutions $x_\varphi(t)$ and $x_\Psi(t)$ are computed at a specific time $t$, the related {\em upgraded} eigenvalues are obtained as a consequence of the Lemma \ref{lemma3}, using the formula
\begin{equation}
\lambda(t)=\frac{\left<x_\Psi(t),Ax_\varphi(t)\right>}{\left<x_\Psi(t),x_\varphi(t)\right>}\qquad \bar{\lambda}(t)=\frac{\left<x_\varphi(t),A^\dagger x_\Psi(t)\right>}{\left<x_\varphi(t),x_\Psi(t)\right>}.
\label{332}\end{equation}
Evaluation of the solution is then stopped when
\begin{equation}
\|Ax_\varphi(t)-\lambda x_\varphi(t)\|<\delta_{tol},\quad \|A^\dag x_\Psi(t)-\bar{\lambda}x_\Psi(t)\|<\delta_{tol},
\label{add1}\end{equation}
where $\delta_{tol}$ is a small tolerance value.
The fixed point approach, described in Section \ref{sec:fixedpoint}, makes use of the map $\mathcal{S}_m$ defined in
Subsection \ref{sec::schwartz}. It requires two initial guess vectors $x_\varphi=\sum_{j=1}^{N}\alpha_j u_j$ and $x_\psi=\sum_{j=1}^{N}\beta_jv_j$, and at the generic iteration $k$ the eigenvalue approximation ($\lambda_k$) used to define $\mathcal{S}_k$ is given by the Schwartz quotient \eqref{squot}.
Then the upgraded eigenvector of $A$ is given by $x_k=\mathcal{S}_kx_{k-1}$, where $x_0=x_\varphi$. Iteration is stopped when, as in (\ref{add1}),
\begin{equation}
\|Ax_k-\lambda_k x_k\|<\delta_{tol}.
\label{add2}\end{equation}
\subsection{A test case: $7\times7$ matrix }\label{sectIV1}
The first numerical experiment (E1), deals with a non hermitian random squared matrix of order 7 of the form
\begin{equation}
A=R+iT,
\label{add3}\end{equation}
where $R$ and $T$ are the random matrices
$$
R=\left(
\begin{array}{ccccccc}
0.445 & -0.219 & 0.489 & 0.770 & 0.589 & -0.00333 & 0.950 \\
0.481 & -0.892 & -0.806 & -0.743 & -0.641 & -0.422 & 0.701 \\
-0.735 & 0.747 & 0.750 & -0.879 & 0.884 & -0.0114 & -0.260 \\
0.528 & 0.357 & 0.707 & 0.986 & 0.201 & 0.320 & 0.207 \\
0.899 & 0.727 & 0.206 & -0.792 & 0.109 & 0.895 & 0.672 \\
-0.400 & -0.259 & -0.988 & 0.459 & 0.681 & 0.843 & 0.788 \\
0.326 & -0.530 & -0.168 & 0.141 & 0.0158 & -0.496 & -0.907 \\
\end{array}
\right)
$$
$$
T=\left(
\begin{array}{ccccccc}
0.959 & 0.314 & -0.237 & 0.232 & -0.608 & 0.199 & -0.164 \\
-0.744 & -0.112 & 0.239 & 0.384 & -0.132 & 0.299 & 0.817 \\
0.921 & 0.681 & -0.302 & 0.942 & -0.781 & 0.908 & -0.0566 \\
0.465 & -0.641 & 0.505 & -0.892 & -0.830 & 0.715 & -0.170 \\
-0.807 & 0.978 & -0.185 & -0.619 & -0.923 & 0.322 & -0.690 \\
0.0672 & 0.893 & 0.620 & 0.711 & -0.631 & -0.636 & -0.211 \\
-0.742 & -0.0257 & 0.536 & -0.952 & -0.325 & 0.0701 & 0.196 \\
\end{array}
\right).$$
The eigenvalues of $A$, as deduced by Matlab, for instance, and ordered from those having largest to those with lowest real parts, are
$\lambda_1=1.5181 - 1.2564i, \lambda_2=0.9604 - 2.2206i,\lambda_3=0.9394 - 0.6078i,\lambda_4=0.8326 + 2.0418i,\lambda_5= -0.7583 - 1.154,\lambda_6= -0.8380 + 0.1978i,\lambda_7=-1.3201 + 1.2896i$. In what follows we will show how these values can be recovered using our strategies.
\subsubsection{Dynamical approach, experiment E1}
Numerical solution of \eqref{32bis} has been carried by fixing the tolerance on the convergence of the solution to
$\delta_{tol}=10^{-8}$. Starting with a random initial conditions, the obtained converging solutions $x_{\varphi_1}^\infty$ and $x_{\Psi_1}^\infty$, related to the first eigenvalues $\lambda_1,\bar\lambda_1$, are
$$x_{\varphi_1}^\infty=\left( \begin{array}{l}
0.1451 + 0.0193i\\
-0.1616 + 0.0317i\\
-0.2025 - 0.0594i\\
0.1936 + 0.1459i\\
0.4491 - 0.2830i\\
0.6958 - 0.2368i\\
0.0554 - 0.1155i
\end{array}\right),\quad
x_{\Psi_1}^\infty=\left( \begin{array}{l}
0.0387 + 0.0163i\\
0.0072 - 0.0091i\\
-0.0274 - 0.0008i\\
0.0256 - 0.0002i\\
0.0234 + 0.0520i\\
0.0452 + 0.0044i\\
0.0317 + 0.01\\
\end{array}\right).
$$
The related eigenvalues, obtained trough \eqref{332}, are $\lambda_1=1.5181 - 1.2564i$ and $\bar{\lambda}_1=1.5181 + 1.2564i$. Notice that $x_{\varphi_1}^\infty$ and $x_{\Psi_1}^\infty$ are not automatically bi-normalized, since $\left<x_{\varphi_1}^\infty,x_{\Psi_1}^\infty\right>=0.0415 + 0.0457i\neq 1$,
and hence they represents the bi-orthonormal vectors $\varphi_1$ and $\Psi_1$ in \eqref{biorto}, respectively, only up to some normalizations.
In Figure \ref{fig:7x7}(a) the convergence of the absolute values of the components of the dynamical solution $x_{\varphi_1}(t)$ to $x_{\varphi_1}^\infty$ is shown (we find a similar behaviour for convergence to $x_{\Psi_1}^\infty$, not shown in figure). Real and imaginary parts of the eigenvalue $\lambda_1$ obtained at each time with
\eqref{332}, are shown in
Figure \ref{fig:7x7}(c).
Concerning the determination of the eigenvectors related the eigenvalue $\lambda_7$ having the smallest real part, we have solved \eqref{32bis} by replacing $A$ and $A^\dagger$ with $-A$ and $-A^\dagger$ respectively.
The solutions $x_{\varphi_7}^\infty$ and $x_{\Psi_7}^\infty$ are
$$x_{\varphi_7}^\infty=\left( \begin{array}{l}
-0.1751 + 0.1561i\\
-0.5680 + 0.0590i\\
-0.0281 + 0.1148i\\
0.1124 - 0.0007i\\
0.1513 + 0.4898i\\
-0.2348 + 0.1173i\\
-0.2529 - 0.4435
\end{array}\right),\quad
x_{\Psi_7}^\infty=\left( \begin{array}{l}
-0.00032 + 0.00015\\
-0.00033 + 0.00089\\
-0.0005 + 0.00025\\
-0.00011 + 0.00016i\\
0.000792 - 0.000228i\\
-0.00055 + 0.00044i\\
-0.001379 - 0.00015\\
\end{array}\right).
$$
The corresponding eigenvalues are $\tilde\lambda_7=1.3201 - 1.2896i$ and $\bar{\tilde\lambda}_7=1.3201 + 1.2896i$ which, as expected, are related to the eigenvalues $\lambda_7=-1.3201 + 1.2896i$ and $\bar\lambda_7=-1.3201 - 1.2896i$ of $A$ and $A^\dagger$ by a simple change of sign.
Again, we see that the solutions we get are not automatically bi-normalized, since $\left<x_{\varphi_7}^\infty,x_{\Psi_7}^\infty\right>=0.00095 - 0.0014i\neq1$, so that $x_{\varphi_7}^\infty$ and $x_{\Psi_7}^\infty$ coincide with $\varphi_7$ and $\Psi_7$ only up to some normalization factor.
However the bi-orthogonality conditions with the vectors $x_{\varphi_1,\Psi_1}^\infty$ are satisfied since
the various scalar products $\left<x_{\varphi_1,\Psi_1}^\infty,x_{\varphi_7,\Psi_7}^\infty\right>$
are well below the tolerance imposed (in general of order the precision of the machine $10^{-15}$): $\left<x_{\varphi_1,\Psi_1}^\infty,x_{\varphi_7,\Psi_7}^\infty\right>\simeq 0$.
\begin{figure}[!ht]
\begin{center}
\hspace*{-2.0cm}\subfigure[Experiment E1: time evolutions of the components of $|x_{\varphi_1}(t)|$ obtained from \eqref{32}]{\includegraphics[width=8.5cm]{eigvmax_7x7eigenv}}
\hspace*{+1.0cm}\subfigure[Experiment E1: time evolutions of the $|x_{\varphi_7}(t)|$ obtained from \eqref{32} with the matrices $-A,-A^\dagger$]{\includegraphics[width=8.5cm]{eigvmin_7x7eigenv}}
\hspace*{-0.4cm}\subfigure[Experiment E1: time evolutions of the real and imaginary parts of the eigenvalues $\lambda_1$ and $\lambda_7$ of $A$]{\includegraphics[width=8.5cm]{7x7eigenv}}
\caption{Numerical results of dynamical approach applied to the experiment E1. }
\label{fig:7x7}
\end{center}
\end{figure}
\subsubsection{Fixed point approach, experiment E1}
In this subsection we show the results concerning the fixed point strategy applied to the same $7\times7$ matrix $A$ as in (\ref{add3}). Starting with a random initial condition $x_0$ and a tolerance $\delta_{tol}=10^{-8}$, the first set of iterations converge toward the vector
$$\tilde\varphi_2=\left( \begin{array}{l}
-0.1366 - 0.2000i\\
0.2704 + 0.0788i\\
-0.5370 + 0.3358i\\
-0.1461 - 0.2711i\\
-0.4839 + 0.0951i\\
0.2303 - 0.0502i\\
-0.1436 - 0.2160i
\end{array}\right).
$$
We call this vector $\tilde\varphi_2$ since it corresponds to the eigenvalue $\lambda_2=0.9604 - 2.2206i$ of $A$ given above, which is the largest in norm.
The convergence of the initial guess to the first eigenvector is shown in Figure \ref{fig:7x7fp}(a), where the norm of the components of the vector are shown. Convergence of the real and imaginary parts of the eigenvalue is shown in Figure \ref{fig:7x7fp}(d).
By applying the {shifted inverse power} method, by picking randomly complex values $q$,
we then obtain the other eigenvectors. The first two of them are in sequence
$$\tilde\varphi_5=\left( \begin{array}{l}
-0.3639 - 0.0758i\\
0.3899 - 0.3802i\\
-0.2303 + 0.3945i\\
0.2895 + 0.3292i\\
0.0127 + 0.2107i\\
-0.2879 - 0.181\\
-0.0623 + 0.0175i
\end{array}\right),\quad
\tilde\varphi_7=\left( \begin{array}{l}
0.2049 - 0.1141i\\
0.5672 + 0.0665i\\
0.0524 - 0.1059i\\
-0.1099 - 0.0238i\\
-0.0407 - 0.5110i\\
0.2547 - 0.0633i\\
0.1500 + 0.4880i
\end{array}\right),
$$
corresponding to the eigenvalues $\lambda_5$ and $\lambda_7$. Notice that with the shifted inverse power method the obtained sequence of eigenvectors do not follow the norm ordering (in fact $\lambda_5$ and $\lambda_7$ are not the greatest eigenvalues in norm after $\lambda_2$). The convergence to the eigenvectors $\tilde\varphi_5$ and $\tilde\varphi_7$ is shown in Figure \ref{fig:7x7fp}(b-c), where the norm of the components of the vectors are shown. Convergence of the real and imaginary parts of the corresponding eigenvalues is shown again in Figure \ref{fig:7x7fp}(d).
\begin{center}
\begin{figure}[!ht]
\hspace*{-2.0cm}\subfigure[Experiment E1: absolute values of the components of the first eigenvector
of $A$]{\includegraphics[width=8.5cm]{phi2_7x7_fp}}
\hspace*{+1.0cm}\subfigure[Experiment E1: absolute values of the components of the second eigenvector
of $A$]{\includegraphics[width=8.5cm]{phi5_7x7_fp}}\\
\hspace*{-2.0cm}\subfigure[Experiment E1: absolute values of the components of the third eigenvector
of $A$]{\includegraphics[width=8.5cm]{phi7_7x7_fp}}
\hspace*{+1cm}\subfigure[Experiment E1:convergence of real and imaginary parts of the three eigenvalues $\lambda_2,\lambda_5$ and $\lambda_7$]{\includegraphics[width=8.0cm]{7x7eigenv_fp}}
\caption{Numerical results of fixed point approach applied to the experiment E1. }
\label{fig:7x7fp}
\end{figure}
\end{center}
\subsection{The Hessenberg matrix}\label{sectIV2}
In this second experiment (E2) we apply our numerical procedures to the finite Hessenberg matrix, an upper matrix with positive subdiagonal. The procedure to construct this kind of matrix is well known and we refer to \cite{Escr,Escr2} for more details.
The finite Hessenberg Matrix of order $n$ we consider is fully defined by a sequence $\{\alpha_k\}_{k\in\mathbb{N}}$ such that $\lim\limits_{k\rightarrow\infty}\alpha_k=0$:
$$
D\{\alpha_k\}=\begin{pmatrix}
-\alpha_1\bar{\alpha_0} & -\frac{k_0}{k_1}\alpha_2\bar{\alpha}_0 & -\frac{k_0}{k_2}\alpha_3\bar{\alpha}_0 & \ldots\ & -\frac{k_0}{k_{n-1}}\alpha_{n}\bar{\alpha}_0\\
\frac{k_0}{k_1} & -\alpha_2\bar{\alpha}_1 & -\frac{k_1}{k_2}\alpha_3\bar{\alpha_1} &\ldots & -\frac{k_1}{k_{n-1}}\alpha_{n}\bar{\alpha}_1\\
0 & \frac{k_1}{k_2} & -\alpha_3\bar{\alpha_2} & \ldots & -\frac{k_2}{k_{n-1}}\alpha_{n}\bar{\alpha}_2 \\
\vdots & \vdots & \vdots & \vdots &\vdots \\
0 & 0 & 0 & \ldots & -\alpha_n\bar{\alpha}_{n-1}&
\end{pmatrix},
$$
with the various $k_j$ related to $\{\alpha_k\}$ as follows: $k_0=1, k_j=k_{j-1}/\sqrt{1-|\alpha_k|^2}$.
{It is well known that the elements of $D\{\alpha_k\}$ converges for large $n$ to those of $S_R$, the (finite) right shift matrix, see \cite{Escr2}. Here $S_R=\{\delta_{j+1,j}\}_{j\in{1,\ldots,n}}$. It can be seen that the faster the sequence $\{\alpha_k\}_{k\in\mathbb{N}}$ converges to zero, the faster the eigenvalues of $D\{\alpha_k\}$ converge also to zero.}
\subsubsection{Dynamical approach, experiment E2}
{We start our analysis on $D\{\alpha_k\}$ by first solving \eqref{32bis} with $A=D\{\alpha_k\}$, with the size of matrix $n=15$, a variable tolerance (depending on the eigenvalue considered\footnote{This is important since the eigenvalues can be very small, as already discussed.})
from $\delta_{tol}=10^{-10}$ to $\delta_{tol}=10^{-14}$, and random initial conditions.
We stress that, for this experiment, due to the particularly small values of the eigenvalues, we need to use smaller tolerance than in experiments E1 above and E3 below.}
Fixing $\delta_{tol}=10^{-10}$, the convergence to the determined eigenvalues with the largest and smallest real part, $\lambda_1$ and $\lambda_{15}$ respectively, are shown in Figs.\ref{fig:Hess_dyna}-\ref{fig:Hess_dynb} respectively for the cases $\{\alpha_k\}_{k\in\mathbb{N}}=\{\exp(-k^2)\}_{k\in\mathbb{N}}$ and
$\{\alpha_k\}_{k\in\mathbb{N}}=\{1/(k^2!)\}_{k\in\mathbb{N}}$: as previously explained the eigenvalue $\lambda_{15}$ is easily retrieved by replacing $A=D\{\alpha_k\}$ with $A=-D\{\alpha_k\}$ in \eqref{32bis}.
{As expected the solutions we obtain converge to eigenvalues which are very small and approach the value $0$: for the matrix $D\{\exp(-k^2)\}$ of order $n=15$, we obtain $\lambda_1=0.0059$
and $\lambda_{15}=-0.0233$, whereas for the matrix $D\{(1/(k^2!)\}$ of order $n=15$, the converging values are $\lambda_1=6.58\cdot 10^{-5}$ and $\lambda_{15}=-0.0417$.
To retrieve the other eigenvalues we have to decrease further the tolerance to
$\delta_{tol}=10^{-14}$ (close to the minimal resolution allowed by the machine) as the other eigenvalues decrease to zero very rapidly: for instance the subsequent eigenvalues retrieved tend to the values $1.81\cdot10^{-8}$ and $1.32\cdot 10^{-11}$, which of course require a very small tolerance to be well determined.
}
\begin{figure}[!ht]
\begin{center}
\hspace*{-2cm}\subfigure[Experiment E2: convergence of real and imaginary parts of the eigenvalues having the largest ($\lambda_1$) and the smallest ($\lambda_{15}$) real part obtained with \eqref{32}
for the Hessenberg matrix $D\{\exp(-k^2)\}$ of order $n=15$. In the inset the early time evolution.]
{\includegraphics[width=8.5cm]{expm2Hess_dyn}\label{fig:Hess_dyna}}
\hspace*{1cm}\subfigure[Experiment E2: convergence of real and imaginary parts of the eigenvalues having the largest ($\lambda_1$) and the smallest ($\lambda_{15}$) real part obtained with \eqref{32}
for the Hessenberg matrix $D\{1/(k^2!)\}$ of order $n=15$. In the inset the early time evolution.]{\includegraphics[width=8.5cm]{factorial2Hess_dyn}\label{fig:Hess_dynb}}
\caption{Numerical results of dynamical approach applied to the experiment E2. }
\end{center}
\end{figure}
\subsubsection{Fixed point approach, experiment E2}
In this subsection we consider the same matrix for the same sequences and the same tolerances, but we use the fixed point strategy.
Convergence of the three largest eigenvalues in norm are shown in Figs.\ref{fig:Hess_fpa}-\ref{fig:Hess_fpnb} (only real part are shown). The eigenvalues we find are $\lambda_1=-0.0233,\lambda_2=0.0059,\lambda_3=-0.00069$ for the matrix $D\{\exp(-k^2)\}$ of order $n=15$ , and $\lambda_1=-0.0417,\lambda_2=6.58\cdot 10^{-5},\lambda_3=1.81\cdot 10^{-8}$ for the
matrix $D\{1/(k^2!)\}$ of order $n=15$.
\begin{figure}[!ht]
\begin{center}
\hspace*{-2cm}\subfigure[Experiment E2: convergence of real parts of the three largest in norm eigenvalues for the Hessenberg matrix $D\{\exp(-k^2)\}$ of order $n=15$ with the fixed point strategy.]
{\includegraphics[width=8.5cm]{Hess_expm2_fp}\label{fig:Hess_fpa}}
\hspace*{1cm}\subfigure[Experiment E2: convergence of real parts of the three largest in norm eigenvalues for
the Hessenberg matrix$D\{1/(k^2!)\}$ of order $n=15$ with the fixed point strategy. ]{\includegraphics[width=8.5cm]{Hess_fact2_fp}\label{fig:Hess_fpnb}}
\caption{Numerical results of fixed point strategy applied to the experiment E2. }
\end{center}
\end{figure}
\subsection{Application to Quantum Mechanics: the truncated Swanson model}
In this third numerical experiment (E3), we work with a finite matrix that has a relevance in the contest of
pseudo-Hermitian Quantum Mechanics. In particular we consider the truncated Swanson model (hereafter TSM),
characterized by a finite Hamiltonian matrix which is not self adjoint, but which still admits only real eigenvalues.
In the following we will briefly recall how this Hamiltonian can be obtained, while we refer the interested reader to \cite{bag2018} for more details and, in particular, for the physical relevance of this model.
The TSM Hamiltonian can be written as
$$
H_{\theta}=\frac{1}{\cos(2\theta)}\left(B_\theta A_\theta+\frac{1}{2}\left(1 \!\! 1-Nk\right)\right),
$$
where $N$ is a non negative integer fixing the dimension of the system,
$\theta\in(-\pi/4,\pi/4)\backslash\{0\}$ is a parameter tuning the non Hermiticity of the system,
$A_\theta$ and $B_\theta$ are operators (matrices) satisfying the commutation rule
$$
\left(A_\theta B_\theta -B_\theta A_\theta\right) f=f-Nkf,\quad \forall f\in \mathbb{C}^N,
$$
and $k$ is a projection operator
which annihilates the vector $e_N$ of the canonical orthonormal basis of $\mathbb{C}^N$, and satisfies $k = k^
2 = k^\dag$ together with $kA_\theta=B_\theta k=0$.
In \cite{bag2018} it is shown that $H_\theta$ is similar to the truncated quantum harmonic oscillator Hamiltonian
$h$,
\begin{equation}
H_\theta=T_\theta h T_\theta^{-1},
\label{add4}\end{equation}
where $T_\theta=\exp\left(i\theta(a^2-(a^\dagger)^2)\right)$ and $a$ is the truncated annihilator operator defined in \cite{batrunc}, which satisfies the ladder equations on the basis $\{e_k\}_{k=1,\ldots N}$:
$$
ae_1=0,\quad a e_k=\sqrt{k}e_{k-1},\quad a^\dag e_k=\sqrt{k+1}e_{k+1}, \quad a^\dag e_N=0.
$$
Equation (\ref{add4}) implies that $H_\theta$ has the same spectrum as $h$, $\mu_k=\frac{2(k-1)+1}{2}, k=1,\ldots,N.$\footnote{Notice that maintaining the same formalism of the previous sections, the numerical eigenvalues, ordered from the one with the largest real part to the lowest, are labelled as $\lambda_1,\ldots,\lambda_N$, so that $\lambda_k$ will correspond to $\mu_{N+1-k}$, $k=1,\ldots,N$. }
For concreteness, we fix now $N=7$ and $\theta=0.4$. Then the TSM Hamiltonian has the following form:
$$
H_\theta=\left(
\begin{array}{ccccccc}
0.348126 & 0 & -0.510541 i & 0 & 0.0140773 & 0 & 0.0216558 i \\
0 & 1.05673 & 0 & -0.805139 i & 0 & -0.145695 & 0 \\
-0.510541 i & 0 & 1.79157 & 0 & -1.01756 i & 0 & -0.372785 \\
0 & -0.805139 i & 0 & 1.9337 & 0 & -2.76093 i & 0 \\
0.0140773 & 0 & -1.01756 i & 0 & 2.0337 & 0 & -4.04439 i \\
0 & -0.145695 & 0 & -2.76093 i & 0 & 7.50957 & 0 \\
0.0216558 i & 0 & -0.372785 & 0 & -4.04439 i & 0 & 9.8266 \\
\end{array}
\right)
$$
\subsubsection{Dynamical approach, experiment E3}
We solve \eqref{32bis} for the matrices $H_\theta$ and $H_\theta^\dag$ with the tolerance
$\delta_{tol}=10^{-8}$. Starting with random initial conditions, the solutions $x_{\varphi_1}^\infty$ and $x_{\Psi_1}^\infty$ related to the first eigenvalues $\lambda_1$, are
$$x_{\varphi_1}^\infty=\left( \begin{array}{l}
-0.0088 + 0.0149i\\
-0.0000003 + 0.000001i\\
-0.1676 - 0.0983i\\
0.0000001 - 0.0000001i\\
0.3206 - 0.5465i\\
0.0000001 + 0.0000002i\\
0.6457 + 0.3789i
\end{array}\right),\quad
x_{\Psi_1}^\infty=\left( \begin{array}{l}
0.00014 - 0.000243444797343i\\
-0.00000136 - 0.000000641i\\
-0.0027 - 0.0016i\\
-0.00000028 + 0.00000065i\\
-0.0052 + 0.0089i\\
0.0000082 + 0.00000038i\\
0.0105 + 0.0061
\end{array}\right).
$$
In Figure \ref{fig:7x7swaa} the convergence of the absolute values of the components of the dynamical solution $x_{\varphi_1}(t)$ converging to $x_{\varphi_1}^\infty$ is shown (we get a similar behaviour for the convergence to $x_{\Psi_1}^\infty$, not shown in figure). Real and imaginary parts of the eigenvalue $\lambda_1$ obtained from
\eqref{332}, are shown in
Figure \ref{fig:7x7swac}, and as expected they both converge to the eigenvalue of $H_\theta$ with the largest real part, that is $\mu_7=6.5$. Considering the results related to the lowest eigenvalue, $\mu_1=0.5$, retrieved by switching to $-H_\theta$ and $-H_\theta^\dag$, the converging solutions $x_{\varphi_7}^\infty$ and $x_{\Psi_7}^\infty$ are
$$x_{\varphi_7}^\infty=\left( \begin{array}{l}
0.8363 - 0.4557i\\
0.0000002 - 0.0000003i\\
0.1362 + 0.2499002i\\
0.0000004 + 0.0000i\\
-0.0908 + 0.0495i\\
-0.0000003 + 0.0000001i\\
-0.0171 - 0.0313i\\
\end{array}\right),\quad
x_{\Psi_7}^\infty=\left( \begin{array}{l}
-0.6568 + 0.3579i\\
-0.0000001 + 0.00000002i\\
0.1070 + 0.1963i\\
0.0000004 + 0.00000003i\\
0.0713 - 0.0389i\\
0.0000005 - 0.00000001i\\
-0.0134 - 0.0246i\\
\end{array}\right).
$$
Convergence of the absolute values of the related dynamical solution $x_{\varphi_7}(t)$ to $x_{\varphi_7}^\infty$ is shown in \ref{fig:7x7swab}, whereas the real and imaginary parts of the corresponding eigenvalue $\lambda_7$ obtained from
\eqref{332} are shown in
Figure \ref{fig:7x7swac}.
\begin{figure}[!ht]
\begin{center}
\hspace*{-1.5cm}\subfigure[Experiment E3: time evolutions of the components of $|x_{\varphi_1}(t)|$ obtained from \eqref{32}]{\includegraphics[width=8.5cm]{eigvecmax_SWA}\label{fig:7x7swaa}}
\hspace*{1cm}\subfigure[Experiment E3: time evolutions of the components of $|x_{\varphi_7}(t)|$ obtained from \eqref{32}]{\includegraphics[width=8.5cm]{eigvecmin_SWA}\label{fig:7x7swab}}
\subfigure[Experiment E3: time evolutions of the real and imaginary parts of the eigenvalues $\lambda_1$ and $\lambda_7$ of $A$]{\includegraphics[width=8.5cm]{swanson_eigenv_dinamyc}\label{fig:7x7swac}}
\caption{Numerical results of dynamical approach applied to the experiment E3. }
\end{center}
\end{figure}
\subsubsection{Fixed point approach, experiment E3}
In this subsection we show the results concerning the fixed point strategy applied to $H_\theta$, again with $N=7$, $\theta=0.4$ and a tolerance $\delta_{tol}=10^{-8}$.
We report in Figure \ref{fig:SWAfp} (for simplicity only the real parts are shown) the convergence of the seven eigenvalues after the applications of the shifted inverse power method. Of course the number of iterations needed for convergence is highly sensitive to the randomly value $q$ used to generate the inverse matrix $A-q1 \!\! 1$, as it is clearly shown in Figure \ref{fig:SWAfp}: some eigenvalue is found after just few iterations, while others need more iterations to be reached.
\begin{figure}[!ht]
\begin{center}
\subfigure[Experiment E3: time evolutions of the real part of the 7 computed eigenvalues $\lambda_1,\ldots,\lambda_7$.]{\includegraphics[width=8.0cm]{swanson_eigenv_fp}}
\caption{Numerical results of fixed point approach applied to the experiment E3. }
\label{fig:SWAfp}
\end{center}
\end{figure}
\section{Conclusions}\label{sect5}
In this paper we have discussed how some standard techniques existing in the literature to compute eigenvalues and eigenvectors of a given Hermitian matrix can be extended to include matrices which are not Hermitian. { In particular we have considered a dynamical approach based on the solution of a system of ODEs, which naturally extends the procedure proposed in \cite{tang}, and a fixed point approach based on the construction of a suitable contraction as in \cite{schae}. Many scientific and data analysis applications require the determination of the eigensystem of Hermitian matrices, while in this work our main interest is moved to non Hermitian matrices. This kind of matrices, if seen as bounded (or even unbounded, if the matrices are infinite) operators, are quite often met in pseudo-hermitian quantum mechanics, where the Hamiltonian of a given system is not required to be Hermitian at all, but rather to satisfy some invariance property, \cite{benbook}.}
In this case, it is very likely that the eigenvalues of the Hamiltonian are complex and the mathematical and numerical setting proposed in \cite{tang} and in \cite{schae} should be adjusted to consider the appearance of biorthogonal sets of eigenvectors.
{In this work we have extended the mathematical procedures presented in the aforementioned papers, and we have applied them to two pedagogical examples generated by a random matrix and by an Hessenberg matrix, and to an Hamiltonian operator obtained from a finite-dimensional version of the Swanson model.
Our methodologies worked very well, and we were able to determine easily the eigensystem of the various matrices.} We should also mention that, as it is well known, when a given matrix possesses some symmetry, its eigenvalues obey some special rule. For instance, if $H$ is a matrix which is similar to a Hermitian operator $H_0$, i.e. if $H=SH_0S^{-1}$ for some invertible matrix $S$, then the eigenvalues of $H$ are all real. This is what happens, for instance, in PT- or pseudo-Hermitian quantum mechanics, \cite{benbook,op3}, and in our Experiment 3. Symmetries have played no role in this paper, but hopefully they will be considered in some future work.
{We stress that our work is mainly intended to provide some rigorous mathematical framework to deal with the eigenvalue problems considered in this paper, and in general for non Hermitian matrices, without focusing on any possible acceleration methods.} Of course,
modern information processing requires the solution of eigenvalue problems for very large matrices, and hence a consistent but also fast method is required. A lot can still be done in trying to accelerate the convergence of the methods considered here. This could involve the application of suitable acceleration procedures to speed up the convergence of the iterations, and this is just a part of our future works. Another relevant extension of our results should include those situations for which the convergences of our approaches fail. This is the case, for instance, when the real parts (resp. the norms) of the eigenvalues coincide, see Section \ref{sect3} (resp. Section \ref{sec:fixedpoint}).
We are planning to consider a possible extension of the dynamical approach adopting (or extending) the fractal variational principle already used in literature for a quite wide class of applications (\cite{frac1,frac2,frac3}).
Also, we are interested in extending our ideas to infinite-dimensional matrices, and to compare our results with those in \cite{laub}. This is particularly interesting for us, in view of our interest for solving Schr\"odinger equations for concrete systems living in infinite-dimensional Hilbert spaces.
\section*{Acknowledgements}
This work was partially supported by the University of Palermo and by the Gruppo Nazionale di Fisica Matematica of Indam.
The authors want to thank Dr. Dario Armanno and the unknown Reviewers for the useful comments and discussions which helped to improve the final version of the paper.
| {
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With Manchester United facing Tottenham Hotspur at White Hart Lane on Sunday, I posed several questions to members of the TottenhamHotspur.tv forum. Read Stretford End Arising's match preview: Tottenham Hotspur vs. Manchester United.
Stretford End Arising: What are your thoughts on Sunday's game? What team is Harry likely to select and what formation will he adopt? Where will the game be won or lost?
CmonyouSpurs: I'm not really looking forward to it to be honest. Every time I have any sort of optimism as we're about to play Man United, your lot are successful in shooting my hopes down quite emphatically. I suppose given our performances this season I should be more hopeful, but it's just not happening for me to be honest. I think I'd be relatively pleased with a draw, elated with a win and disappointed, yet not terribly surprised if we lost the game. As for formation, I'm guessing we'll stick to a 4-4-1-1: Gomes – Hutton, Dawson, Gallas, Assou-Ekotto – Lennon, Modric, Jenas/Palacios, Bale – van der Vaart – Defoe. Where will the game be won or lost? On the scoreboard methinks. But on a more serious note, obviously the midfield is the key area where pretty much every game is won or lost. I'm confident with 4 our of our 5 midfield slots, the only one that makes me nervous being the CM slot partnering Modric. Be it Palacios or Jenas, I have reason to not be confident in either one, as Palacios, as good as a ball-winner as he is, can't pass to save his life, and Jenas, as good as he has been when he's come in this season, is very much prone to errors and just switching off in general which could be costly. Hopefully we can have the better of you down the flanks this time though.
HeroesComesandGoes: It would be a good game. It's a must win for both teams. The team would be the same as the line up against Aston Villa (away). I can see us winning this to be honest.
SEA: David Beckham's loan deal – yes please? Or no thank you? If the loan deal goes through where will Beckham fit into the squad/team? Your thoughts on recent rumours linking Spurs with Phil Neville?
CyS: Definite yes please to Becks. Mainly for his experience, attitude and mentality rather than what he'd offer on the pitch. We need a good experienced midfielder who's pretty much been there and done it all. It would be fantastic if he could teach Azza to cross a ball too. Aside from that, I still think he has something to offer on the pitch as well. If we bring him on in the last 20 minutes or so with us winning, he could help calm us down and retain possession better, and if we're losing he can create more chances from the wide areas by constantly bombarding the opposition with whipped crosses in. Add his set-piece threat and it seems like we'd be spoilt for choice when it comes to set-piece takes in the team with the likes of Bale, Modric, VDV and Becks all ready to take them.
HCaG: Yes please. David Beckham is a true legend and his free kicks will be deadly. He may play centre midfield or come on as a sub for Lennon. Phil Neville would be another good signing, he is versatile and experienced – which the team needs.
SEA: Redknapp has recently stated Real Madrid and Barcelona are on a different planet to United. What are your thoughts on the current United team/squad? Is Harry incorrect? Or is Fergie's team over-achieving to remain undefeated and top of the table?
CyS: I'd have to agree with Harry to be honest. That top Spanish duo is quite simply on another level to any other club side in the world today I feel. When you consider Barca spanked Madrid 5-0, it also shows how bloody great Barca are. Man Utd's team is quite strong all-round at the minute I guess. Defence seems as solid as ever, with Rio, Vidic and Evra being as solid as ever, Rafael seemingly coming on in leaps and bounds, and with the likes of Smalling, Evans, Brown, Neville, O'Shea, Fabio and others able to do a job as cover, though Evans has been a bit shaky this year I've heard, and Neville seems to be on his last legs. Midfield could seemingly use some strengthening, with Hargreaves being perma-crocked, Fletcher seemingly underperforming this year, Giggs and Scholes nearing retirement and Gibson looking lacking in certain areas of his game. Carrick seems quite solid though, most of the time, if not quite outstanding, and Anderson has seemingly improved quite a lot this season. The wings seem well covered though, with Nani upping his game, Valencia due to return from injury soon and Park always being able to do an efficient job whenever called upon. Forward areas could perhaps use some strengthening as Rooney hasn't quite been firing on all cylinders this season and Berbatov bring inconsistent in terms of scoring goals. Plenty of good prospects coming through though, with Chicharito, Macheda, Welbeck, Obertan and perhaps Diouf I could all see being important players for you in the next decade and more. Shame things haven't worked out for Mickey Owen though. Can't see him staying at your club beyond this summer myself. I think you have over-achieved slightly to be undefeated at this stage of the season, however seeing you top of the table at the moment isn't a massive shock, although I expected Chelsea to be far better challengers than they have been so far.
HCaG: Madrid and Barcelona both possesses a great squad. United on the other hand need to invest, particular with the likes of Scholes and Giggs who are old. So, I'm with Harry here.
SEA: Spurs are currently fourth in the Premier League, one point ahead of Chelsea and four behind Arsenal. Will finishing out of the top four after finishing fourth last season be classed as failure?
CyS: I guess so, but I don't think it would be disastrous. I'm convinced that whether we finish in the top 4 again or not, the likes of Modric, Bale, VDV, Lennon etc. will all still remain at the club unless silly money is offered. It would be disappointing to finish out of the top 4 this year, but when you look at how strong City have gotten and with you lot, Arsenal and Chelsea being as strong as ever (I'm convinced Chelsea will click back into something resembling true form soon enough), then it isn't inconceivable that we'll drop out this year.
SEA: The last time we spoke Stephen Caulker and Danny Rose were two young players recommended to keep an eye on. How have they progressed? Which other young players have impressed recently? At the start of the season I watched a friendly between the two academy teams and was impressed with Harry Kane and Oyenuga. What is the latest on the pair?
CyS: Well Caulker has seemingly progressed very well. He won the Football League Young Player of the Month for November 2010 for his performances, and Bristol City fans really think highly of him, with some even going as far to think that they have a future England international currently on loan at the club. Danny Rose is at the stage of his career where he should probably be playing more first team football, and it doesn't look like he's progressed well enough to get that at Spurs just yet, so there's a chance he may look for a move away soon. I'll always remember the lad fondly obviously though. He came in for one game against our arch rivals and his wonder goal against them helped us beat them. Can't really have asked for much more than that. As for Kane and Oyenuga, well Kane went off on loan to Leyton Orient last week and there are high hopes for his future, so hopefully he'll gain some valuable experience and playing time with the O's. Haven't really heard much about Oyenuga recently though, but he's also a highly-rated player for the future.
HCaG: Andros Townsend's performance against Charlton was amazing. He is a left footed winger.
SEA: You must be excited by the Champions League tie against AC Milan? How will Spurs fare? You must be fancying your chances after the two performances against Inter Milan.
CyS: Oh I am most definitely excited about the game vs Milan. I truly believe that we can have what it takes to beat them. Our main advantage is that our strengths are going to come up directly against their weaknesses, with Bale and Lennon bombing down the flanks against Abate and Zambrotta or Antonini. There is no doubt in my mind that we will get the better of them out wide, so it's all about the quality of our delivery and who's there to get on the end of our crosses into the area. I think we have enough in our midfield to win the midfield battle too, though it won't be easy for Modric and Huddlestone against Pirlo, Gattuso and Ambrosini. Their attack is very potent though, and we must remain wary of that, as the likes of Robinho, Cassano, Ibrahimovic, Pato and Boateng are good enough to pierce any defence on the planet in my opinion. However, I'm confident that over the two legs, we will prove that we are better than them and we will progress into the Champions League Quarter Finals.
SEA: Thank you CmonyouSpurs and HeroesComesandGoes!
Discuss in the Stretford End Arising forum.
"the retarded and insulting idea that Lennon has anything to learn from Beckham" which planet do you hail from? | {
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{"url":"http:\/\/math.stackexchange.com\/questions\/93580\/name-for-a-bipartite-graph-in-which-one-vertex-set-has-maximal-degree-1","text":"# Name for a bipartite graph in which one vertex set has maximal degree 1?\n\nI'm looking for a specific name for a bipartite graph $(U,V,E)$ in which there is at most one edge incident to each vertex $u \\in U$. That is, $|E_u| \\le 1$ for all $u \\in U$, where $E_u = \\{(u,v) \\in E\\}$.\n\nThe best I have been able to think of is \"star forest\", but this term seems to be used specifically for subgraphs.\n\nIt would be helpful if there was also terminology for the vertex sets $U$ (with maximal degree 1) and $V$ (with arbitrary degree).\n\nBackground: the application is in parallel computing where each vertex has a canonical owner, but may be \"ghosted\" in other memory spaces. Sometimes the term \"local-to-global map\" is used for this, but the concept is more general.\n\n-\nYou could use \"disjoint union of stars\". Also \"star forest\" sounds fine, as long as you define the term before using it. \u2013\u00a0 Yuval Filmus Dec 22 '11 at 22:25\n@Yuval: I think you can post your comment as answer. \u2013\u00a0 Paul Dec 23 '11 at 0:17\n\nYou could use \"disjoint union of stars\". Also \"star forest\" sounds fine, as long as you define the term before using it.\n\n-\n\nI talked to a graph theorist friend who also affirmed that \"star forest\" is a good, unambiguous name. She pointed out that in the other instances where this term is used, the authors really meant spanning star forest. Here is a draft of some notes I'm writing on the communication model. An implementation in terms of MPI one-sided operations is in the development version of PETSc.\n\n-","date":"2015-08-01 01:57:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7961058020591736, \"perplexity\": 350.66876147038977}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-32\/segments\/1438042988399.65\/warc\/CC-MAIN-20150728002308-00295-ip-10-236-191-2.ec2.internal.warc.gz\"}"} | null | null |
Bonjour my Beauties, so today's post is all about Valentine's day. Whether you are celebrating with your love or your friends, it's the perfect time to dress up romantic and no it doesn't have to be in red.
I got the pleasure of shooting with Paola, a friend who is a fashion blogger as well and we both agreed that red is a little cliché, so why not think outside the box and inspire you with outfits you can rock for Valentine's day. She wore this beautiful romper from Nasty gal, I swear I kept checking her out, it looked so beautiful, loved the flowy feel every time she walked, you will definitely attract attention wherever you go wearing this romper. I will put the link of her outfit below. | {
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\section{Introduction}
\label{sec:1}
\paragraph{Brief history}
The study of nonlinear wave equations has been an active research field since decades ago, and tremendous results have been obtained in $\RR^{3+1}$.
It is well-known, for example see the examples by John \cite{John2, John}, that wave equations with quadratic nonlinearities might not have global-in-time solutions. Later on, the celebrated breakthrough by Klainerman \cite{Klainerman80, Klainerman852, Klainerman86} relying on the vector field method and Christodoulou \cite{Christodoulou} relying on the conformal method, showed that global-in-time solutions exist for wave equations with null nonlinearities. We also recall other work on the three dimensional wave equations \cite{L-R-cmp, L-R-annals} and \cite{P-S-cpam}, which generalise the notion of null forms. As an application, many physical models, like Dirac equations, Maxwell equations, Einstein equations, are proved to be stable under small perturbations.
Due to the fact that waves in $\RR^{2+1}$ do not decay fast enough, the classical null condition cannot guarantee that semilinear wave equations in $\RR^{2+1}$ with null quadratic nonlinearities have global-in-time solutions.
The existence results of global-in-time solutions to quadratic nonlinear wave equations in $\RR^{2+1}$ was first obtained by Godin \cite{Godin} in the semi linear case and by Alinhac \cite{Alinhac1, Alinhac2} in the quasilinear case, where the author showed that a class of quasilinear wave equations of the form $-\Box u + g^{\alpha \beta \gamma} \del_\gamma u \del_{\alpha \beta} u = 0$ satisfying the null condition are stable under the assumption that the initial data are small and compactly supported. Later on Zha \cite{Zha} had a thorough study on a large class of wave equations in $\RR^{2+1}$. We also remind one the study of the system of coupled quasilinear wave and Klein-Gordon equations in $\RR^{2+1}$ by Ma \cite{YM1, YM2, YM0}, using the hyperboloidal foliation method \cite{LM0} which dates back to \cite{Klainerman85}. But the compactness assumption on the initial data is needed in all of the above results.
Without the compactness restriction on the initial data, Katayama \cite{Katayama} obtained the global solutions to a class of semilinear wave equations in $\RR^{2+1}$.
Later on, Cai, Lei, and Masmoudi \cite{Cai} removed the compactness assumption on the initial data after \cite{Alinhac1}, and obtained the global well-posedness result for the scalar wave equation which is fully nonlinear.
There is a large literature on the study of wave maps, we are not going to be exhaustive. We only mention the work \cite{Tao2001, Wong} on wave maps in $\RR^{2+1}$.
\paragraph{Model of interest and the main difficulties}
We will consider the following system of semilinear wave equations
\bel{eq:model}
\aligned
- \Box u_i &= R^{jkl}_i u_j Q_0 (u_k, u_l),
\\
u_i (t_0, \cdot) = &u_{i0},
\qquad
\del_t u_i (t_0, \cdot) = u_{i1},
\endaligned
\ee
in which $\Box = \del_\alpha \del^\alpha = -\del_t \del_t + \del_a \del^a$, and $Q_0 (v, w) := - \del_\alpha v \del^\alpha w~ or ~ \del_\alpha v \del_\beta w - \del_\alpha w \del_\beta v$ is the null form.
In the above, $t_0$ is the initial time taken to be 0, and $i, j, k, l \in \{ 1, 2, \cdots, n_0 \}$ with $n_0\geq 1$ the number of unknowns (equations). Greek indices $\alpha, \beta, \cdots$ run in $\{0, 1, 2\}$, Latin letters $a, b, \cdots$ run in $\{1, 2\}$, and the Einstein summation convention is adopted unless specified. Besides, we use $A \lesssim B$ to denote $A \leq C B$ with $C$ a generic constant.
Compared to the existing results in \cite{Alinhac1, Alinhac2, Cai, YM1, YM2}, the main difference is that the nonlinearities in the model problem \eqref{eq:model} include the potential $u_k$ (with no derivatives), instead of only $\del u_k$ and $\del \del u_j$. Recall the standard energy of wave equation is of the form
$$
\sum_{i, \alpha} \int_{\RR^2} | \del_\alpha u_i |^2 \, dx,
$$
which does not include the $L^2$ norm of the potential
$$
\sum_i \int_{\RR^2} | u_i |^2 \, dx.
$$
Thus it requires us to bound $\| u \|_{L^2(\RR^2)}$.
The Hardy inequality
$$
\Big\| {w \over r} \Big\|_{L^2(\RR^n)}
\leq C \| \nabla w \|_{L^2(\RR^n)}
$$
is only true for $n \geq 3$, and the $L^2$ norm of the potential can be bounded by the conformal energy only when $n \geq 3$, see the remark in \cite{Wong}. The $L^2$ estimates (possibly with weight) for waves in $\RR^{2+1}$ were obtained in \cite{Wong, Liu, YM0}, but the compact support assumption on the initial data is required.
In order to conquer that difficulty and bound $\sum_i \int_{\RR^2} | u_i |^2 \, dx$, we write the wave equations in the Fourier space, and derive the formulation for $u_i$ by solving the ordinary differential equation (see for example \cite{Dong-zeromass}), then a careful treatment on each terms in the formulation gives us the desired result, see Lemma \ref{lem:linear}. However, there is a "loss of derivative" problem occurring when we estimate the highest order energy, see the proof of Proposition \ref{prop:E} for example. We find that this "loss of derivative" problem can be overcome by an observation on the estimates of the null forms and by the aid of the ghost weight energy estimates \cite{Alinhac1}.
\paragraph{Main theorem}
Our goal is to obtain global-in-time solutions to the system \eqref{eq:model}, which is stated now.
\begin{theorem}\label{thm:main}
Consider the system of coupled wave equations \eqref{eq:model}, and let $N \geq 1$ be a sufficiently large integer. The parameters $R^{j k l}_i$ are taken to be constants. Then there exists a small $\eps_0 > 0$, such that the Cauchy problem \eqref{eq:model} admits a global-in-time solution $(u_i)$ as long as the initial data $(u_{i0}, u_{i1})$ satisfy the smallness condition
$$
\sum_{|I| \leq N+1}\| \Lambda^I u_{i0} \|_{L^2(\RR^2)} + \sum_{|J|\leq N}\| \Lambda^J u_{i1} \|_{L^2(\RR^2) \cap L^1(\RR^2)}
< \eps,
\qquad
\text{for any } \eps \in (0, \eps_0),
$$
with $\Lambda = \del_a, r\del_r, \Omega_{ab}$.
Moreover, the solution decays almost sharply with
\be
|u(t, x)|
\lesssim (1+t+|x|)^{-1/2} (1+|t-|x||)^{-1/2} \log(1+t).
\ee
\end{theorem}
In general the smallness condition on $\| \Lambda^J u_{i1} \|_{L^1(\RR^2)}$ is not assumed, but it will be used in the proof of Lemma \ref{lem:linear}.
\paragraph{Organisation}
In Section \ref{sec:pre}, we revisit some preliminaries of the wave equations and the vector field method. Next in Section \ref{sec:linear}, we prove the $L^2$ norm of the potential solving the linear wave equation. Then we give the proof to Theorem \ref{thm:main} in Section \ref{sec:proof}.
\section{Preliminaries}\label{sec:pre}
We work in the $(2+1)$ dimensional spacetime with signature $(-, +, +)$. A point in $\RR^{2+1}$ is denoted by $(x_0, x_1, x_2) = (t, x_1, x_2)$, and its spacial radius is denoted by $r = \sqrt{x_1^2 + x_2^2}$. We will use
\be
E(w, t)
:=
\int_{\RR^2} \Big(|\del_t w|^2 + \sum_a |\del_a w|^2 \Big) \, dx
\ee
to denote the energy of a sufficiently nice function $w = w(t, x)$ on the constant time slice.
The vector fields we will use in the following analysis include:
\bei
\item Translations: $\del_\alpha$, \quad $\alpha = 0, 1, 2$.
\item Rotations: $\Omega_{ab} = x_a \del_b - x_b \del_a$, \quad $a, b = 1, 2$.
\item Lorentz boosts: $L_a = x_a \del_t + t \del_a$, \quad $a = 1, 2$.
\item Scaling vector field: $L_0 = t \del_t + r \del_r$.
\eei
We will use $\Gamma$ to denote the vector fields in
$$
V := \{ \del_\alpha, \Omega_{ab}, L_a, L_0 \}.
$$
The following well-known results of commutators will be also frequently used.
\begin{lemma}
For any $\Gamma', \Gamma'' \in V$ we have
\be
[\Box, \Gamma'] = C \Box,
\qquad
[\Gamma', \Gamma''] = \sum_{\Gamma \in V, C_\Gamma} C_\Gamma \Gamma,
\ee
with $C, C_\Gamma$ some constants.
\end{lemma}
In order to estimate null forms and to overcome the problem of "loss of derivative", we need the following lemma which gives very detailed estimates on the null forms and can be found in \cite{Sogge} for example.
\begin{lemma}\label{lem:null}
It holds that
\be
(1+t) |Q_0(v, w)|
\lesssim
\sum_{|I| = 1} | \Gamma^I v | \sum_{a, \alpha} \big( | L_a w | + | \del_\alpha w| \big).
\ee
Besides, if we act the vector field $\Gamma^I$ on the null form $Q_0 (v, w)$, a similar result holds
\be
\aligned
(1+t) |\Gamma^I Q_0(v, w)|
&\lesssim
\sum_{|I_1| \leq |I|, a, \alpha} \big(| \Gamma^{I_1} v | + |L_a \Gamma^{I_1} v | + |\del_\alpha \Gamma^{I_1} v | \big) \sum_{1\leq |I_2| \leq |I|/2 + 1} | \Gamma^{I_2} w |
\\
&+ \sum_{ |I_1| \leq |I|, a, \alpha} \big( | \Gamma^{I_1} w | + |L_a \Gamma^{I_1} w | + |\del_\alpha \Gamma^{I_1} w | \big) \sum_{1\leq |I_2| \leq |I|/2 + 1} | \Gamma^{I_2} v |.
\endaligned
\ee
\end{lemma}
We next recall the Klainerman-Sobolev inequality (see \cite{Sogge} for example) in $\RR^{2+1}$.
\begin{proposition}\label{prop:Sobolev}
It holds that
\be
\langle t+r \rangle^{1/2} \langle t-r \rangle^{1/2} |u|
\lesssim
\sum_{|I| \leq 2} \| \Gamma^I u \|_{L^2(\RR^2)},
\ee
with $\langle a \rangle = \sqrt{1 + |a|^2}$.
\end{proposition}
The following energy estimates of the ghost weight method by Alinhac \cite{Alinhac1} will play a vital role in compensating the "loss of derivative" issue.
\begin{proposition}
Let $w$ be the solution to
$$
- \Box w = f,
$$
then it holds
\be
\aligned
&E_{gst1} (w, t)
\leq
\int_{\RR^2} e^{q} \big( |\del_t w|^2 + \sum_a |\del_a w|^2 \big) \, dx (0)
+
2 \int_0^t \int_{\RR^2} f \del_t w e^q \, dxdt,
\endaligned
\ee
in which $q = \arctan (r-t)$, and
\be
E_{gst1} (w, t)
=
\int_{\RR^2} e^q \big( |\del_t w|^2 + \sum_a |\del_a w|^2 \big) \, dx (t)
+
\sum_{a} \int_0^t \int_{\RR^2} {e^q \over r^2 \langle r-t \rangle^2} \big| (x_a \del_t + r \del_a) w \big|^2 \, dxdt.
\ee
\end{proposition}
Since $-\pi/2 \leq q \leq \pi/2$, we equivalently have
\bel{eq:ghost1}
\aligned
E_{gst2} (w, t)
\lesssim
\int_{\RR^2} \big( |\del_t w|^2 + \sum_a |\del_a w|^2 \big) \, dx (0)
+
\int_0^t \int_{\RR^2} f \del_t w \, dxdt,
\endaligned
\ee
with
\be
E_{gst2} (w, t)
=
\int_{\RR^2} \big( |\del_t w|^2 + \sum_a |\del_a w|^2 \big) \, dx (t)
+
\sum_{a} \int_0^t \int_{\RR^2} {1 \over r^2 \langle r-t \rangle^2} \big| (x_a \del_t + r \del_a) w \big|^2 \, dxdt.
\ee
\section{$L^2$ estimates on wave equation}\label{sec:linear}
We have the following lemmas which help bound the $L^2$ norm of the solution (with no derivatives in front) to wave equations.
\begin{lemma}\label{lem:linear}
Let $w$ be the solution to the linear wave equation
\bel{eq:w}
\aligned
&- \Box w = f,
\\
w(1, \cdot) = &w_0,
\quad
\del_t w(1, \cdot) = w_1.
\endaligned
\ee
We assume that
\be
\| w_0 \|_{L^2(\RR^2)} + \| w_1 \|_{L^2(\RR^2) \cap L^1(\RR^2)}
< +\infty,
\ee
as well as either
\be
\| f(t, \cdot) \|_{L^2(\RR^2) \cap L^1(\RR^2)}
< C_f (1+t)^{-1 + \beta},
\qquad
\beta \in (-\infty, 1),
\ee
or
\be
\int_0^t \| f(t', \cdot) \|_{L^2(\RR^2) \cap L^1(\RR^2)} \, dt'
\lesssim
C_f (1+t)^\beta,
\qquad
\beta \in [0, 1).
\ee
Then the following $L^2$ norm bound is valid
\begin{eqnarray}\label{eq:l2bound}
\| u \|_{L^2(\RR^2)}
\lesssim
\left\{
\begin{array}{lll}
&
\| w_0 \|_{L^2(\RR^2)}
+
\log^{1/2} (2+t) \| w_1 \|_{L^2(\RR^2) \cap L^1(\RR^2)}
+
\log^{1/2} (2+t) C_f,
\quad
& \beta < 0,
\\
&
\| w_0 \|_{L^2(\RR^2)}
+
\log^{1/2} (2+t) \| w_1 \|_{L^2(\RR^2) \cap L^1(\RR^2)}
+
\log^{3/2} (2+t) C_f,
\quad
& \beta = 0,
\\
&
\| w_0 \|_{L^2(\RR^2)}
+
\log^{1/2} (2+t) \| w_1 \|_{L^2(\RR^2) \cap L^1(\RR^2)}
+
(1+t)^\beta \log^{1/2} (2+t) C_f,
\quad
& 0< \beta <1.
\end{array}
\right.
\end{eqnarray}
\end{lemma}
\begin{proof}
$Step ~1.$ Expressing the solution in Fourier space.
We write the equation \eqref{eq:w} in the Fourier space to get
$$
\aligned
\del_{tt} \wh(t, \xi) + |\xi|^2 \wh(t, \xi) = \fh(t, \xi),
\\
\wh(1, \cdot) = \wh_0,
\qquad
\del_t \wh(1, \cdot) = \wh_1.
\endaligned
$$
We solve the ordinary differential equation in $t$ to arrive at the expression of the solution $w$ in Fourier space
$$
\wh(t, \xi)
=
\cos (t |\xi|) \wh_0
+
{\sin (t |\xi|) \over |\xi|} \wh_1
+
\int_0^t {\sin \big( (t-t') |\xi|\big) \over |\xi|} \fh(t') \, dt'.
$$
Thus the $L^2$ norm of $w$ can be bounded by the following three terms
\be
\aligned
\|w\|_{L^2(\RR^2)}
&\lesssim
\|w_0\|_{L^2(\RR^2)}
+
\Big\|{\sin (t |\xi|) \over |\xi|} \wh_1 \Big\|_{L^2(\RR^2)}
+
\int_0^t \Big\|{\sin \big( (t-t') |\xi|\big) \over |\xi|} \fh(t') \Big\|_{L^2(\RR^2)} \, dt'
\\
& =: \|w_0\|_{L^2(\RR^2)} + A_1 + A_2.
\endaligned
\ee
The last two terms $A_1, A_2$ needs a more careful treatment.
$Step ~2.$ Estimating the term $A_1$.
We first bound the term $A_1^2$
$$
\aligned
A_1^2
&=
\int_{\{\xi : |\xi| \leq 1\}} {\sin^2 (t |\xi|) \over |\xi|^2} |\wh_1|^2 \, d\xi
+
\int_{\{\xi : |\xi| \geq 1\}} {\sin^2 (t |\xi|) \over |\xi|^2} |\wh_1|^2 \, d\xi
\\
&\leq
|| \wh_1 ||^2_{L^\infty(\RR^2)} \int_{\{\xi : |\xi| \leq 1\}} {\sin^2 (t |\xi|) \over |\xi|^2} \, d\xi
+
\| \wh_1 \|_{L^2(\RR^2)}^2.
\endaligned
$$
Next we proceed by estimating
$$
\aligned
\int_{\{\xi : |\xi| \leq 1\}} {\sin^2 (t |\xi|) \over |\xi|^2} \, d\xi
&=
\int_{S^1} dS^1 \int_0^{1} {\sin^2 (t |\xi|) \over |\xi|} \, d|\xi|
\\
&\lesssim
\int_0^{t} {\sin^2 p \over p} \, d p
\\
&\lesssim
\int_0^{1} 1 \, d p
+
\int_{1}^{t+2} {1 \over p} \, d p,
\endaligned
$$
where we used the simple fact that $\sin |p| \leq |p|$ and $\sin p \leq 1$.
By gathering the above results, we obtain
$$
A_1^2
\lesssim
\log(2+t) \| w_1 \|_{L^2(\RR^2) \cap L^1(\RR^2)}^2,
$$
which gives us
\be
A_1
\lesssim
\log^{1/2}(2+t) \| w_1 \|_{L^2(\RR^2) \cap L^1(\RR^2)}.
\ee
$Step~3.$ Estimating the term $A_2$.
By the analysis in $Step~2$, we have
$$
\aligned
\Big\|{\sin \big( (t-t') |\xi|\big) \over |\xi|} \fh(t') \Big\|_{L^2(\RR^2)}
&\lesssim
\log^{1/2}(2 + t- t') \| f(t') \|_{L^2(\RR^2) \cap L^1(\RR^2)}
\\
&\leq
C_f \log^{1/2}(2+t - t') \, (1+t')^{-1 + \beta},
\endaligned
$$
in which we used the decay assumption on $ \| f(t') \|_{L^2(\RR^2) \cap L^1(\RR^2)}$ in the last step.
Hence we have
$$
A_2
\lesssim
\log^{1/2} (2+t) \int_0^t (1+t')^{-1 + \beta} \, dt'.
$$
By calculating the integral in terms of the values of $\beta$, and gathering the results in the previous steps, we thus complete the proof.
\end{proof}
This lemma is of vital importance in the proof of Theorem \ref{thm:main}. Recall that the $L^2$ norm estimates on solutions to wave equations in $\RR^{2+1}$ have been obtained before \cite{Wong, Liu, YM0}, but the compactness assumption on the initial data is needed (although the compactness assumption can be removed in the theorem in \cite{Wong}). To the best of our knowledge, the estimates in Lemma \ref{lem:linear} is the first such result where no compactness assumptions are imposed on the initial data.
In Lemma \ref{lem:linear} there is a $\log$ or polynomial growth in the bounds of the $L^2$ norms. As far as we understand, such growth also exists in \cite{Wong, Liu, YM0}.
\begin{remark}
Recall that by using the conformal energy estimates in $\RR^{d+1}$ with $d \geq 3$, the energy
$$
\sum_{|I| \leq 1} \| \Gamma^I w \|_{L^2(\RR^d)}
$$
can be bounded by the conformal energy.
However we note that in Lemma \ref{lem:null}, we only have the upper bound for
$$
\| w \|_{L^2(\RR^2)},
$$
which means we "lose" one order of derivative, and that is what we interpret as the issue of "loss of derivative".
\end{remark}
\section{Proof of the main theorem}\label{sec:proof}
Relying on a standard bootstrap argument, we are going to give the proof of Theorem \ref{thm:main}.
According to the smallness of the initial data, we can assume the following bounds hold in the time interval $[t_0, T)$ with $T> t_0$
\bel{eq:BA}
\aligned
\sum_i E_{gst2} ( \Gamma^I u_i, t)^{1/2}
&\leq C_1 \eps,
\qquad
|I| \leq N,
\\
\sum_i \big\| \Gamma^I u_i \big\|_{L^2(\RR^2)}
&\leq C_1 \eps \log(1+t),
\qquad
|I| \leq N-1,
\\
\sum_i \big\| \Gamma^I u_i \big\|_{L^2(\RR^2)}
&\leq C_1 \eps (1+t)^{1/4 + 2\delta},
\qquad
|I| = N.
\endaligned
\ee
In the above, $\delta>0$ is some small constant, $C_1$ is some large constant (to ensure $T>t_0$) which is to be determined, and $T$ is defined by
\bel{eq:T-def}
T := \{ t > t_0 : \eqref{eq:BA} ~holds\}.
\ee
Our goal is to show the following refined estimates
\bel{eq:BA-refined}
\aligned
\sum_i E_{gst2} ( \Gamma^I u_i, t)^{1/2}
&\leq {1\over 2} C_1 \eps,
\qquad
|I| \leq N,
\\
\sum_i \big\| \Gamma^I u_i \big\|_{L^2(\RR^2)}
&\leq {1\over 2} C_1 \eps \log(1+t),
\qquad
|I| \leq N-1,
\\
\sum_i \big\| \Gamma^I u_i \big\|_{L^2(\RR^2)}
&\leq {1\over 2} C_1 \eps (1+t)^{1/4 + 2\delta},
\qquad
|I| = N.
\endaligned
\ee
The time bound $T$ has two kinds of possible values: $+\infty$ or some finite number. If $T = + \infty$, then Theorem \ref{thm:main} is proved. If $T < + \infty$ is some finite number and the refined estimates in \eqref{eq:BA-refined} are also established, then this contradicts the definition of $T$ in \eqref{eq:T-def}, which means $T$ must be $+\infty$. In either case, we are led to $T = +\infty$, which implies Theorem \ref{thm:main}.
A direct result of the Sobolev-Klainerman inequality in Proposition \ref{prop:Sobolev} gives us the following pointwise decay results.
\begin{lemma}\label{lem:sup}
Let the bootstrap assumptions in \eqref{eq:BA} be true, then we have
\be
\aligned
\langle t+r \rangle^{1/2} \langle t-r \rangle^{1/2} \sum_i | \del \Gamma^J u_i |
&\lesssim C_1 \eps,
\qquad
|J| \leq N-3,
\\
\langle t+r \rangle^{1/2} \langle t-r \rangle^{1/2} \sum_i | \Gamma^J u_i |
&\lesssim C_1 \eps \log(1+t),
\qquad
|J| \leq N-3.
\endaligned
\ee
\end{lemma}
\begin{proposition}[Improved estimates on the energy norm $E$]\label{prop:E}
Under the bootstrap assumptions in \eqref{eq:BA}, we have
\be
\sum_i E_{gst2}( \Gamma^I u_i, t)^{1/2}
\lesssim \eps + (C_1 \eps)^2,
\qquad
|I| \leq N.
\ee
\end{proposition}
\begin{proof}
Recall the system of equations \eqref{eq:model}, and we act the vector filed $\Gamma^I$ with $I \leq N-1$ to get
$$
\aligned
- \Box \Gamma^I u_i
= \sum_{|I_1| + |I_2| \leq N-1} R^{jkl}_i \Gamma^{I_1} u_j \Gamma^{I_2} Q_0 (u_k, u_l).
\endaligned
$$
By using the ghost weight energy estimate \eqref{eq:ghost1}, we have
$$
\aligned
\sum_i E_{gst2}( \Gamma^I u_i, t)
&\lesssim
\int_{\RR^2} \big( |\del_t \Gamma^I u|^2 + \sum_a |\del_a \Gamma^I u|^2 \big) \, dx (0)
\\
&+
\sum_{|I_1| + |I_2| \leq N-1} \int_0^t \int_{\RR^2} \Big| R^{jkl}_i \Gamma^{I_1} u_j \Gamma^{I_2} Q_0 (u_k, u_l) \del_t \Gamma^I u_i \Big| \, dxdt'.
\endaligned
$$
We proceed by estimating the last term
$$
\aligned
&\sum_{|I_1| + |I_2| \leq N-1} \int_0^t \int_{\RR^2} \Big| R^{jkl}_i \Gamma^{I_1} u_j \Gamma^{I_2} Q_0 (u_k, u_l) \del_t \Gamma^I u_i \Big| \, dxdt'
\\
\lesssim
& \sum_{|I_1| + |I_2| + |I_3| \leq N-1} \int_0^t \int_{\RR^2} {1\over 1+t'} |\Gamma^{I_1} u| | \Gamma^{I_2} u| |\Gamma^{I_3} u| | \del_t \Gamma^I u| \, dxdt'
\\
\lesssim
&\sum_{|I_1| \leq N, |I_2| \leq N-3} \int_0^t {1\over 1+t'} \big\| \Gamma^{I_1} u \big\|_{L^2(\RR^2)}^2 \big\| \Gamma^{I_2} u \big\|_{L^\infty(\RR^2)}^2 \, dt'
\\
\lesssim
& (C_1 \eps)^4 \int_0^t (1+t')^{-5/4} \, dt'
\lesssim
(C_1 \eps)^4.
\endaligned
$$
Thus we easily get
\be
\sum_i E_{gst2}( \Gamma^I u_i, t)^{1/2}
\lesssim \eps + (C_1 \eps)^2,
\qquad
|I| \leq N-1.
\ee
Next, we look at the case of $|I| = N$, and we have
$$
\aligned
\sum_i E_{gst2}( \Gamma^I u_i, t)
&\lesssim
\int_{\RR^2} \big( |\del_t \Gamma^I u|^2 + \sum_a |\del_a \Gamma^I u|^2 \big) \, dx (0)
\\
&+
\sum_{|I_1| + |I_2| \leq N} \int_0^t \int_{\RR^2} \Big| R^{jkl}_i \Gamma^{I_1} u_j \Gamma^{I_2} Q_0 (u_k, u_l) \del_t \Gamma^I u_i \Big| \, dxdt'.
\endaligned
$$
We first divide the last estimate into two parts
$$
\aligned
&\sum_{|I_1| + |I_2| \leq N} \int_0^t \int_{\RR^2} \Big| R^{jkl}_i \Gamma^{I_1} u_j \Gamma^{I_2} Q_0 (u_k, u_l) \del_t \Gamma^I u_i \Big| \, dxdt'
\\
\leq
&\sum_{|I_1| + |I_2| \leq N, |I_2| \leq N-1} \int_0^t \int_{\RR^2} \Big| R^{jkl}_i \Gamma^{I_1} u_j \Gamma^{I_2} Q_0 (u_k, u_l) \del_t \Gamma^I u_i \Big| \, dxdt'
\\
+
&\sum_{ |I_2| = N} \int_0^t \int_{\RR^2} \Big| R^{jkl}_i u_j \Gamma^{I_2} Q_0 (u_k, u_l) \del_t \Gamma^I u_i \Big| \, dxdt'
:= P_1 + P_2.
\endaligned
$$
We notice that the estimate of $P_1$ part follows from what we have done for the case of $|I| \leq N-1$ with
$$
P_1 \lesssim (C_1 \eps)^4
$$
so we only need to estimate $J_2$.
We split the space domain into two parts for each fixed $t$
$$
S_{int} := \{ x: |x| \leq (1+t)^{9/8} \},
\qquad
S_{ext} := \{ x: |x| \geq (1+t)^{9/8} \},
$$
and the term $P_2$ can be divided into two parts
$$
\aligned
P_2
&=
P_2(S_{int}) + P_2(S_{ext}),
\\
P_2(S)
&:=
\sum_{ |I_2| = N} \int_0^t \int_{S} \Big| R^{jkl}_i u_j \Gamma^{I_2} Q_0 (u_k, u_l) \del_t \Gamma^I u_i \Big| \, dxdt'.
\endaligned
$$
In the region $S_{ext}$, it holds $(1+t)^{1/8} \lesssim \langle r-t \rangle$, and we do not even need the null structure to have the bound
$$
\aligned
P_2(S_{ext})
&\lesssim
\sum_{|I_1| \leq N, |I_2| \leq N-3} \int_0^t \big\| \Gamma^{I_1} u \big\|_{L^2(\RR^2)}^2 \big\| \Gamma^{I_2} u \big\|_{L^\infty(S_{ext})}^2 \, dt'
\\
&\lesssim
(C_1 \eps)^4 \int_0^t (1+t')^{-9/8} \, dt'
\lesssim (C_1 \eps)^4.
\endaligned
$$
Then we only need to estimate the term $P_2(S_{int})$.
By recalling the estimates for null forms in Lemma \ref{lem:null}, it holds
$$
\aligned
P_2(S_{int})
&\lesssim
\sum_{|I_1| = N, a} \int_0^t \int_{S_{int}} {1\over 1+t'} |u| \big| L_a \Gamma^{I_1} u \big| \big| \Gamma u \big| \big| \del_t \Gamma^I u \big| \, dxdt'
\\
&+ \sum_{|I_1| = N, \alpha} \int_0^t \int_{S_{int}} {1\over 1+t'} |u| \big| \del_\alpha \Gamma^{I_1} u \big| \big| \Gamma u \big| \big| \del_t \Gamma^I u \big| \, dxdt'
\\
&+ \sum_{\substack{|I_1| \leq N, |I_2| \leq N\\ |I_1| + |I_2| \leq N+2}} \int_0^t \int_{S_{int}} {1\over 1+t'} |u| \big| \Gamma^{I_1} u \big| \big| \Gamma^{I_2} u \big| \big| \del_t \Gamma^I u \big| \, dxdt'
:= P_{21} + P_{22} + P_{23}.
\endaligned
$$
It is not hard to see that the terms $P_{22}, P_{23}$ can be estimated in the same fashion as the term $P_1$ with
$$
P_{22} + P_{23} \lesssim (C_1 \eps)^4,
$$
so we only focus on the estimate of $P_{21}$.
We decompose the Lorentz boosts $L_a$ as
$$
L_a = x_a \del_t + t \del_a
= \big( x_a \del_t + r \del_a \big) + (t-r) \del_a.
$$
Hence we have
$$
\aligned
P_{21}
&\lesssim
\sum_{|I_1| = N, a} \int_0^t \int_{S_{int}} {1\over 1+t'} |u| \big| ( x_a \del_t + r \del_a) \Gamma^{I_1} u \big| \big| \Gamma u \big| \big| \del_t \Gamma^I u \big| \, dxdt'
\\
&+
\sum_{|I_1| = N, a} \int_0^t \int_{S_{int}} {1\over 1+t'} |u| \big| (t'-r) \del_a \Gamma^{I_1} u \big| \big| \Gamma u \big| \big| \del_t \Gamma^I u \big| \, dxdt'
\\
&\lesssim
\sum_{|I_1| = N, a} \int_0^t \int_{S_{int}} {\big| ( x_a \del_t + r \del_a) \Gamma^{I_1} u \big| \over (1+t') \langle r-t' \rangle } \langle r-t \rangle |u| \big| \Gamma u \big| \big| \del_t \Gamma^I u \big| \, dxdt'
\\
&+
\sum_{|I_1| = N, a} \int_0^t \int_{S_{int}} \langle r-t' \rangle {|u| \big| \Gamma u \big| \over 1+t'} \big| \del_a \Gamma^{I_1} u \big| \big| \del_t \Gamma^I u \big| \, dxdt'
\\
&\lesssim
\sum_{|I_1| = N, a} \int_0^t \Big\| {\big| ( x_a \del_t + r \del_a) \Gamma^{I_1} u \big| \over (1+t')^{9/8} \langle r-t' \rangle } \Big\|_{L^2(S_{int})} \big\| \del_t \Gamma^I u \big\|_{L^2(\RR^2)} (1+t')^{1/8} \big\| \langle r-t' \rangle |u| \big| \Gamma u \big| \big\|_{L^\infty(\RR^2)} \, dt'
\\
&+ \sum_{|I_1| = N, a} \int_0^t \big\| \del_a \Gamma^{I_1} u \big\|_{L^2(\RR^2)}^2 \Big\| \langle r-t' \rangle {|u| \big| \Gamma u \big| \over 1+t'} \Big\|_{L^\infty(\RR^2)} \, dt' =: P_{211} + P_{212}.
\endaligned
$$
Successively, we deduce
$$
P_{212}
\lesssim
\sum_{|I_1| = N} \int_0^t (C_1 \eps)^4 (1+t')^{-3/2} \, dt'
\lesssim
(C_1 \eps)^4.
$$
In order to estimate the term $P_{211}$, we note that in the region $S_{int}$, it holds $1/(1+t)^{9/8} \leq 1/ |x|$, and we thus have
$$
\aligned
P_{211}
&\lesssim
\sum_{|I_1| \leq N, a} \int_0^t \Big\| {\big| ( x_a \del_t + r \del_a) \Gamma^{I_1} u \big| \over r \langle r-t' \rangle } \Big\|_{L^2(\RR^2)} (C_1 \eps)^3 (1+t')^{-5/8} \, dt'
\\
&\lesssim
\sum_{|I_1| \leq N, a} (C_1 \eps)^3 \Big( \int_0^t \Big\| {\big| ( x_a \del_t + r \del_a) \Gamma^{I_1} u \big| \over r \langle r-t' \rangle } \Big\|_{L^2(\RR^2)}^2 \, dt' \Big)^{1/2} \Big( \int_0^t (1+t')^{-5/4} \, dt' \Big)^{1/2}
\\
&\lesssim
(C_1 \eps)^4.
\endaligned
$$
By gathering the estimates above, the proof is complete.
\end{proof}
\begin{proposition}[Improved estimates on the $L^2$ norm]\label{prop:L2}
Under the bootstrap assumptions in \eqref{eq:BA}, the following estimates hold
\bel{eq:L2Improved}
\aligned
\sum_i \big\| \Gamma^I u_i \big\|_{L^2(\RR^2)}
&\leq \eps \log(1+t) + (C_1 \eps)^2 \log(1+t),
\qquad
|I| \leq N-1,
\\
\sum_i \big\| \Gamma^I u_i \big\|_{L^2(\RR^2)}
&\leq \eps (1+t)^{1/4 + 2\delta} + (C_1 \eps)^2 (1+t)^{1/4 + 2\delta},
\qquad
|I| = N.
\endaligned
\ee
\end{proposition}
\begin{proof}
According to Lemma \ref{lem:linear}, in order to show the first improved estimate in \eqref{eq:L2Improved} it suffices to show for each $i$ that
$$
\Big\| \Gamma^{I} \Big( R^{jkl}_i u_j Q_0 (u_k, u_l) \Big) \Big\|_{L^2 (\RR^2) \cap L^1 (\RR^2)}
\lesssim (C_1 \eps)^2 (1+t)^{-9/8},
$$
for all $|I| \leq N-1$.
It is easy to see that
$$
\aligned
\Big\| \Gamma^I \big( R^{jkl}_i u_j Q_0 (u_k, u_l) \big) \Big\|_{L^2(\RR^2) \cap L^1(\RR^2)}
&\lesssim
{1\over 1 + t} \sum_{i, |I_1| \leq N } \big\| \Gamma^{I_1} u_i \big\|_{L^2(\RR^2)}^2 \sum_{|I_2| \leq N - 3} \| \Gamma^{I_2} u_i \|_{L^\infty(\RR^2)}
\\
&\lesssim (C_1 \eps)^2 t^{-5/4 + 4\delta}
\leq (C_1 \eps)^2 (1+t)^{-9/8},
\endaligned
$$
where we have used Lemma \ref{lem:null}.
Next we consider the case of $|I| = N$, and the proof in Proposition \ref{prop:E} indicates that it is easy to have
$$
\sum_{|I_1| + |I_2| \leq N, |I_2| \leq N-1} \Big\| \Big( R^{jkl}_i \Gamma^{I_1} u_j \Gamma^{I_2} Q_0 (u_k, u_l) \Big) \Big\|_{L^2 (\RR^2) \cap L^1 (\RR^2)}
\lesssim (C_1 \eps)^3 (1+t)^{-5/4},
$$
which is integrable.
Then in order to estimate the rest term
$$
\int_0^t \Big\| \Big( R^{jkl}_i u_j \Gamma^{I} Q_0 (u_k, u_l) \Big) \Big\|_{L^2 (\RR^2) \cap L^1 (\RR^2)} \, dt',
$$
we split the space region into two parts
$$
S_1 := \{ x: \langle t - r \rangle \leq (1+t)^{1/2} \},
\qquad
S_2 := \{ x: \langle t - r \rangle \geq (1+t)^{1/2} \}.
$$
We easily have in the region $S_2$ that
$$
\aligned
&\int_0^t \Big\| R^{jkl}_i u_j \Gamma^{I} Q_0 (u_k, u_l) \Big\|_{L^2 (S_2) \cap L^1 (S_2)} \, dt'
\\
\lesssim
&\sum_{|I_1| \leq N, \alpha} \int_0^t \| \del_\alpha \Gamma^{I_1} u \|^2_{L^2(\RR^2)} \| u \|_{L^\infty(S_2)} \, dt'
\\
\lesssim
&
(C_1 \eps)^3 \int_0^t (1+t')^{-3/4} \log(1+t') \, dt'
\lesssim (C_1 \eps)^3 (1+t)^{1/4} \log(1+t).
\endaligned
$$
We next estimate
$$
\aligned
&\int_0^t \Big\| R^{jkl}_i u_j \Gamma^{I} Q_0 (u_k, u_l) \Big\|_{L^2 (S_1) \cap L^1 (S_1)} \, dt'
\\
\lesssim
&\sum_{|I_1| \leq N} \int_0^t {|u| \over 1+t'} \big\| \Gamma^{I_1} u \big\|_{L^2(\RR^2)}^2 \, dt'
+
\sum_{|I_1| = N, a} \int_0^t \Big\| {|u| \over 1+t'} \big(|L_a \Gamma^{I_1} u| \big) \big| \Gamma u \big| \Big\|_{L^1 (S_1)} \, dt'
\\
&+
\sum_{|I_1| = N, \alpha} \int_0^t \Big\| {|u| \over 1+t'} \big( |\del_\alpha \Gamma^{I_1} u| \big) \big| \Gamma u \big| \Big\|_{L^1 (S_1)} \, dt'
\\
&\lesssim
(C_1 \eps)^3
+
\sum_{|I_1| = N, a} \int_0^t \Big\| {|u| \over 1+t'} \big(|(t'\del_a + r \del_t) \Gamma^{I_1} u| \big) \big| \Gamma u \big| \Big\|_{L^1 (S_1)} \, dt'
\\
&+
\sum_{|I_1| = N, \alpha} \int_0^t \Big\| {|u| \over 1+t'} \big( \langle r-t' \rangle |\del_\alpha \Gamma^{I_1} u| \big) \big| \Gamma u \big| \Big\|_{L^1 (S_1)} \, dt'.
\endaligned
$$
Since it holds $\langle r-t \rangle \leq (1+t)^{1/2}$ in the region $S_1$, we obtain
$$
\sum_{|I_1| = N, \alpha} \int_0^t \Big\| {|u| \over 1+t'} \langle r-t' \rangle |\del_\alpha \Gamma^{I_1} u| \big| \Gamma u \big| \Big\|_{L^1 (S_1)} \, dt'
\lesssim
(C_1 \eps)^3.
$$
We are left with the estimate
$$
\aligned
&\sum_{|I_1| = N, a} \int_0^t \Big\| {|u| \over 1+t'} \big(|(t'\del_a + r \del_t) \Gamma^{I_1} u| \big) \big| \Gamma u \big| \Big\|_{L^1 (S_1)} \, dt'
\\
=
&\sum_{|I_1| = N, a} \int_0^t \Big\| {|(t'\del_a + r \del_t) \Gamma^{I_1} u| \over \langle r-t' \rangle (1+t')} \langle r-t' \rangle |u| \big| \Gamma u \big| \Big\|_{L^1 (S_1)} \, dt'
\\
\lesssim
& \sum_{|I_1| = N, a} \int_0^t \Big\| {(t'\del_a + r \del_t) \Gamma^{I_1} u \over \langle r-t' \rangle r} \Big\|_{L^2(\RR^2)} (C_1 \eps) \log(1+t') \big\| \langle r-t' \rangle u \big\|_{L^\infty (S_1)} \, dt'
\\
\lesssim
& \sum_{|I_1| = N, a} \int_0^t \Big\| {(t'\del_a + r \del_t) \Gamma^{I_1} u \over \langle r-t' \rangle r} \Big\|_{L^2(\RR^2)} (C_1 \eps) (1+t')^{-1/4 + \delta} \, dt',
\endaligned
$$
where we have applied the fact that $r \leq 2(1+t)$ within the region $S_1$ in the second step, and the simple relation $\log^2(1+t) \lesssim (1+t)^\delta$ for $t, \delta > 0$ in the last step, and then Cauchy-Schwarz inequality gives us
$$
\sum_{|I_1| = N, a} \int_0^t \Big\| {|u| \over 1+t'} \big(|(t\del_a + r \del_t) \Gamma^{I_1} u| \big) \big| \Gamma u \big| \Big\|_{L^1 (S_1)} \, dt
\lesssim
(C_1 \eps)^3 (1+t)^{1/4 + \delta}.
$$
The proof is done.
\end{proof}
With Proposition \ref{prop:E} and Proposition \ref{prop:L2} prepared, we are now ready to give the proof of Theorem \ref{thm:main}.
\begin{proof}[Proof of Theorem \ref{thm:main}]
Recall the refined estimates in proposition \ref{prop:E} and \ref{prop:L2}, and if we choose $C_1$ very large (say three times larger than the implicit constant in $\lesssim$), and $\eps$ small enough such that $(C_1 \eps)^2 \leq \eps$, then we obtain the refined estimates in \eqref{eq:BA-refined}. Then we know $T>t_0$ cannot be finite, that is $T$ must be $+\infty$, and Theorem \ref{thm:main} is thus verified.
\end{proof}
\section*{Acknowledgments} This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program Grant agreement No 637653, project BLOC "Mathematical Study of Boundary Layers in Oceanic Motion". The author was partially supported by the Innovative Training Networks (ITN) grant 642768 entitled ModCompShock. The author would like to thank Allen Fang (Sorbonne University) for letting the author know the reference \cite{Liu} and helpful discussions, and to thank Siyuan Ma (Sorbonne University) for helpful discussions. The author also owes great thanks to Dongbing Zha (Donghua University), who informed the author the references \cite{Godin, Katayama, Zha}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,459 |
Q: How do I add multiple optional parameters in Express in same route Big fan of Stackoverflow and this is my first question ever on it! I'm working on a javascript express project and trying to figure out if I can do this under one get request. I'm having a hard time figuring this out
app.get("/test/:q1&:q2&:q3", (req, res) => {
console.log(req.params)
})
Basically, when the user looks up the following routes, I want the following result in req.params when these get requests are made. I want to keep it under one get request if possible.
TO CLARIFY. The above code CAN change to meet the functional requirements BELOW. Below CANNOT change
"http://localhost:3000/test/hello" --->
req.params will equal { q1: "hello" }
"http://localhost:3000/test/hello&world" --->
req.params will equal { q1: "hello", q2: "world" }
"http://localhost:3000/test/hello&world&foobar" --->
req.params will equal { q1: "hello", q2: "world", q3: "foobar" }
I tried using the "?" in the query but it seems to only work on new routes followed by a "/", not within the same route. I'm thinking I'll have to go with the quick and dirty solution of making 2 more get requests for each scenario. Currently when I try putting in "/test/hello&world" or "/test/hello", it will crash and will only work if I have all 3 parts filled.
A: Can you try this?
app.get('/test/:string', (req, res) => {
const str = req.params.string
const params = str.split('&')
.reduce((acc, curr, i) => ({ ...acc, [`q${i + 1}`]: curr }), {})
// params => { q1: 'hello', q2: 'world', q3: 'foobar' }
})
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,963 |
VeloNews News Mountain
Horgan-Kobelski favored for this weekend's MTB nats
Jeremy Horgan-Kobelski is making room in his closet for another stars and stripes jersey. Maybe two. Less than one month after taking the marathon cross-country national championship race, held at the Firecracker 50 race in Breckenridge, Colorado, Horgan-Kobelski comes into this weekend's USA Cycling mountain bike national championships in nearby Granby as the odds-on favorite to win.
By Fred Dreier
JHK at the Sand Creek International ProXCT race in Colorado last month.
Photo: Brad Kaminski
Jeremy Horgan-Kobelski is making room in his closet for another stars and stripes jersey.
Maybe two.
Less than one month after taking the marathon cross-country national championship race, held at the Firecracker 50 race in Breckenridge, Colorado, Horgan-Kobelski comes into this weekend's USA Cycling mountain bike national championships in nearby Granby as the odds-on favorite to win.
The question is whether JHK will finish the month of July owning all three endurance mountain bike titles: marathon, cross-country and short track. No one has achieved the trifecta since USA Cycling introduced the marathon championships in 2005.
"Yeah, I've been thinking that (winning three titles) is possible, but everything would have to come together perfectly," Horgan-Kobelski told VeloNews. "I felt really good at marathon nats, and I'd be really happy to feel that good again this weekend."
Horgan-Kobelski dominated the 50-mile long marathon championships, climbing away from the field on the opening lap and never looking back. The Coloradan crossed the line nearly 12 minutes ahead of defending champion Jeremiah Bishop (Monavie-Cannondale), and his finishing time of 3:28:16 set a new course record.
The win added an eighth stars-and-stripes jersey to Horgan-Kobelski's collection. He already owns four cross-country and three short-track titles.
It also marked Horgan-Kobelski's first national title since 2005.
But Horgan-Kobeslki knows this weekend's events, held at Sol Vista Basin ski area, pose a more serious challenge than the Firecracker 50. Saturday's cross-country and Sunday's short track will showcase the heaviest hitters in American mountain bike racing, including two-time defending cross-country champ Adam Craig (Giant), short track champ Ryan Trebon (Kona) and World Cup heavy hitter Todd Wells (Specialized), among others.
Horgan-Kobelski said Saturday's cross-country race takes the priority.
"I'm not really even thinking about the short track, I'm just going to see what happens," Horgan-Kobelski said. "The cross-country is the biggest race. It's the one that everyone is after."
Bishop and Horgan-Kobelski are the only two men to have won two jerseys in the same year since USA Cycling debuted the single-day national championship format in 2004. JHK took the cross-country and short track titles that year, when the championships were held at the high-altitude resort in Mammoth, California.
Bishop won the marathon and short track titles last year.
The main advantage Horgan-Kobelski owns on his competitors, however, is home-field advantage. Horgan-Kobelski and his wife, Subaru-Gary Fisher rider Heather Irmiger, own a condo in nearby Winter Park and spend the summers living at altitude and training in the Fraser Valley.
Horgan-Kobelski said he's done two reconnaissance rides at Sol Vista to check out the newly cut cross-country course.
"The trails aren't as fun as some of the ones elsewhere in the valley, but the course is going to determine a worthy national champion," Horgan-Kobelski said. "The climb is really demanding and the descent is freshly cut trail and super bumpy."
Horgan-Kobelski said he plans to race on his Gary Fisher Superfly 29er hardtail. The light bike with big wheels, he said, is the tool for the job on the climb-heavy course.
"This is my highest priority race of the year," Horgan-Kobelski said. "I started my career off as a 15-year-old racing at Winter Park. It would be cool to win this year."
Could signing Pauline Ferrand-Prévot convince Ineos Grenadiers to invest more in women's cycling?
Taylor Phinney: I thought Badlands would destroy me but I feel rejuvenated
Daniel Benson
Hannah Otto: 'Happy tears and a million stories to tell' from Leadville Trail 100 winner
Hannah Otto (Finchamp)
Who to watch at this year's Cape Epic
Betsy Welch | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,263 |
The Lady Plainsmen Are Headed To State Basketball
David Settle Published: March 3, 2017
For the first time in four years the Laramie Lady Plainsmen are going back to the 4A State Basketball tournament in Casper.
Lexi Pulley scored a career-high 24 points to lead Laramie past Cheyenne South, 60-34, in a first round, loser-out, 4A East Regional tournament game Thursday in Gillette. It will mark the first trip to state for the six seniors on Laramie's roster.
Laramie head coach Laura Pollard said her team performed consistently.
"I think they were confident; knew what they had to do. Had some usual turnovers, our own turnovers in the third quarter. Overall it was a good performance."
Kade Setright added 11 points, while Jennifer Aadland contributed 7 points and 14 rebounds.
The game was drawn out, as the two teams combined for 52 fouls and 66 free throw attempts.
Aadland said with all the fouls, "It was really tough because I felt like the clock was constantly getting stopped. We just had to keep playing through it because I felt like the game got dragged on."
Setright hit the only three-pointer of the game for the Lady Plainsmen. That gave LHS a 3-0 lead, and they never trailed. Laramie got out to a quick 15-4 lead after the first quarter. The Bison trimmed it to 19-11 in the second. The Lady Plainsmen scored six of the last seven points in the half and led 25-12 at the break.
Pollard said, "Mixing up our press a little bit, we knew handling the ball wasn't going to be easy for 'em."
Laramie had some turnover issues in the third quarter. Nine turnovers kept South in the game, as the Bison drew within 38-26 after three. The Lady Plainsmen pulled away in the fourth quarter, as they out-scored South 22-8 for the 26-point victory.
Laramie (16-6) shot 36 percent in the game. They out-rebounded the Bison 46-36 and capitalized on their 27 turnovers. The Lady Plainsmen went 25-37, 68 percent, at the charity stripe. That included 19-of-23, 83 percent, in the second half at the line.
Cheyenne South (2-20) was held to 22 percent from the field. They were led by Karli Noble with 13 points and Felizia Sena with 10. The Bison were only 10-29, 34 percent, at the foul line.
The Lady Plainsmen will move on in the 4A East Regional Championship bracket and face No. 2 seed and second-ranked Cheyenne East in the semifinals Friday at 4 p.m.
Pollard added, "Everything from here on out, since we reached our long-term goal, is gravy."
KOWB (AM 1290) will air the game live starting at 3:15 p.m. David Settle and VerDon Hoopes will call the action.
Regional Tournament Preview
Listen to Lady Plainsmen Basketball Live!
Filed Under: 4A East Regional Championships, Cheyenne South Bison, Lady Plainsmen Basketball, Laramie Basketball, Laramie High School Sports, Laramie Lady Plainsmen, Wyoming High School Basketball
Indoor Track Season Kicks off in Casper
Wyoming High School Boys Basketball Standings: Jan. 8, 2023
Wyoming High School Girls Basketball Standings: Jan. 8, 2023
Laramie Makes it Six Straight 4A Swim Titles in 2022 [VIDEOS] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,789 |
Turn left: From the intersection, the track leads down the ridgeline, keeping the houses and railway behind and the creek below on the left. The track winds down the rocky ridge to a rock surface with a view over the Nepean River.
Around 210m of this walk has gentle hills with occasional steps, whilst the remaining 170m has short steep hills.
Around 210m of this walk follows a rough track, where fallen trees and other obstacles are likely, whilst the remaining 170m follows a formed track, with some branches and other obstacles.
Around 210m of this walk has minimal directional signs, whilst the remaining 170m has directional signs at most intersection.
Around 330m of this walk has limited facilities (such as not all cliffs fenced), whilst the remaining 50m is close to useful facilities (such as fenced cliffs and seats).
A list of walks that share part of the track with the Lapstone Lookout walk.
A list of walks that start near the Lapstone Lookout walk.
http://new.wildwalks.com/wildwalks_custom/includes/walk_fire_danger.php?walkid=nsw-bmnp-ll Each park may have its own fire ban, this rating is only valid for today and is based on information from the RFS Please check the RFS Website for more information. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,886 |
Cuando hice el post de inspiración para crear un Working Space os prometí que os enseñaría el mío cuando lo tuviera. Lo tengo desde hace bastante, pero parece que ya he hecho las paces con la cámara para poder enseñaros algunas fotos medio decentes.
When I made this inspirational post to create your own working space I promised I'd show mine when I had it. I have had it for a while, but finally stopped fighting with my camera to take some decent pics to show.
Lo primero y más importante es buscar un lugar tranquilo y luminoso como esta esquinita que he hecho mía.
First of all you need a quite with a lot of light, just like this little corner I made mine.
As you can see I've got all I need. My plant, my notebooks, laptop, magazines and a quite poor moodboard that I'll complete for sure with thousands of ideas.
Yo trabajo más y mejor, así que lo recomiendo 100%.
Many people laughed at my working space but they finally got me lots of stuff to make it funnier I work harder and better so I totally recommend to have your own. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,071 |
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<h1>Retrieving from database</h1>
<p> Note: we are going to be adding all these code into <code>lib/slackir_web/channels/random_channel.ex</code> </p>
<p>Everytime when you refresh your page, you lose the message history and that is not ideal if you want to know what has been discussed.
We have all these stored messages in our database, and we want to retrieve it when someone logs in. Therefore, we will need to retrieve these messages and display them in the chat dialog.</p>
<h2>Perform a callback after join</h2>
<p>Go to the <strong>join</strong> function, and add the line just below <code>if authorized?(payload) do</code></p>
<p>We are going to send a message contaning our self <b>PID(Process Identifier)</b> as the first parameter. The second parameter is a message to run a function called <code>after_join</code>. This function is identified by an atom type.</p>
<div class="highlight"><pre class="highlight elixir"><code> <span class="k">def</span> <span class="n">join</span><span class="p">(</span><span class="sd">"</span><span class="s2">random:lobby"</span><span class="p">,</span> <span class="n">payload</span><span class="p">,</span> <span class="n">socket</span><span class="p">)</span> <span class="k">do</span>
<span class="k">if</span> <span class="n">authorized?</span><span class="p">(</span><span class="n">payload</span><span class="p">)</span> <span class="k">do</span>
<span class="n">send</span> <span class="n">self</span><span class="p">(),</span> <span class="ss">:after_join</span> <span class="c1">#<-- Add this line</span>
<span class="p">{</span><span class="ss">:ok</span><span class="p">,</span> <span class="n">socket</span><span class="p">}</span>
<span class="k">else</span>
<span class="p">{</span><span class="ss">:error</span><span class="p">,</span> <span class="p">%{</span><span class="ss">reason:</span> <span class="sd">"</span><span class="s2">unauthorized"</span><span class="p">}}</span>
<span class="k">end</span>
<span class="k">end</span>
</code></pre></div>
<h4>What does send do?</h4>
<p><code>send</code> sends a message, <code>:after_join</code>, with it's PID generated by the function ,<code>self()</code>. We want to be able to run the after join task asynchronously. Lets look at how we should receive the <code>:after_join</code> message</p>
<h1>Receiving a message in a callback</h1>
<p>We now know that a message identified by the atom <code>:after_join</code> is being sent.
Phoenix provides us with a function called <strong>handle_info</strong> to be able to receive that message and do something with it.
Insert the following code after the <em>join</em> function </p>
<div class="highlight"><pre class="highlight elixir"><code> <span class="k">def</span> <span class="n">handle_info</span><span class="p">(</span><span class="ss">:after_join</span><span class="p">,</span> <span class="n">socket</span><span class="p">)</span> <span class="k">do</span>
<span class="n">messages</span> <span class="o">=</span>
<span class="no">Slackir</span><span class="o">.</span><span class="no">Conversations</span><span class="o">.</span><span class="n">list_messages</span><span class="p">()</span>
<span class="o">|></span> <span class="no">Enum</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">fn</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">-></span> <span class="p">%{</span><span class="ss">message:</span> <span class="n">m</span><span class="o">.</span><span class="n">message</span><span class="p">,</span> <span class="ss">name:</span> <span class="n">m</span><span class="o">.</span><span class="n">name</span><span class="p">}</span> <span class="k">end</span><span class="p">)</span>
<span class="n">push</span> <span class="n">socket</span><span class="p">,</span> <span class="sd">"</span><span class="s2">messages_history"</span><span class="p">,</span> <span class="p">%{</span><span class="ss">messages:</span> <span class="n">messages</span><span class="p">}</span>
<span class="p">{</span><span class="ss">:noreply</span><span class="p">,</span> <span class="n">socket</span><span class="p">}</span>
<span class="k">end</span>
</code></pre></div>
<h1>Retrieving all messages from our database</h1>
<p>Phew, that looked like a lot is going on in the <code>handle_info</code> function. Lets breakdown the code to understand what is going on.</p>
<p>
As we have seen previously, context <code>Slackir.Conversations</code> provides us with many useful functions to communicate with the database. Thanks to <code>list_messages()</code> function we can retrieve all the messages from our database.
</p>
<p>
We then, pipe the retrieved <b>messages</b> into a <code>map()</code> function provided by the <code>Enum</code> module.
<code>map</code> applies this anonymous function <code>&(%{message: &1.message, name: &1.name})</code> onto every element in <b>messages</b>
</p>
<p><em>Note: Please ask your mentor to explain how <code>Enum.map</code> works if you are unsure about it :)</em></p>
<p>
There is of course the <b>&</b> operator that is used as a shorthand to convert the expression into a function. We are trying to format the messages that we have retrieved into a proper structure.
</p>
<h1>Passing the messages back to the browser</h1>
<p>Finally, we want to push our list of messages to the socket via the <strong>push</strong> function.</p>
<p>We are using <em>push</em> instead of <em>broadcast</em>, because we only want to display it to the current user that has just joined the channel</P>
<h1>Render the messages in the browser</h1>
<p>Add the below code to the <code>assets/js/socket.js</code> file</p>
<div class="highlight"><pre class="highlight javascript"><code> <span class="nx">channel</span><span class="p">.</span><span class="nx">on</span><span class="p">(</span><span class="s1">'messages_history'</span><span class="p">,</span> <span class="nx">messages</span> <span class="o">=></span> <span class="p">{</span>
<span class="kd">let</span> <span class="nx">messages_list</span> <span class="o">=</span> <span class="nx">messages</span><span class="p">[</span><span class="s2">"messages"</span><span class="p">];</span>
<span class="nx">messages_list</span><span class="p">.</span><span class="nx">forEach</span><span class="p">(</span> <span class="kd">function</span><span class="p">(</span><span class="nx">msg</span><span class="p">)</span> <span class="p">{</span>
<span class="nx">list</span><span class="p">.</span><span class="nx">append</span><span class="p">(</span><span class="s2">`<b></span><span class="p">${</span><span class="nx">msg</span><span class="p">[</span><span class="s2">"name"</span><span class="p">]</span> <span class="o">||</span> <span class="s1">'Anonymous'</span><span class="p">}</span><span class="s2">:</b> </span><span class="p">${</span><span class="nx">msg</span><span class="p">[</span><span class="s2">"message"</span><span class="p">]}</span><span class="s2"><br>`</span><span class="p">);</span>
<span class="nx">list</span><span class="p">.</span><span class="nx">prop</span><span class="p">({</span><span class="na">scrollTop</span><span class="p">:</span> <span class="nx">list</span><span class="p">.</span><span class="nx">prop</span><span class="p">(</span><span class="s2">"scrollHeight"</span><span class="p">)});</span>
<span class="p">});</span>
<span class="p">});</span>
</code></pre></div> <p>This is the final step, really. We need display the messages in our browser by iterating through our list
You should have the history of messages displayed to you when you join the channel. Try opening multiple tabs connecting to http://127.0.0.1/4000
<p>You will be able to see that all the messages gets rendered everytime you join</p>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,874 |
The leaves of Autumn have begun to turn, as have the pages of the Autumn edition of 2600, which is hitting the stands worldwide as you read this. One can be sent to your mailbox easily and cheaply by subscribing through our online store.
But it doesn't end there. You can wander on down to your local bookstore or newsstand and pick up the latest issue if we're carried there. And if we're not, you can let us know and we'll take steps to fix it.
And, of course, our digital options abound. You can subscribe via Kindle in either the U.S. or the U.K. , while obtaining individual issues for other areas (see our guide here ). You can have issues sent to your phone or computer via Google Play . You can also subscribe through the Barnes and Noble Nook . And if all of this makes you hunger for more, you can peruse our digital digests from years past. | {
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Pieter Gerardus van Os ( - ) est un peintre et graveur néerlandais, membre de la célèbre famille d'artistes Van Os.
Biographie
Né à La Haye, fils de Jan van Os, peintre de fleurs et poète, il y étudie avec son père et de 1794 à 1795 à l'Académie royale des beaux-arts de La Haye. Pendant cette période, il copie des œuvres de Paulus Potter et Charles Dujardin. Il aime particulièrement peindre les animaux et a fait une si excellente copie d'une des œuvres de Potter Jeune taureau, qu'elle est achetée par Guillaume V d'Orange-Nassau.
Après avoir terminé sa formation, il part pour Amsterdam, où il vit principalement en peignant des miniatures de portraits plutôt médiocres, et en donnant des cours de dessin. Il a deux fils, George Johannes Jacobus van Os et Pieter Frederik van Os (1808–1892) devenu peintre, et qui a eu pour élève Anton Mauve.
Chevalier dans l'ordre du Dutch Lion, il est soldat et capitaine des volontaires (1813-1814) au siège de Naarden.
Il devient un membre de quatrième classe de l'Académie royale en 1820.
En il se brise le bras droit et apprend à peindre de la main gauche.
Il meurt à La Haye en 1839.
Œuvre
Vers 1805, il commence à se consacrer à la production de peintures de paysages où figure son sujet favori, le bétail. Il est encore fortement influencé par les maîtres hollandais du . En 1808, son Paysage vallonné avec du bétail remporta le prix Louis Bonaparte du meilleur paysage lors de la première exposition publique d'art contemporain néerlandais à Amsterdam.
Paysage avec bétail (1806), huile sur panneau, , Rijksmuseum, Amsterdam
En 1813 et 1814, il s'entraîne en tant que capitaine de volontaires et s'essaye à des sujets militaires et un tableau de ce genre lui a été acheté pour son palais de Saint-Pétersbourg, par .
Avant-poste cosaque en 1813 (1813-1815), huile sur panneau, , Rijksmuseum, Amsterdam
L'Arrivée des cosaques à Utrecht en 1813 (1816), huile sur toile, , Centraal Museum, Utrecht
La Traversée du Karnemelksloot près de Naarden () (1814-1815), huile sur toile, , Rijksmuseum, Amsterdam
Vue lointaine des prés à 's-Graveland (1817), huile sur toile, , Rijksmuseum, Amsterdam
Le Canal à 's-Graveland (1818), huile sur toile, , Rijksmuseum, Amsterdam
Notes
Bibliographie
Liens externes
Œuvres de Pieter Van Os au Musée de Nouvelle-Zélande Te Papa Tongarewa
Naissance en octobre 1776
Décès en mars 1839
Peintre néerlandais du XIXe siècle
Pages avec des traductions non relues
Décès à 62 ans | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,441 |
«На исходе дня» ( хи но хатэ) — японский чёрно-белый фильм, военная драма 1954 года выпуска. Постановка осуществлена известным представителем независимого кино Японии режиссёром Сацуо Ямамото. Экранизация одноимённого романа , впервые изданного в 1948 году.
Сюжет
В апреле 1945 года японские войска были на грани развала. В находящемся на острове Лусон подразделении растёт число дезертиров. Среди убежавших в джунгли оказался и военный хирург Ханада, захвативший с собой влюблённую в него местную девушку Тиэ. Вместе со своим подчинённым, сержантом Такасиро, поручик Удзи начинает преследование беглецов. Когда они находят их убежище, Удзи овладевает девушкой в отсутствие Ханады. На следующий день Удзи поведал об этом военврачу. Ханада избивает Тиэ, но сразу после этого случая они снова скрываются вдвоем. Поручик Удзи уже не хочет далее преследовать беглецов, однако сержант Такасиро настроен на их поимку. Когда двое мужчин, беспрестанно споря друг с другом, вышли на берег озера, находящегося в окружении девственного леса, внезапно появляются Ханада и Тиэ с оружием в руках. Ханада стреляет, но в его пистолете закончились пули. Ответным выстрелом сержант Такасиро убивает Ханаду. Тиэ расстреливает своих преследователей, но умирающий Такасиро успевает выпустить пулю в неё.
В ролях
Кодзи Цурута — поручик Удзи
Эйдзи Окада — Ханада, военврач
Юкико Симадзаки — Тиэ
Ясуми Хара — сержант Такасиро
Ютака Фукуда — сержант Мацуо
Дзюнкити Оримото — солдат Ямамура
Сусуму Мидзусима — солдат Фудзита
Киндзо Син — солдат Симидзу
Ёси Като — командующий
Такаси Канда — адъютант Одзава
Тайдзи Тонояма — Саката
Исао Кимура — солдат
Нобуо Канэко — солдат в заброшенном доме
Тосио Такахара — парнишка
Харуэ Тонэ — Суми
Премьеры
— национальная премьера фильма состоялась 20 февраля 1954 года.
Примечания
Ссылки
Литература
Ивасаки, Акира. «Современное японское кино», 1958, (перевод с японского 1962, Переводчики: Владимир Гривнин, Л. Левин), — М.: Искусство, 1962, С.524.
Акира Ивасаки. «История японского кино», 1961 (перевод с японского 1966, Переводчики: Владимир Гривнин, Л. Левин и Б. Раскин). — М. : Искусство, 1966, С.320.
Фильмы Японии 1954 года
Чёрно-белые фильмы Японии
Военные фильмы Японии
Фильмы-драмы Японии
Экранизации литературных произведений
Фильмы Сацуо Ямамото | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,466 |
require 'jenkins_api_client'
require 'thor'
require 'nokogiri'
require 'parallel'
require 'leeroy_jenkins/version'
require 'leeroy_jenkins/cli'
require 'leeroy_jenkins/result'
require 'leeroy_jenkins/job_finder'
require 'leeroy_jenkins/jenkins_client_builder'
require 'leeroy_jenkins/job_updater'
require 'leeroy_jenkins/job_backupper'
require 'leeroy_jenkins/job_restorer'
module LeeroyJenkins
def self.invalid_xml_document?(raw_xml)
begin
Nokogiri::XML(raw_xml, &:strict)
rescue Nokogiri::XML::SyntaxError => e
return e
end
false
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 96 |
La est une gare ferroviaire située à Higashimurayama, à Tokyo au Japon. La gare est exploitée par la compagnie Seibu.
Situation ferroviaire
La gare est située au début de la ligne Seibu Yamaguchi et la fin de la ligne Seibu Tamako.
Historique
La gare est inaugurée le .
Service des voyageurs
Accueil
La gare dispose d'un bâtiment voyageurs, avec guichets, ouvert tous les jours.
Desserte
Ligne Seibu Tamako :
voies 1 et 2 : direction Kokubunji
Ligne Seibu Yamaguchi :
voie 3 : direction Seibu-Kyūjō-mae
Notes et références
Voir aussi
Lien externe
La gare sur le site de la Seibu
Gare à Tokyo
Gare Seibu
Gare mise en service en 1936 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,882 |
Q: Create Sqlite database in iOS I am unable to create sqlite database in my documents directory.
Here is the code:
NSString *fileDir;
NSArray *dirPaths;
//Get the documents directory
dirPaths = NSSearchPathForDirectoriesInDomains(NSDemoApplicationDirectory, NSUserDomainMask, YES);
fileDir = [dirPaths objectAtIndex:0];
// Build the database path
databasePath = [[NSString alloc]initWithString:[fileDir stringByAppendingPathComponent:@"student.sql"]];
NSFileManager *fileMgr = [NSFileManager defaultManager];
if([fileMgr fileExistsAtPath:databasePath] == NO)
{
const char *dbPath = [databasePath UTF8String];
if (sqlite3_open(dbPath, &database) == SQLITE_OK)
{
char *errMsg;
const char *sql_stmt = "CREATE TABLE IF NOT EXISTS CONTACTS(ID INTEGER PRIMARY KEY , NAME TEXT, ADDRESS TEXT, MOBILE INTEGER)";
if (sqlite3_exec(database, sql_stmt, NULL, NULL, &errMsg) != SQLITE_OK) {
_status.text = @"Failed to create table";
}
sqlite3_close(database);
}
else
{
_status.text = @"Failed to open/create database";
}
}
I have debug the code and found that the compiler is not going under this condition.
sqlite3_open(dbPath, &database) == SQLITE_OK
I don't know what i am doing wrong.
Any help will be appreciated...
Thanks,
A: Check you code to get dirPath, you are getting right path or not, I used following way in my code and its working for me :
dirPaths = NSSearchPathForDirectoriesInDomains(NSDocumentDirectory, NSUserDomainMask, YES);
// dirPaths = NSSearchPathForDirectoriesInDomains(NSDemoApplicationDirectory, NSUserDomainMask,YES);
fileDir = dirPaths[0];
// Build the database path
databasePath = [[NSString alloc]initWithString:[fileDir stringByAppendingPathComponent:@"student.sqlite"]];
A: this is how i managed it.
DataBaseAccess.m
static sqlite3 *database=nil;
-(id)init
{
if(self=[super init])
{
self.user_data=@"user_data.db";
}
return self;
}
-(void)createUserDataDatabase
{
BOOL success;
NSFileManager *fileManager = [NSFileManager defaultManager];
NSError *error;
NSArray *paths = NSSearchPathForDirectoriesInDomains(NSDocumentDirectory, NSUserDomainMask, YES);
NSString *documentsDirectory = [paths objectAtIndex:0];
NSString *writableDBPath = [documentsDirectory stringByAppendingPathComponent:user_data];
success = [fileManager fileExistsAtPath:writableDBPath];
if (success) return;
// construct database from external ud.sql
NSString *filePath=[[NSBundle mainBundle]pathForResource:@"ud" ofType:@"sql"];
NSString *sqlStatement=[NSString stringWithContentsOfFile:filePath encoding:NSUTF8StringEncoding error:&error];
if(sqlite3_open([writableDBPath UTF8String], &database)==SQLITE_OK)
{
sqlite3_exec(database, [sqlStatement UTF8String], NULL, NULL, NULL);
sqlite3_close(database);
}
}
Your external sql file must contain the sql queries:
CREATE TABLE quantityInSubCountries (
refID INT,
quantity INT
);
CREATE TABLE quantityInSubRegions (
refID INT,
quantity INT
); ....
Hope it will help.
A: This is common methods used for Database
-(void)updateTable:(NSString *)tableName setname:(NSString *)Name setImagePath:(NSString *)imagePath whereID:(NSInteger)rid{
NSString *sqlString=[NSString stringWithFormat:@"update %@ set name='%@' where id=%ld",tableName,Name,rid];
char *error;
if (sqlite3_exec(db, [sqlString UTF8String], NULL, NULL, &error)!=SQLITE_OK) {
[self closeDatabase];
NSLog(@"Faield to update");
}
else{
NSLog(@"update successfully");
}
}
-(void)deleteFrom:(NSString *)tablename whereName:(NSInteger )rid {
NSString *sqlString=[NSString stringWithFormat:@"delete from %@ where id=%ld",tablename,(long)rid];
char *error;
if (sqlite3_exec(db, [sqlString UTF8String], NULL, NULL, &error)!=SQLITE_OK) {
[self closeDatabase];
NSLog(@"faield to Delete");
}
else{
NSLog(@"Deleted successfully");
}
}
-(void)insertInTable:(NSString *)tableName withName:(NSString *)name withImagePath:(NSString *)imagePath
{
NSString *sqlString=[NSString stringWithFormat:@"insert into %@(name,path)values('%@','%@')",tableName,name,imagePath];
char *error;
if (sqlite3_exec(db, [sqlString UTF8String], NULL, NULL, &error)!=SQLITE_OK) {
[self closeDatabase];
NSLog(@"Failed to insert");
}
else{
NSLog(@"Inserted succesfully");
}
}
-(NSString *)path
{
NSArray *paths=NSSearchPathForDirectoriesInDomains(NSDocumentDirectory, NSUserDomainMask, YES);
NSString *documentDir=[paths objectAtIndex:0];
return [documentDir stringByAppendingPathComponent:@"Storage.db"];
}
-(void)open{
if(sqlite3_open([[self path] UTF8String], &db)!=SQLITE_OK)
{
sqlite3_close(db);
NSLog(@"your database table has been crash");
}
else{
NSLog(@"Database open successfully");
}
}
-(void)closeDatabase
{
sqlite3_close(db);
NSLog(@"Database closed");
}
-(void)copyFileToDocumentPath:(NSString *)fileName withExtension:(NSString *)ext{
NSString *filePath=[self path];
NSFileManager *fileManager=[NSFileManager defaultManager];
if (![fileManager fileExistsAtPath:filePath]) {
NSString *pathToFileInBundle=[[NSBundle mainBundle] pathForResource:fileName ofType:ext];
NSError *err=nil;
BOOL suc=[fileManager copyItemAtPath:pathToFileInBundle toPath:filePath error:&err];
if (suc) {
NSLog(@"file copied successfully");
}
else
{
NSLog(@"faield to copied");
}
}
else
{
NSLog(@"File allready present");
}
}
-(NSMutableArray *)AllRowFromTableName:(NSString *)tableName{
NSMutableArray *array=[[NSMutableArray alloc] init];
NSString *sqlString=[NSString stringWithFormat:@"select *from %@",tableName];
sqlite3_stmt *statement;
if (sqlite3_prepare_v2(db, [sqlString UTF8String], -1, &statement, nil)==SQLITE_OK) {
while (sqlite3_step(statement)==SQLITE_ROW) {
Database *tempDatabase=[[Database alloc] init];
tempDatabase.Hid=sqlite3_column_int(statement, 0);
tempDatabase.Hname =[[NSString alloc] initWithUTF8String:(char *)sqlite3_column_text(statement, 1)];
tempDatabase.Hpath=[[NSString alloc] initWithUTF8String:(char *)sqlite3_column_text(statement, 2)];
[array addObject:tempDatabase];
}
}
return array;
}
-(void)test
{
//Pet photos
NSString *sqlString=[NSString stringWithFormat:@"create table if not exists StorageTable(id integer primary key autoincrement,name text,path text)"];
char *error;
if (sqlite3_exec(db, [sqlString UTF8String], NULL, NULL, &error)!=SQLITE_OK) {
[self closeDatabase];
NSLog(@"Faield to blanck 1 %s",error);
}
else{
NSLog(@"Test On StorageTable Database successfully");
}
}
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} | 5,206 |
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"redpajama_set_name": "RedPajamaC4"
} | 3,122 |
{"url":"http:\/\/mathoverflow.net\/revisions\/51984\/list","text":"MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06\/24\/2013) at approximately 9:00 PM Eastern time (UTC-4).\n\n5 added 212 characters in body\n\nChapter 2 of Harer's paper \"The cohomology of the moduli space of curves\" is good.\n\nThe point is that there is the \"arc complex\", a simplicial complex which gives a suitable triangulation of Teichm\u00fcller space which is compatible with the action of the mapping class group. The simplices of the simplicial complex correspond to \"arc systems\", which are certain collections of curves $C_i$ on an oriented surface $\\Sigma$ with boundary $\\partial \\Sigma$ that begin and end on the boundary, and which decompose the surface into discs.\n\nYou can define a ribbon graph to be a graph together with a cyclic ordering of the edges around each vertex. To get a ribbon graph corresponding to an arc system, take the dual graph of the arc system, that is take the graph whose vertices are the components of $\\Sigma \\setminus (\\bigcup_i C_i)$, and the edges are as you'd guess (I can't think of a nice terse way to put this into words, but it's easy to explain with a picture...), and then give the edges around each vertex a cyclic orientation ordering via the orientation of the surface.\n\nBeware that Harer's paper doesn't mention ribbon graphs, nor do some of the other standard references. This had confused me for a while (an embarrassingly long time, in fact) when I was trying to read about this in the literature. But as I said, just know that the ribbon graph picture is just the dual picture to the arc system picture.\n\nP.S. Some more references here.\n\nChapter 2 of Harer's paper \"The cohomology of the moduli space of curves\" is good.\n\nThe point is that there is the \"arc complex\", a simplicial complex which gives a suitable triangulation of Teichm\u00fcller space which is compatible with the action of the mapping class group. The simplices of the simplicial complex correspond to \"arc systems\", which are certain collections of curves $C_i$ on an oriented surface $\\Sigma$ with boundary $\\partial \\Sigma$ that begin and end on the boundary, and which decompose the surface into discs. To get a ribbon graph corresponding to an arc system, take the dual graph of the arc system, that is take the graph whose vertices are the components of $\\Sigma \\setminus (\\bigcup_i C_i)$, and the edges are as you'd guess, and then give the edges around each vertex a cyclic orientation via the orientation of the surface.\n\nBeware that Harer's paper doesn't mention ribbon graphs, nor do some of the other standard references. This had confused me for a while (an embarrassingly long time, in fact) when I was trying to read about this in the literature. But as I said, just know that the ribbon graph picture is just the dual picture to the arc system picture.\n\nP.S. Some more references here.\n\nThe point is that there is the \"arc complex\", a simplicial complex which gives a triangulation of Teichm\u00fcller space. The simplices of the simplicial complex correspond to \"arc systems\", which are certain collections of curves $C_i$ on an oriented surface $\\Sigma$ with boundary $\\partial \\Sigma$ that begin and end on the boundary. To get a ribbon graph corresponding to an arc system, take the dual graph of the arc system, that is take the graph whose vertices are the components of $\\Sigma \\setminus (\\bigcup_i C_i)$, and the edges are as you'd guess, and then give the edges around each vertex a cyclic orientation via the orientation of the surface.","date":"2013-06-20 02:47:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7312979102134705, \"perplexity\": 210.140399207312}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368710115542\/warc\/CC-MAIN-20130516131515-00092-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
Life in Bronze: Lawrence M. Ludtke, Sculptor
Amy L. Bacon (Author), H. Ross Perot (Introduction author) and James R. Reynolds (Introduction author)
A disciple of Classical sculpture in a time of pervasive abstract modernism, Lawrence M. Ludtke (1929–2007) of Houston imbued his creations with a sense of movement and realism through his attention to detail, anatomy, and proportion.
As a skilled athlete who played professional baseball for the Brooklyn Dodgers organization, Ludtke brought to his art a fascination with musculature and motion that empowered him to capture the living essence of his subjects. As author Amy L. Bacon shows in this sensitive biography, Ludtke's gentle humanity and sensitivity shines through his work; his sculpture truly projects character, purpose, and personality.
Ludtke, a Fellow in the National Sculpture Society (US) and a Corresponding Member of the Royal Academy of British Sculptors, became well-known for his portrait and figurative art. His works grace the halls and grounds of the United States Air Force Academy, Johns Hopkins Medical School, Rice University, Texas A&M University, CIA headquarters, the National Cowboy Hall of Fame, the Pentagon, Lyndon Baines Johnson Presidential Library, and the National Battlefield Park at Gettysburg, Pennsylvania. He has also created significant liturgical art, most notably a life-size Pieta for St. Mary's Seminary in Houston and a Christ and Child for Travis Park Methodist Church in San Antonio.
Based on personal interviews with the artist as well as his family, friends, colleagues, and patrons such as H. Ross Perot, Life in Bronze: Lawrence M. Ludtke, Sculptor places Ludtke's art within the context of the American figurative art tradition. The author explains how Ludtke was influenced by Italian-born Pompeo Coppini, whose monumental art has especially marked Texas and whose clay Ludtke inherited and used as his own favored modeling medium. Bacon meticulously details how Ludtke's research into the lives and careers of his subjects was married to his attention to technique and talent. His own life story figures crucially in the creation of those character studies his sculptures so beautifully represent.
As a skilled athlete who played professional baseball for the Brooklyn Dodgers organization, Ludtke brought to his art a fascination with musculature and motion that empowered him to capture the living essence of his subjects. As author… (more) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,429 |
{"url":"https:\/\/www.physicsforums.com\/threads\/maximum-minimum-problem.147657\/","text":"# Maximum\/Minimum Problem\n\n1. Dec 11, 2006\n\n### skies222\n\nSorry to post another problem, but hey, the more the marrier, right?\n\n1. The problem statement, all variables and given\/known data\n\nmaxima\/minima problem\n2)a cylinder, including top and bottom, is made from material which costs \"x\" dollars per square inch. Suppose there is an additional cost of fabrication given by \"n\" dollars per inch of the circumfrence of the top and bottom of the can. find an algerbraic equation whose solution would be he radius ,r, of the can of given volume, V, whose cost is a minimum.\n* Do not have to solve equation, just find a algerbraic solution\n\n2. Relevant equations\n\n3. The attempt at a solution\n\nI tried this one, and I think I may be overthinking it. Would the equation just consists of the volume of the cone (equalling the money for the material) plus the additional amount of the circumfrence of top and bottom (so two times the circumfrence) for the entire cost of the can?\n\nOnce again, thanks again!\n\n2. Dec 11, 2006\n\n### slearch\n\nBy the way it's worded it looks like your cost will depend on the total surface area of the closed cylinder - that is twice the area of the circles and the area of the rest. That part about the cost being n times the circumfrence also seems ambiguous - I can't really tell if they want the total circumfrence of the top and bottom or just the circumfrence of either.\n\n3. Dec 11, 2006\n\n### skies222\n\nI get what you mean, and agree about the surface area. It seems I was underestimating the problem before, haha. The circumference aspect asks for both, the top and bottom, (so 2 times that). I've been messing around with it and I know that once you differntiate the problem, you should just be left with pi,r,n and x in the answer since V is a constant and the problem doesn't want anything else in the answer.\n\n4. Dec 11, 2006\n\n### chaoseverlasting\n\nHere, the cost of material: $$((2\\ pi)r^2+ (2 \\pi)rl)x$$\ncost of fabrication: $$(2 \\pi r)n$$\nHence, the total cost is the sum of the above;\nIt is also given that the volume of the cylinder is constant, so\n$$V=\\pi r^2 l$$\nfrom which you get $$l= \\frac{V}{\\pi r^2}$$\nsubstituting the value of l in the initial equation gives you your equation.\n\nYou can differentiate this equation w.r.t r and get your solution.\n\nLast edited: Dec 11, 2006\nKnow someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook","date":"2017-06-24 17:54:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7596288323402405, \"perplexity\": 743.628294665674}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-26\/segments\/1498128320270.12\/warc\/CC-MAIN-20170624170517-20170624190517-00608.warc.gz\"}"} | null | null |
\section{Introduction}
\vspace{-2.5mm}
Charged particles accelerated in the vicinity of a rapidly rotating neutron star, or pulsar,
flow out into the interstellar medium and encounter the supernova ejecta from the pulsar's birth
event and form a shock. The shock may further
enhance the acceleration of the particles which can then attain relativistic speeds.
This interaction between the accelerated charged particles and the
surrounding medium produces a pulsar wind nebula (PWN). PWNe are often
observable at wavelengths from the radio through the $\gamma$-ray. Around the youngest, most
energetic pulsars, the radio emitting regions of these nebulae are rather
amorphous, whereas the X-ray emitting regions can be highly structured
\cite{ref1}. The high spatial resolution of the {\it Chandra} X-ray
Observatory has made it possible to resolve the structures of PWNe.
The presence of a PWN can also be inferred spectrally. For instance, a
non-thermal component is often seen in the ASCA and INTEGRAL
observations of pulsars, such as PSR~B1509-58 and PSR~B1046-58.
This seems to suggest that PWNe are a common phenomenon for all
energetic pulsars \cite{ref2}.
\begin{table*}[t]
\centering
\begin{tabular}{l|cccccccc}\hline
Pulsar & PWN & P (ms) & B ($10^{12}$G) & t (kyr) &
$\rm \dot{E}$ ($10^{36}$erg/s) & d (kpc) & $\rm F_\gamma$($10^{-11}$erg/s) \\ \hline
J2021+3651 & G75.2+0.1 & 104 & 3.2 & 17 & 3.4 & 12 & 7 \\
J2229+6114 & G106.6+2.9 & 51.6 & 2 & 10.5 & 22 & 3 & 4 \\
J0205+6449 & 3C58 & 65 & 3.6 & 5 & 27 & 3.2 & 0.8 \\ \hline
\end{tabular}
\caption{Physical parameters of observed pulsars and their PWNe. }
\label{t1}
\vspace{-4mm}
\end{table*}
It was widely believed that PWNe are potential sources of
VHE $\gamma$-ray emission. The emission probably arises from inverse Compton (IC)
scattering of low-energy photons by the relativistic electrons, while the X-ray
emission is associated with the synchrotron radiation from the same population of electrons.
The best example of a PWN is the Crab Nebula, which is an established source of
pulsed $\gamma$-ray emission up to a few GeV detected by EGRET, as well
as a source of steady TeV $\gamma$ rays observed by a
number of ground-based Cherenkov detectors and recently with the VERITAS
array of four imaging atmospheric Cherenkov telescopes \cite{ref3}. The
TeV emission is thought to originate at the base of its PWN. The H.E.S.S. detector,
located in the southern hemisphere, discovered a number of previously unknown
$\gamma$-ray sources in the VHE domain above 100 GeV. A total of five of these
new sources (PSR~B1509-58, Vela~X, ``Kookaburra'', SNR~G0.9+0.1,
PSR~B1823-13) are apparently associated with PWNe. Such associations
rest on a positional and morphological match of the VHE $\gamma$-ray source to a
known PWN at lower energies. It is worth noting that in
all cases the pulsar is significantly offset from the center of the VHE $\gamma$-ray
source. This offset could be attributed to the interaction between the PWN and the
SNR ejecta \cite{ref5}.
A detailed survey of the inner part of the Galactic Plane at VHE
$\gamma$-ray energies has been carried out with H.E.S.S. Fourteen previously unknown,
extended sources were detected with high significance \cite{ref6}.
Some of these sources have fairly well-established counterparts at longer wavelengths,
based exclusively on positional coincidence, but others have none at all. A number
of models have been proposed regarding the nature of these unidentified VHE $\gamma$-ray
sources. At present, PWNe and shell-type SNRs are considered the most plausible
counterparts for the remaining unidentified VHE $\gamma$-ray sources amongst the
numerous possibilities that have been put forward.
\vspace{-2mm}
\section{Targets}
\vspace{-2mm}
Motivated by the growing catalog of TeV PWNe, VERITAS, in 2006, observed
three northern sky PWNe, G75.2+0.1, G106.6+2.9, and 3C58, associated with young,
energetic pulsars (see Table~\ref{t1}).
\noindent
{\bf PSR~J2021+3651}. A {\it Chandra} observation showed this
pulsar to be embedded in a compact, bright X-ray PWN (PWN G75.2+0.1) with the standard torus and
jet morphology \cite{ref7}. Its X-ray spectrum is well fit by a power-law model with photon
index $\Gamma$=1.7 and a corresponding 0.3-10~keV flux of
1.9$\times 10^{-12}\, \rm erg\, cm^{-2}\, s^{-1}$. This young Vela-like pulsar is coincident
with the EGRET $\gamma$-ray source GeV~2020+3651. Recently, the Milagro
$\gamma$-ray observatory detected an extended source or multiple unresolved
sources of $\gamma$ rays at a median-detected energy of 12~TeV \cite{ref8} coincident with
the same region.
The radio dispersion measure suggests a distance to PWN G75.2+0.1 $\rm d \geq 10$~kpc,
but this measurement could have been contaminated by the gas in the Cygnus
region and the true distance may be in fact substantially closer. Presently PWN G75.2+0.1
is considered to be one of the best candidates for the Milagro source (MGRO J2021+37) and it is
likely to be seen in the energy range covered by VERITAS.
\noindent
{\bf PSR~J2229+6114}. The {\it Chandra} X-ray image of PWN G106.6+2.9 shows
an incomplete elliptical arc and a possible jet, similar to the Vela PWN \cite{ref9}.
PSR~J2229+6114 is a compelling counterpart of the EGRET source
3EG~J2227+6122. This young, energetic pulsar is second only to the Crab pulsar in spin-down
power, and it is substantially more luminous than the Vela pulsar. Given the relatively
small distance of 3~kpc this pulsar has a very high rank among all pulsars
in the discriminant $\dot{E}/d^2$. Part of this flux can be converted into a high flux of VHE
$\gamma$ rays.
\noindent
{\bf 3C58}. 3C58 is a young Crab-like SNR generally accepted as being the
remnant of the historical supernova SN 1181. A compact object (nebula) at the center of the
SNR has been resolved in {\it Chandra} X-ray data \cite{ref10}, and is centered on
PSR~J0205+6449. Given its very high spin-down power, the pulsar is capable of
supplying the energy of the X-ray nebula, $L_x = 2.9\times 10^{34}\, \rm ergs\, s^{-1}$,
and may have substantial VHE $\gamma$-ray emission.
The TeV $\gamma$-ray fluxes expected from the PWNe around both PSR~J2021+3651 and
PSR J2229+6114 in terms of a hadronic-leptonic model for the high-energy
processes inside the PWNe \cite{ref11} exceed 10\% of the Crab Nebula
flux above 200~GeV (see Table~1). This suggests that both PSR J2021+3651 and
PSR J2229+6114 should be detectable with VERITAS
after rather short exposures. A somewhat lower $\gamma$-ray flux
of a few percent of the Crab Nebula was predicted for 3C58 \cite{ref11},
however it is still well above the sensitivity limit of the VERITAS detector for a reasonable exposure.
\vspace{-3mm}
\section{VERITAS Observations and Analysis}
\vspace{-2mm}
\begin{table*}[!t]
\centering
\begin{tabular}{lcccccccccc}\hline
Pulsar & $\rm N_{tel}$ & $\rm N_{runs}$ & $\rm R$ (Hz) & T (hr) & $\Theta$ ($^\circ$) & On & Off & $\alpha_{\small Li\&Ma}$ & S/N ($\sigma$) & {\small U.L. (Crab)} \\ \hline
{\small J2021+3651} & 2 & 23 & 100 & 8.4 & 30.5 & 189 & 512 & 0.33 & 1.19 & 4.6\% \\
{\small J2229+6114} & 2 & 32 & 88 & 12 & 31.8 & 151 & 543 & 0.25 & 0.19 & 2.7\% \\
{\small J0205+6449} & 3 & 17 & 147 & 5.3 & 34.2 & 109 & 406 & 0.25 & 0.32 & 2.4\% \\ \hline
\end{tabular}
\caption{Summary of data. }
\label{t2}
\vspace{-4mm}
\end{table*}
VERITAS is an array of four imaging Cherenkov telescopes sited in Amado,
Arizona, and dedicated to the detection of VHE $\gamma$ rays with
energies above 100 GeV. Each telescope has a tessellated mirror with an area of
$\simeq 110$~m$^2$ and a camera consisting of 499 photomultiplier tubes. The
first telescope in the array has been operating since February 2005. First stereo
observations with two telescopes began in April 2006, and the full array of four
telescopes has been operational since January 2007. A full VERITAS array has the sensitivity of
7~mCrab (5$\sigma$ detection over 50 hour exposure).
The angular resolution
of better than $0.14^\circ$ and a 3.5$^\circ$ field of view enable VERITAS to detect
and study a variety of compact galactic $\gamma$-ray sources like PWNe.
The VERITAS observations of three PWNe were made while the system was
under construction. Observations of PSR~J2021+3651 and PSR~J2229+6114 in
November 2006 were made with a two telescope system. Later a third
telescope was added to the system and observations of 3C58 in December 2006
were made with three telescopes. The data were taken mostly in 20~minute runs
with a few runs of 28~minutes using the {\it wobble} mode. In this mode, the source
direction is positioned $\pm 0.3^\circ$ (a $\pm 0.5^\circ$ offset was used for
later observations) in declination or right ascension relative to the center of the
camera field of view. The sign of the offset was altered in successive runs to
reduce systematic effects. The {\it wobble} mode allows on-source observation
and simultaneous estimation of the background induced by charge cosmic-ray
particles. This eliminates the need for off-source observations and consequently
doubles the amount of available on-source time.
For final analysis only those runs passing the data quality criteria are used.
The images are calibrated and then cleaned using a two-threshold
picture/boundary selection procedure which requires a pixel to have a signal
greater than 5.0 pedestal variances (PV) and a neighboring pixel to have a signal
larger than 2.5 PV. The pixels with a signal greater than 2.5 PV are included only
if they have a neighbor with a signal greater than 5.0 PV. After image cleaning
the shower images are parameterized using a standard second-moment approach.
The shower geometry is reconstructed using stereoscopic techniques with a typical
angular resolution of about 0.14$^\circ$ and an average accuracy of better than 20 m
in the determination of the shower core location. To ensure that images are not
truncated by the camera edge, only images with the center of gravity less than
1.3$^\circ$ from the center of the camera are used in the reconstruction. In addition,
at least two images are each required to exceed a minimum total signal of 400
digital counts of the respective flash analog-to-digital converter to ensure that the
showers are well reconstructed.
After the shower reconstruction, the cosmic-ray background events are rejected
using standard cuts on mean scaled width and mean scaled length parameters. The number
of events passing cuts in a circle of standard angular size around the source
position gives the number
of on-source (On) counts. The background is estimated using all events passing cuts in a
number of non-overlapping circles of the same size. The centers of these circles
are positioned at the wobble offset from the tracking position. The number of
background regions may vary depending on the actual wobble offset used in the
observation. The use of a larger background region reduces the relative
statistical error on the background measurement. For a given number of on-source
and background counts acquired after event selection the significance of the excess is calculated
following the method of Equation (17) in the Li \& Ma technique \cite{ref12}.
It is worth noting that the data have been analyzed using independent analysis packages
(see \cite{ref16, ref17} for details on the analyses). All of these analyses yield consistent results.
\vspace{-3mm}
\section{Results and Conclusion}
\vspace{-1mm}
Table~2 summarizes the results of the VERITAS observations of each of the
individual sources. These objects have been observed with VERITAS for rather
limited exposure times. The longest exposure (T) of 12~hrs was for PSR~J2229+6114.
All observations were made at the median zenith angle ($\Theta$) of $\sim 30^\circ$.
Parameter $\alpha_{Li\&Ma}$ used in the Li \& Ma technique is also given in Table~2. No
significant excess suggesting TeV $\gamma$-ray emission is evident for any of the observed
PWN. The 99\% confidence level flux upper limits \cite{ref13} at the pulsar position expressed
in the flux of the Crab Nebula are shown in Table~\ref{t2}.
Present VERITAS upper limits for a sample of northern PWNe contradict
predictions of high VHE $\gamma$-ray fluxes made in \cite{ref11}, which might possibly
constrain the choice of magnetic field strength within the nebulae. For PSR~J2021+3651
a very hard $\gamma$-ray spectrum of hadronic origin with a peak at $\sim$10~TeV, which
is the median-energy of the MILAGRO detection, and sharp fall off at higher energies as
suggested in \cite{ref14} would still satisfy the VERITAS upper limit. A similar spectrum
has been observed from PWN Vela X by H.E.S.S. \cite{ref15}. Note, however, that
present VERITAS upper limits were derived assuming a point like $\gamma$-ray source.
Estimating the VERITAS upper limit for an extended source seems to be premature
given that the exact localization and angular extent of the VHE $\gamma$-ray source detected
by the MILAGRO $\gamma$-ray observatory are still in the process of final evaluation.
\vspace{-2mm}
\subsection*{Acknowledgments}
\vspace{-1mm}
VERITAS is supported by grants from the U.S. Department of Energy, the
U.S. National Science Foundation and the Smithsonian Institution, by NSERC in
Canada, by PPARC in the U.K. and by Science Foundation Ireland.
\vspace{-3mm}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,155 |
\section{Introduction}
\label{Sec:Intro}
Observables in relativistic heavy ion experiments are often reported as functions of measured total inelastic cross section fraction (centrality) and are related to the initial collision geometry using Monte Carlo Glauber (MCG) or optical Glauber models~\cite{starmc,glauber-rev,glauber}. An example is charged particle multiplicity $N_{ch}$ versus number of participating nucleons $N_{part}$. It is therefore imperative that the level of accuracy of the Glauber models be understood and that relevant experimental information be used to estimate collision centrality. For instance, previous studies~\cite{starmc,glauber-rev,starspec} concluded that the uncertainties in relating peripheral collision data to the corresponding initial collision geometry were very large, thus contributing to the omission of those data from publications~\cite{starmc}. In another example, studies of high-$p_t$ suppression of charged particle production for peripheral to mid-central collisions are presently limited by large uncertainties in the estimated number of binary collisions which is used as a scaling variable~\cite{highpt}.
In this work we re-examine the accuracy of Monte Carlo Glauber model descriptions of minimum-bias multiplicity frequency distribution data from RHIC using updated density and multipliciity production model parameters and a sensitive power-law inspired representation introduced by Trainor and Prindle (TP)~\cite{powerlaw}. After demonstrating the accuracy of our Monte Carlo Glauber model in describing RHIC data we then use the power-law analysis method of Ref.~\cite{powerlaw} to estimate the uncertainties in the mapping relationships between initial stage collision geometry quantities and centrality for heavy ion collision systems relevant to the RHIC program. We find that previous uncertainties~\cite{starmc,glauber-rev,starspec} in centrality measures are too pessimistic.
The centrality analysis method of Trainor and Prindle~\cite{powerlaw}
exploits the
approximate power-law dependence of the multiplicity frequency
distribution data $dN_{evt}/dN_{ch}$ ($N_{evt}$ is the number of triggered
events) for minimum-bias~\footnote{A minimum-bias trigger
typically refers to the detection of forward going
spectator fragments from both colliding nuclei plus
a minimal requirement for particle
production transverse to the beam direction.
Each of the four RHIC experiments utilize a common lowest level trigger
detector system based on calorimetric detection of neutrons at zero degree
scattering angle using two zero-degree calorimeters (ZDC) placed
symmetrically upstream and downstream from the beam-beam intersection
region. The minimum-bias trigger systems for the STAR, PHENIX, PHOBOS and
BRAHMS experiments are described in Nucl. Instrum. Meth. A {\bf 499}, Nos. 2-3
(2003).} collisions and uses
proton-proton (p-p) multiplicity production data to constrain the peripheral
collision end-point of the $dN_{evt}/dN_{ch}$ distribution.
Their analysis demonstrates that uncertainties in centrality determination and
MCG geometry measures can be significantly reduced, particularly in
the peripheral region.
In this paper MCG results and errors for centrality bin average quantities
$\langle b \rangle$ (mean impact parameter), $\langle N_{part} \rangle$,
$\langle N_{bin} \rangle$, and $\nu = \langle N_{bin} \rangle/(\langle N_{part} \rangle /2)$ ~\cite{tomnu}
are presented for minimum-bias Au-Au
collisions at $\sqrt{s_{NN}}$ = 20, 62, 130 and 200~GeV and for Cu-Cu at
62 and 200~GeV. In this study centrality is based on
charged particle production at midrapidity. Other centrality definitions
appear in the literature but will not be considered here.
However, the MCG model
and analysis methods presented here and in Ref.~\cite{powerlaw} can be readily
extended to any practical centrality determination method.
The MCG model and updated input parameters are discussed in Sec.~\ref{Sec:model}.
The accuracy of the model for describing
$dN_{evt}/dN_{ch}$ data from RHIC is demonstrated in Sec.~\ref{Sec:datafit}.
MCG predictions and errors are presented and discussed in
Sec.~\ref{Sec:results}. Comparison of the MCG results and analytic parametrizations from Ref.~\cite{powerlaw} are presented in Sec.~\ref{Sec:analytic}. In Sec.~\ref{Sec:discussion} we demonstrate via simulations
how energy deposition data from transverse
particle production, which are generally available at the trigger level from the RHIC experiments~\cite{dunlop},
can be used to further reduce
the impact of background contamination and to mitigate the effects of event loss due to
trigger and collision vertex
finding inefficiencies.
A summary and conclusions are presented
in Sec.~\ref{Sec:summary}. Computational details are contained in three
appendixes at the end.
\section{Monte Carlo Glauber Model}
\label{Sec:model}
The Monte Carlo Glauber collision model used here is based on a standard
set of assumptions~\cite{glauber} which are appropriate for ultra-relativistic heavy ion
collisions. These assumptions include the characterization of the collision
in terms of a classical impact parameter $(b)$, straight-line propagation of
each incident nucleon through the oncoming nucleus, and a fixed transverse
interaction range determined by the nucleon-nucleon (N-N) total inelastic cross
section ($\sigma_{inel}$) in free space.
The impact parameter was selected at random and the positions of the nucleons
relative to the geometrical centers of the colliding nuclei were randomly
distributed according to a spherical density $\rho(r)$. Center-of-mass constraints were not imposed on the nucleon
positions. Nucleon pairs in the colliding nuclei were assumed to interact hadronically
if their relative impact parameter was $\leq \sqrt{\sigma_{inel}/\pi}$.
Charged hadron multiplicity was assigned using the phenomenological
two-component model of Kharzeev and Nardi~\cite{kn} for ``soft'' plus
``hard'' particle production processes where the mean number of charged
hadrons in the acceptance ($\bar{N}_{ch}$) per unit pseudorapidity ($\eta$)
was computed according to
\begin{eqnarray}
\frac{d\bar{N}_{ch}}{d\eta} & = & (1-x)n_{pp}\frac{N_{part}}{2} +
x n_{pp} N_{bin}.
\label{Eq1}
\end{eqnarray}
Parameters $n_{pp} \equiv d\bar{N}_{ch}(pp)/d\eta$ and $x$
depend on collision energy.
Event-wise multiplicities were obtained by sampling the Gaussian
distribution~\cite{kn}
\begin{eqnarray}
{\cal P}(N_{ch},\bar{N}_{ch}) & = & \frac{1}{\sqrt{2\pi a \bar{N}_{ch}}}
\exp \left( -\frac{(N_{ch} - \bar{N}_{ch})^2}{2a\bar{N}_{ch}} \right)
\label{Eq2}
\end{eqnarray}
where $a$ is a multiplicity fluctuation width parameter. Alternate distributions
({\em e.g.} Poisson, negative binomial~\cite{nbd}) produce quantitative effects on the
multiplicity frequency distribution for very peripheral collisions but do not
affect the present estimates of systematic errors.
$N_{bin}$ and $N_{ch}$ were both required to be $\geq 1$ in order for the simulated
collision to be used in the analysis.
Centrality bins were defined using the multiplicity frequency distribution
$dN_{evt}/dN_{ch}$. Centrality bin average quantities, $\langle b \rangle$, $\langle N_{part} \rangle$,
$\langle N_{bin} \rangle$ and
$\nu \equiv 2 \langle N_{bin} \rangle / \langle N_{part} \rangle$,
were calculated using the events within each bin. The acceptance for this
study was $|\eta| \leq 0.5$ and full $2\pi$ in azimuth.
\subsection{Matter Densities}
\label{Subsec:matterdensity}
Monte Carlo Glauber simulations require the distribution of the centers of the
nucleons in the nuclear ground state, $\rho_{pt,m}(r)$, the point-matter density.
For $^{63}$Cu and $^{197}$Au these
were estimated using the measured charge densities~\cite{dejager}
and the Hartree-Fock calculations of Negele using the Density
Matrix Expansion (DME) framework~\cite{dme}
for the neutron - proton density differences.
The charge densities for $^{63}$Cu and $^{197}$Au were represented by a
Woods-Saxon distribution,
\begin{eqnarray}
\rho(r) & = & \rho_0 \{ 1 + \exp [(r-c)/z] \}^{-1},
\label{Eq3}
\end{eqnarray}
where the radius and diffuseness parameters are listed in Table~\ref{TableI}.
\begin{table}[h]
\caption{\label{TableI}
Charge and point matter density Woods-Saxon parameters for $^{63}$Cu and
$^{197}$Au in fm.}
\begin{indented}
\item[]\begin{tabular}{@{}cllll
\br
Density & \multicolumn{2}{c}{$^{63}$Cu} & \multicolumn{2}{c}{$^{197}$Au} \\
Parameter & Empirical$^{\rm a}$
& DME$^{\rm b}$ & Empirical$^{\rm a}$ & DME$^{\rm b}$ \\
\mr
$c_{chrg}$ & 4.214$\pm$0.026 & 4.232 & 6.38$\pm$0.06 & 6.443 \\
$z_{chrg}$ & 0.586$\pm$0.018 & & 0.535$\pm$0.027 & \\
$\langle r^2_{chrg} \rangle ^{1/2}$ & 3.925$\pm$0.022 & 3.899 & 5.33$\pm$0.05 & 5.423 \\
\mr
$c_{pt,m}$ & 4.195$\pm$0.085 & 4.213 & 6.43$\pm$0.10 & 6.495 \\
$z_{pt,m}$ & 0.581$\pm$0.031 & & 0.568$\pm$0.047 & \\
$\langle r^2_{pt,m} \rangle^{1/2}$ & 3.901 & 3.875 & 5.41 & 5.502 \\
\br
\end{tabular}
\item[] $^{\rm a}$ Charge density results based on electron scattering
analysis~\cite{dejager}; estimates of point matter densities as discussed in the text.
\item[] $^{\rm b}$ Density Matrix Expansion predictions~\cite{dme}.
\end{indented}
\end{table}
The
point matter densities assumed for the present analysis are spherically
symmetric with a Woods-Saxon radial distribution where the half-density
and rms radii were estimated by adding the Hartree-Fock DME point matter --
charge distribution differences to the measured charge density radii
where,
\begin{eqnarray}
c_{pt,m} & = & c_{chrg} + \left[c_{pt,m} - c_{chrg} \right]_{\rm DME}, \nonumber \\
\langle r^2_{pt,m} \rangle ^{1/2} & = & \langle r^2_{chrg} \rangle ^{1/2}
+ \left[ \langle r^2_{pt,m} \rangle ^{1/2} - \langle r^2_{chrg} \rangle ^{1/2}
\right]_{\rm DME},
\label{Eq4}
\end{eqnarray}
where $c_{chrg}$ and $\langle r^2_{chrg} \rangle ^{1/2}$ denote the measured
radii from electron scattering while the quantities in the square brackets
are the DME predictions. The diffusivity parameter $z_{pt,m}$ was obtained
from the quantities in Eq.~(\ref{Eq4}) and the approximate relation,
$\langle r^2\rangle \cong \frac{3}{5} c^2 [1 + \frac{7}{3} (\pi z/c)^2 ]$~\cite{velicia}.
The nominal radius and diffusivity parameters assumed here for Au and Cu
are listed in Table~\ref{TableI}.
The uncertainties in $c_{pt,m}$ and $z_{pt,m}$ were obtained by summing
the independent errors in the proton~\cite{dejager} and neutron point matter densities in
quadrature. The latter
is constrained by
theoretical and experimental information about the neutron - proton
density difference. Theoretical nuclear structure model predictions
for the neutron$-$proton rms radii differences for isotopes in the Cu and Au mass range agree to
within about $\pm 0.06$~fm~\cite{cls}. In general the theoretical
predictions agree with analyses of medium energy proton-nucleus elastic
scattering data~\cite{rayden} which are typically uncertain by about
$\pm 0.07$~fm for isotopes in the Cu and Au mass range.
The uncertainty in the neutron density rms radii relative to the proton
density was therefore assumed to be $\pm\sqrt{0.06^2 + 0.07^2}{\rm ~fm}
\approx \pm 0.09$~fm and the corresponding uncertainty in the matter rms radii
was $\pm(N/A)0.09$~fm, where $N,A$ are the
number of neutrons and nucleons in the isotope. Theoretical
contributions to the errors in $c_{pt,m}$ and $z_{pt,m}$ were conservatively
estimated by requiring each to independently account for the $\pm$(N/A)0.09~fm
error. The latter theoretical errors were combined in quadrature with
the corresponding errors for $c_{chrg}$ and $z_{chrg}$ from analysis of
electron scattering data to obtain the final errors listed in
Table~\ref{TableI}.
\subsection{N-N Inelastic Cross Section}
\label{Subsec:cross-section}
The N-N total inelastic cross sections used here were based on total cross
section measurements for p-p collisions ($\pm$1~mb uncertainty) and
elastic total cross sections for p-p and p-$\bar{\rm p}$
($\pm$0.5~mb error)~\cite{pdg}.
Proton-neutron total cross section data are not available in the energy
range studied here.
The results for energies $\sqrt{s}$ = 20, 62, 130 and 200~GeV are respectively
33, 35.3, 38.7 and 41.7~mb, each being uncertain by $\pm$~1.1~mb.
It is possible that the effective interaction cross section between colliding
nucleons inside a nucleus differs from that in free
space (density dependence)
or that the strength and range of the effective
N-N interaction changes with each successive collision
as in the ``used'' nucleon
scenario~\cite{signneff} or the ``strict'' participant scaling
model~\cite{powerlaw} (limiting case of the used-nucleon model in which nucleons
interact only once). The study of density dependent
effects is well beyond the scope
of the present analysis.
\subsection{Two-Component Multiplicity Production Model}
\label{Subsec:twocomponent}
Parameters $n_{pp} = d\bar{N}_{ch}(pp)/d\eta|_{\eta=0}$ at $\sqrt{s}$ = 62,
130 and 200~GeV are 2.01, 2.25 and 2.43 ($\pm0.08$ error for each),
respectively, using the energy dependent parametrization
of the UA5~\cite{ua5} and CDF data
given in Ref.~\cite{abe}. For the $\sqrt{s}$ = 20~GeV data, which is outside
the energy range parametrized in~\cite{abe}, the average of ISR~\cite{isr-thome}
and FNAL~\cite{fnal} measurements summarized in Ref.~\cite{ua1} was assumed where
$n_{pp} = 1.4 \pm 0.12$.
Binary scattering parameter $x$ in Eq.~(\ref{Eq1}) was estimated by fitting
$(dN_{ch}/d\eta)/(0.5N_{part})$ data from STAR~\cite{starspec,molnar},
PHENIX~\cite{phenixdata}, and PHOBOS~\cite{phobos2002,phobos2006}
versus $\nu$ with Eq.~(\ref{Eq1}) rewritten as,
\begin{eqnarray}
\frac{2}{N_{part}} \frac{d\bar{N}_{ch}}{d\eta} & = & n_{pp}[1+x(\nu-1)],
\label{Eq5}
\end{eqnarray}
assuming the above values for $n_{pp}$. The resulting values of $x$ are
0.07, 0.09, 0.09 and 0.13 ($\pm$0.03 errors for each) for the 20, 62, 130
and 200~GeV data, respectively. Analysis of the 19.6~GeV Au-Au
data by the PHOBOS experiment~\cite{phobos2002} assumed $n_{pp} = 1.27 \pm 0.13$~\cite{isr-thome}
and obtained $x = 0.12$, leading to claims in the literature that hard
scattering contributions to multiplicity do not change with collision
energies from 20 to 200~GeV~\cite{phobos2002,phobos2006,loizides,stephans}.
A more recent compilation~\cite{ua1} of $n_{pp}$ measurements in this
energy range indicates a larger value of $n_{pp} = 1.4 \pm 0.12$. Fitting
Eq.~(\ref{Eq5}) to the combined PHENIX~\cite{phenixdata} and
PHOBOS~\cite{phobos2002} 19.6~GeV data with $n_{pp} = 1.4$ resulted in a
smaller $x = 0.07$. PHOBOS 62~GeV Au-Au data~\cite{phobos2006}, when plotted
versus $\nu$, do not linearly extrapolate to $n_{pp}$ in contrast to
what is expected from
Fig.~4 of Ref.~\cite{phobos2006} and Fig.~1 of Ref.~\cite{stephans} from
the PHOBOS experiment.
However, 62~GeV Au-Au results from the STAR experiment linearly extrapolate to $n_{pp}$ at $\nu=1$,
resulting in the fitted value $x = 0.09$ used here. $x$ parameters from the three RHIC experiments
generally agree for the 130 and 200~GeV data; the PHOBOS values of
$x = 0.09$~\cite{phobos2002} and 0.13~\cite{phobos2002} at 130 and 200~GeV,
respectively, were confirmed and used here.
It is possible that parameter $x$
is affected by other processes
in addition to hard and semi-hard partonic scattering in the initial collision
stage where the latter mechanisms would be expected to follow a $\log\sqrt{s}$ dependence.
The variance of the multiplicity distribution ${\cal P}(N_{ch},\bar{N}_{ch})$
in Eq.~(\ref{Eq2}) is given by $a\bar{N}_{ch}$ where values of parameter
$a<1$ or $>1$ represent multiplicity fluctuation suppression or excess,
respectively, relative to pure statistical fluctuations ($a=1$). In general
a non-vanishing integral~\cite{clt,delsign,ptscale} of two-particle correlations~\cite{axialci}
over the acceptance requires $a \neq 1$. In principle, correlation measures
of the type reported in Ref.~\cite{axialci} could be used to determine parameter $a$
using the relationship between fluctuations and correlations developed
in Refs.~\cite{clt,ptscale} and discussed in Appendix~A.
On the other hand, the Kharzeev and Nardi two-component
multiplicity model with distribution ${\cal P}(N_{ch},\bar{N}_{ch})$
constitutes a phenomenology for describing event-wise multiplicity frequency
distribution data. The phenomenological approach based on fits to multiplicity
distribution data, shown in the next section, was used to estimate parameter
$a$ in the present analysis. The uncertainty in the width of ${\cal P}(N_{ch},\bar{N}_{ch})$ was estimated by fitting the data. The uncertainty in the
analytic form of the multiplicity fluctuation distribution was accounted for
by comparing Monte
Carlo Glauber results assuming the Gaussian ${\cal P}(N_{ch},\bar{N}_{ch})$ in Eq.~(\ref{Eq2})
(with $a=1$) with results assuming a negative binomial distribution
(NBD)~\cite{nbd} in place of Eq.~(\ref{Eq2})
as explained in Appendix~A.
\section{Fits to 130 GeV Au-Au Data}
\label{Sec:datafit}
The 130~GeV Au-Au minimum-bias negatively charged hadron multiplicity frequency
distribution data from STAR~\cite{starspec} for 60K events
were fit by
adjusting parameters $n_{pp}$ (for negative hadrons) and $a$ where
$x$ was fixed to 0.09 as discussed above. The acceptance was defined by transverse momentum
$(p_t) > 0.1$~GeV/$c$, $|\eta| < 0.5$, and $\Delta\phi = 2\pi$.
The fit (solid histogram) is shown in
the left panel of Fig.~\ref{Fig1}
in comparison with data (solid dots), where the optimum values of $n_{pp}$ and
$a$ are $1.110\pm0.004$ (consistent with $n_{pp} = 2.25\pm0.08$ for
charged particle yields) and $1.04\pm0.10$, respectively. Similar fits
(not shown) to the charged particle minimum-bias multiplicity distribution
from the same data set resulted in $a = 0.94\pm0.15$. Parameter $a$ was therefore
set to $1.0\pm0.2$ for all four energies.
\begin{figure*}[t]
\includegraphics[keepaspectratio,width=6.3in]{Ray_figure1}
\caption{\label{Fig1}
Negative hadron multiplicity frequency distributions
for Au-Au minimum-bias collisions at $\sqrt{s_{NN}}$ = 130~GeV normalized
to the total inelastic cross section in barns.
Left panel (a): semi-log plot comparing STAR data~\cite{starspec}
(solid dots) with the
Monte Carlo Glauber model fit (histogram). Middle panel (b): same
data and Monte Carlo fit on a log-log plot showing
the power-law dependence and the exponent (slope)
of approximately -3/4.
Right panel (c): same data (solid dots) and Monte Carlo fit (solid line) plotted as $d\sigma/dN_{h^-}^{1/4}$ versus
$N_{h^-}^{1/4}$.}
\end{figure*}
In the TP analysis~\cite{powerlaw} it was shown that
minimum-bias multiplicity
frequency distributions for relativistic heavy ion collision
experiments and Monte Carlo Glauber models approximately follow
a power-law distribution. This is illustrated by plotting
the data and MCG fit from the left panel of Fig.~\ref{Fig1} on log-log axes
in the middle panel.
Except near the end-points the data can be described to within 10\% with a
slope (exponent) of approximately $-3/4$.
The collision event yield is approximately proportional
to $N_{h^-}^{-3/4}$.
Therefore distribution $d\sigma/dN_{h^-}^{1/4}$
versus $N_{h^-}^{1/4}$ is approximately constant, where
\begin{eqnarray}
\frac{d\sigma}{dN_{h^-}^{1/4}} & = & \frac{dN_{h^-}}{dN_{h^-}^{1/4}}
\frac{d\sigma}{dN_{h^-}} = 4N_{h^-}^{3/4}\frac{d\sigma}{dN_{h^-}}
\approx {\rm const.}
\end{eqnarray}
as shown in the right-hand panel of Fig.~\ref{Fig1} for
data (solid dots) and MCG fit (solid line). The power-law plotting format in the right-hand panel enables a more sensitive comparison between the model fit and data than in the usual semi-log format (left panel). The MCG model fit is consistent with the data except for the first two data points at low $N_{h^-}^{1/4}$. Except near the end-points the $d\sigma/dN_{h^-}^{1/4}$ data are
constant on $N_{h^-}^{1/4}$ to within an overall variation of 20\% in comparison to the
$d\sigma/dN_{h^-}$ data which span nearly two orders of magnitude within the
same multiplicity range.
The results in Fig.~\ref{Fig1} demonstrate the efficacy of the Monte Carlo Glauber model with
two-component multiplicity production for accurately describing the measured
multiplicity frequency distributions at RHIC.
Based on this outcome we conclude that the present model is reasonable to use in
estimating centrality bin average quantities and their
systematic errors for RHIC data.
\section{Results and Error Analysis}
\label{Sec:results}
Ensembles of one-million, minimum-bias ({\em i.e.} random impact parameter)
Monte Carlo collisions were generated for each of the six systems studied here.
Centrality bin averaged quantities $\langle N_{ch} \rangle$,
$\langle b \rangle$, $\langle N_{part} \rangle$,
$\langle N_{bin} \rangle$ and $\nu$
are listed in Tables~\ref{TableII}-\ref{TableVII}
using the nominal parameter values
discussed above.
Statistical errors are typically 0.1-0.2\% and always $<0.5$\% of the
nominal bin averages and in all instances are much less than the systematic
errors discussed below.
Results in Table~\ref{TableII} for $\langle N_{part} \rangle$ for Au-Au collisions at 200~GeV
are systematically larger than published Monte Carlo
Glauber predictions in Ref.~\cite{starmc} by approximately 5\% for peripheral
centrality bins. This systematic increase is primarily caused by multiplicity fluctuations in the present model
and basing centrality on $N_{ch}$ here rather than on $N_{part}$ as was done
in Ref.~\cite{starmc}.
Average charged particle multiplicity per participant pair for $|\eta| < 0.5$
at mid-pseudorapidity as a function of centrality ($\nu$) is shown in
Fig.~\ref{Fig2} for Monte Carlo simulated Au-Au collisions at
$\sqrt{s_{NN}}$ = 200~GeV (solid dots). The p-p limit, $n_{pp} = 2.43 \pm 0.08$,
is indicated by the solid square symbol.
The data display a linear dependence on $(\nu - 1)$ as expected from Eq.~(\ref{Eq5}) except for the most-peripheral 90-100\% centrality bin where multiplicity
fluctuations significantly reduce the average $N_{ch}$. The slope agrees with
$n_{pp} x = 0.32$; the linear extrapolation to the p-p limit is evident.
\begin{figure}[htb]
\begin{center}
\includegraphics[keepaspectratio,width=3.0in]{Ray_figure2}
\end{center}
\caption{\label{Fig2}
(color online) Monte Carlo Glauber results for
$\langle N_{ch} \rangle / \langle N_{part}/2 \rangle$ for
one-million Au-Au collisions at $\sqrt{s_{NN}}$ = 200~GeV
using the nominal parameters discussed in the text (solid dots).
Linear fit (dashed line) to the Au-Au results (excluding the 90-100\%
centrality bin) accurately extrapolates to the p-p multiplicity from
UA5~\cite{ua5,abe}. Monte Carlo data for Au-Au include statistical errors only which are
smaller than the symbols.}
\end{figure}
\subsection{Error Estimation Method}
\label{Subsec:errorestimate}
The sources of uncertainty which cause systematic error in the collision
geometry quantities can be organized into three categories.
The first includes fitting uncertainties that occur when the corrected,
multiplicity frequency distribution data are described by adjusting the
parameters of the multiplicity production model in Eqs.~(\ref{Eq1}-\ref{Eq2}).
The second includes the uncertainties in the matter density, nucleon-nucleon
total inelastic cross section, and functional representation of the
multiplicity fluctuation distribution ${\cal P}(N_{ch},\bar{N}_{ch})$. The
third includes the uncertainty in the corrected multiplicity frequency
distribution data $dN_{evt}/dN_{ch}$. The latter
arise from trigger inefficiency, collision
vertex finding inefficiency, background contamination, and particle
tajectory finding inefficiency.
Systematic errors in centrality bin average quantities due to fitting uncertainty were estimated by comparing the nominal results to that obtained
with parameters $x$ and $a$ in Eqs.~(\ref{Eq1}-\ref{Eq2}) individually varied
within their respective fitting errors (Secs.~\ref{Subsec:twocomponent} and \ref{Sec:datafit}). Uncertainty in parameter
$n_{pp}$ directly affects $\langle N_{ch} \rangle$ but does not affect $\langle b \rangle$, $\langle N_{part} \rangle$,
$\langle N_{bin} \rangle$ or $\nu$.
Error estimates due to uncertainties in the matter density radius and diffuseness parameters and the N-N inelastic cross section require a refitting of the $dN_{evt}/dN_{ch}$ distribution. This was accomplished via a phenomenological adjustment of the multiplicity production model described in Appendix~B. Fit recovery via $|\chi|^2$ minimization for MCG simulations with ample statistics (of order $10^6$ collisions) is computationally intensive. The method in Appendix~B is fast and accurate. Errors due to the uncertainty in the mathematical representation of ${\cal P}(N_{ch},\bar{N}_{ch})$ were estimated by comparing the nominal MCG results using the Gaussian distribution in Eq.~(\ref{Eq2}) with results assuming a negative binomial distribution~\cite{ua5,nbd} as explained in Appendix~A.
Uncertainties in the $dN_{evt}/dN_{ch}$ data arising from trigger inefficiency, collision
vertex finding inefficiency, and background contamination mainly affect the
low multiplicity region. Minimum-bias trigger efficiencies at RHIC are
$92.2^{+2.5}_{-3.0}$\%~\cite{phenixdata,phenixspec},
$94\pm2$\%~\cite{starspec}, 96\%~\cite{brahmsspec}
and $97\pm3$\%~\cite{phobos2002,phobosspec,phobosspec3}. Trigger inefficiency
causes the lower $N_{ch}$ half-max point of the $dN_{evt}/dN_{ch}^{1/4}$
distribution to shift to larger $N_{ch}^{1/4}$. The position of this point for the
$dN_{evt}/dN_{ch}^{1/4}$ distribution when corrected for trigger inefficiency
(assuming the power-law behavior) has a relative uncertainty of about
$\pm$10\%.~\footnote{Trigger efficiencies are uncertain by about $\pm$2 to $\pm$3\%. The fractional
uncertainty in the lower half-max position is $\pm$(0.02 to 0.03)$(N_{ch,max}^{1/4} - N_{ch,min}^{1/4})/N_{ch,min}^{1/4} \approx 10$\% where
$N_{ch,min}^{1/4}$ and $N_{ch,max}^{1/4}$ are the lower and upper half-max points of the $dN_{evt}/dN_{ch}^{1/4}$ distribution.} However, knowledge that the lower half-max point is
constrained by p-p scattering allows the relative uncertainties of the
lower half-max point on $N_{ch}^{1/4}$ to be reduced to 1/4 of the uncertainties in $n_{pp}$,
or to $\pm$0.8\%, $\pm$0.9\%, $\pm$1\%, and $\pm$2\% for the 200, 130, 62
and 20~GeV data, respectively.
Primary collision vertex reconstruction efficiency is approximately 100\%
for events with $N_{ch}$ of order a few tens and greater but falls
precipitously for $N_{ch}<10$~\cite{calderon}. PHOBOS~\cite{phobos2002,phobosspec3}
and STAR~\cite{starspec,calderon} both report uncertainties in their overall
vertex finding efficiencies of about $\pm2$\% for minimum-bias
collisions. Vertex finding inefficiency increases the slope and half-max
position of the low multiplicity edge of the $dN_{evt}/dN_{ch}^{1/4}$ distribution.
An uncertainty of $\pm2$\% in the overall efficiency results in about
$\pm$10\% uncertainty in the low $N_{ch}^{1/4}$ half-max position.
However, the p-p data constrain this uncertainty. In this
analysis we assumed 100\% vertex finding efficiency for collisions producing
$N_{ch} \geq 14$~\cite{calderon}, or $N_{ch}^{1/4} \geq 1.93$, and allowed
the slope of the lower edge of the $dN_{evt}/dN_{ch}^{1/4}$ distribution to vary such that the half-max position varied by $\pm$0.8\%, $\pm$0.9\%, $\pm$1\%, and $\pm$2\% for the 200, 130, 62
and 20~GeV data, respectively, as in the preceding paragraph.
The principle sources of background contamination are from
ultra-peripheral two-photon interactions~\cite{upc}
and beam-gas collisions. The former process corresponds to coherent photon-photon
interactions which excite both nuclei, followed by neutron decay (which activates the minimum-bias trigger detectors) and accompanied by
resonance(s) production which decays into charged particles
transverse to the beam direction. Transverse particle multiplicities from
UPC events are typically $\leq 2$ ({\em e.g.} $\rho$-meson
decay) and generally $\leq 4$~\cite{upc} for $|\eta|<1$ at midrapidity.
UPC backgrounds are therefore restricted to the most-peripheral
(90-100\%) centrality bin. UPC yields should be approximately proportional to
$(Z_1 Z_2)^2$~\cite{upc} (charge numbers for colliding ions 1 and 2) whereas beam-gas
contamination should scale with beam current. Other background events,
{\em e.g.} mutual Coulomb dissociation processes such as $\gamma$ + A $\rightarrow ~ {\rm A}^{\star}
~ \rightarrow$ B + n for both nuclei, can be eliminated by requiring minimum
transverse particle production. Remaining background events will appear near
the lower $N_{ch}$ edge of the $dN_{evt}/dN_{ch}^{1/4}$ distribution.
Estimates of background contamination in minimum-bias trigger data at RHIC
range from $1\pm1$\%~\cite{phenixspec} to 6\%~\cite{starmc,starspec}
overall for Au-Au at $\sqrt{s_{NN}}$ =130~GeV corresponding to 0-20\% and
60\%, respectively, of the hadronic collision event yield in the 90-100\% centrality
bin. Most of the UPC events occur at $N_{ch} < n_{pp}$ and can be
eliminated by cuts on the number of transverse charged particles.
The remaining background contributions for $N_{ch} > n_{pp}$ are less than
the amounts listed above. For the present analysis background contamination
was assumed to be dominated by UPC events and to diminish in magnitude with
collision energy and $(Z_1 Z_2)^2$. In the present analysis background
contamination was applied to the nominal 90-100\% centrality bin assuming
3\%, 2\% and 1\% overall contamination levels in the Au-Au data at
$\sqrt{s_{NN}}$ = 200, 130 and 62~GeV, respectively.
UPC contamination for Au-Au collisions at 20~GeV and for Cu-Cu at 200 and 62~GeV
was estimated to be negligible.
However, calculations were done for the latter
three systems assuming a 1\% overall background contamination (10\%
contamination within the nominal 90-100\% centrality bin)
in order to provide a reference for further systematic error estimation.
The above sources of systematic uncertainty primarily affect the position, slope and shape of the $dN_{evt}/dN_{ch}^{1/4}$ distribution at the lower $N_{ch}^{1/4}$ end-point. These changes impact the MCG model which must describe those data and, in turn, the collision geometry quantities like $\langle N_{part} \rangle$. The phenomenological method in Appendix~B was used to estimate the changes in the collision geometry quantities relative to the nominal values. The latter differences were taken as the estimated errors.
Background levels in collider experiments vary significantly depending
on beam quality, beam current and interaction rate.
Excessive trigger backgrounds beyond
that considered here may result
from beam-gas interactions during periods of high integrated beam
currents. Collision event pile-up in the particle tracking detectors
during periods of high luminosity adversely affect collision vertex finding. Either condition may be so severe as to
preclude access to the low
multiplicity range of the minimum-bias distribution. Even so, the power-law
dependence and p-p end-point constraints enable accurate centrality estimates
and collision geometry assignments to be made for the remaining minimum-bias data.
Charged particle trajectory reconstruction efficiencies in the large acceptance
tracking detectors at RHIC are typically
70 - 95\%~\cite{starspec,phenixdata,phobos2002} and
decrease approximately linearly with particle density in the detectors by
about 20\% from most-peripheral to most-central collisions~\cite{calderon}.
Uncertainties in the assumed track reconstruction efficiencies
were estimated by comparing corrected data for
$(dN_{ch}/d\eta|_{\eta=0})/(N_{part}/2)$ versus $N_{part}$
between the RHIC experiments for Au-Au collisions at 20~GeV~\cite{phenixdata,phobos2002}
130~GeV~\cite{starspec,phenixdata,phobos2002} and
200~GeV~\cite{phenixdata,phobos2002}. The comparisons indicate
uncertainties in both the overall tracking efficiencies and in the dependence on particle density in the tracking detectors. The latter variation in efficiency from peripheral to central collisions is about $20\pm8$\%.
Overall changes in tracking efficiency are compensated in the MCG model by
multiplicative adjustments to parameter $n_{pp}$ and therefore have no effect on
the centrality measures reported here.
Changes in the assumed tracking efficiency dependence on $N_{ch}$ affect
the lower and upper end-point positions of the $dN_{evt}/dN_{ch}^{1/4}$
distribution and distort its shape. The distortions must be accounted for
by the MCG model in order to determine the net effect on the collision geometry measures. The effects of the
8\% uncertainty in the multiplicity dependence of the trajectory reconstruction efficiency were estimated by
generalizing parameter $n_{pp}$ in Eq.~(\ref{Eq1}) to
$n_{pp}(1 + \alpha N_{part})$ where $\alpha = \pm 0.00023$ and
$\alpha N_{part} = \pm 0.08$ for most-central Au-Au collisions. The same
value for $\alpha$ was assumed for Cu-Cu. Systematic errors were estimated
as the differences between the nominal centrality bin averages and those
obtained assuming $\alpha = \pm 0.00023$. The choice
to maximize multiplicity variation
for most-central collisions is arbitrary. As a result systematic errors in centrality bin average
multiplicities were not included in Tables~\ref{TableII}-\ref{TableVII}.
However, this ambiguity does not affect the resulting systematic errors
in the other centrality measures reported here.
\subsection{Error Results}
\label{Subsec:errorresults}
The combined systematic errors (all components added in quadrature)
in both magnitude and relative percent (given in parentheses) are listed
in Tables~\ref{TableII} - \ref{TableVII} for the six collision systems
studied here. Impact parameter uncertainty is about $\pm(2\, {\rm to}\, 3)$\%
for Au-Au and $\pm(2\, {\rm to}\, 6)$\% for Cu-Cu.
Uncertainty in $\langle N_{part} \rangle$ is about $\pm(4\, {\rm to}\, 8)$\%
for Au-Au and $\pm(3\, {\rm to}\, 7)$\% for Cu-Cu for peripheral centralities,
reducing to about $\pm1$\% or less for central collisions where full
geometrical overlap of the two nuclei suppresses the dependence of
$N_{part}$ on variations in the nuclear surface geometry. Systematic
errors in $\langle N_{bin} \rangle$ vary from about $\pm(4\, {\rm to}\, 12)$\%
for Au-Au collisions and $\pm(5\, {\rm to}\, 10)$\% for Cu-Cu
for central to peripheral collisions.
Errors in $\nu$
($\pm4$\% or less) are suppressed due to covariation of $\langle N_{part} \rangle$ and $\langle N_{bin} \rangle$.
Individual contributions to the systematic errors in collision geometry bin
averages $\langle b \rangle$, $\langle N_{part} \rangle$,
$\langle N_{bin} \rangle$ and
$\nu$ for Au-Au and Cu-Cu collisions at $\sqrt{s_{NN}}$ = 200~GeV are
summarized in Tables~\ref{TableVIII} and \ref{TableIX}, respectively.
Similar results were obtained for the other four collision systems.
Errors are given in percent where the
absolute values of positive and negative errors were averaged together.
The three numbers listed for each instance correspond
to the average errors within the 60-100\%, 20-60\%
and 0-20\% centrality bins, respectively.
The dominant errors are due to uncertainties in the matter density
and the N-N total inelastic cross section.
Errors due to uncertainties in the analytic form of the phenomenological
multiplicity fluctuation model are only significant for
peripheral collisions. Errors due to uncertainties in the multiplicity dependent particle track reconstruction
efficiency and in the trigger and vertex reconstruction inefficiencies
are negligible when constrained by p-p data
as shown previously
by Trainor and Prindle~\cite{powerlaw}.
Percent uncertainties due to possible background contamination are
listed for all six collision systems in Table~\ref{TableX}.
The largest errors occur in the most-peripheral centrality bin as
expected. The absolute magnitudes of the errors decline rapidly
with increasing centrality and are negligible for the centrality bins
not listed in the table. Reference errors for Au-Au at 20~GeV and Cu-Cu
at $\sqrt{s_{NN}}$ = 200 and 62~GeV were based on an assumed 10\% background
contamination in the 90-100\% centrality bin.
Estimates of total systematic error when background contamination
differs from that assumed here can be obtained by removing the
error contributions in Table~\ref{TableX} from the
total errors in Tables~\ref{TableII} - \ref{TableIV} for Au-Au
collisions at 200, 130 and 62~GeV, respectively, and then adding (in
quadrature) the appropriately scaled background errors from Table~\ref{TableX}.
For Au-Au collisions at 20~GeV and Cu-Cu collisions at 200 and 62~GeV
the scaled errors from Table~\ref{TableX} should be combined in quadrature
with the total systematic errors in Tables~\ref{TableV} - \ref{TableVII}.
If much larger backgrounds
than those assumed here are encountered, then
the present Monte Carlo results should not be scaled; rather the errors
should be recalculated.
Overall, the systematic errors in collision geometry bin
averages for non-peripheral centralities are dominated by uncertainties in the nuclear geometry
and $\sigma_{inel}$. Errors in the more peripheral bins are
dominated by background contamination and ambiguities in the analytic form of the
multiplicity fluctuation model.
\section{Analytic Parametrizations}
\label{Sec:analytic}
The power-law description of heavy-ion collision centrality developed by Trainor and
Prindle~\cite{powerlaw} prescribes simple, analytic parametrizations of the $N_{part}$,
$N_{bin}$, $\nu$ and $N_{ch}$ dependences on total inelastic cross section
fraction $\sigma/\sigma_0$. For $N_{part}$ the running integral relation on
$(1 - \sigma/\sigma_0)$ is accurately given by~\cite{powerlaw}
\begin{eqnarray}
(1 - \sigma/\sigma_0) & = & \frac{(N_{part}/2)^{\frac{1}{4}}
- (N_{part,min}/2)^{\frac{1}{4}}}
{(N_{part,max}/2)^{\frac{1}{4}} - (N_{part,min}/2)^{\frac{1}{4}}},
\label{Eq7}
\end{eqnarray}
where
\begin{eqnarray}
(N_{part}/2)^{\frac{1}{4}}
& = &
(N_{part,min}/2)^{\frac{1}{4}} \sigma/\sigma_0
+ (N_{part,max}/2)^{\frac{1}{4}} (1 - \sigma/\sigma_0).
\label{Eq8}
\end{eqnarray}
Similarly
\begin{eqnarray}
N_{bin}^{\frac{1}{6}} & = & N_{bin,min}^{\frac{1}{6}} \sigma/\sigma_0
+ N_{bin,max}^{\frac{1}{6}} (1 - \sigma/\sigma_0),
\label{Eq9}
\end{eqnarray}
where subscripts $min$ and $max$ refer respectively to the lower and upper
half-max end-point positions of the $d\sigma/d(N_{part}/2)^{1/4}$ and
$d\sigma/dN_{bin}^{1/6}$ distributions.
The above parametrizations are shown as the solid lines in the two
upper
panels of Fig.~\ref{Fig3} in comparison with the present MCG model results
(solid dots) from Table~\ref{TableII} for Au-Au collisions at $\sqrt{s_{NN}}$
= 200~GeV. The end-point parameters for the power-law parametrizations were $N_{part,min}/2 = 0.75$,
$N_{part,max}/2 = 189$, $N_{bin,min} = 0.75$ and $N_{bin,max} = 1144$.
The parametrization for $\nu = N_{bin}/(N_{part}/2)$ is compared with the MCG
result in the
lower-left
panel. The average multiplicities using Eq.~(\ref{Eq5}),
the values for $n_{pp}$ and $x$ from Sec.~\ref{Subsec:twocomponent},
and the above parametrizations for $N_{part}$ and $\nu$ (solid curve) is
compared with the MCG result in the
lower-right
panel. The simple power-law
parametrizations introduced in Ref.~\cite{powerlaw} quantitatively describe the
MCG results, thus confirming their utility and precision.
\begin{figure*}[t]
\begin{center}
\includegraphics[keepaspectratio,width=2.5in]{Ray_figure3a}
\includegraphics[keepaspectratio,width=2.5in]{Ray_figure3b}
\includegraphics[keepaspectratio,width=2.3in]{Ray_figure3c}
\includegraphics[keepaspectratio,width=2.3in]{Ray_figure3d}
\end{center}
\caption{\label{Fig3}
Comparison of Monte Carlo Glauber centrality quantities (solid dots) for Au-Au minimum-bias collisions at $\sqrt{s_{NN}}$ = 200~GeV from Table~\ref{TableII} with analytic parametrizations from Trainor and Prindle~\cite{powerlaw} based on the power-law description of heavy-ion collisions as discussed in the text. The four panels show from
upper-left to lower-right
the dependences of $(N_{part}/2)^{1/4}$, $N_{bin}^{1/6}$, $\nu$ and
$N_{ch}^{1/4}$ on relative cross section fraction $(1 - \sigma/\sigma_0)$.}
\end{figure*}
Calculation of the analytic power-law parametrizations require the upper half-max end-point values for $N_{part}$ and $N_{bin}$ from the MCG model. These quantities are listed in Table~\ref{TableXI} for the six collision systems studied here.
\section{Discussion}
\label{Sec:discussion}
Very peripheral collision data from the four RHIC experiments have generally
remained unpublished due to concerns with possibly large, and not well
understood background contamination, trigger inefficiencies,
and primary collision
vertex finding inefficiencies. Trainor and Prindle~\cite{powerlaw}
originally showed, and the present analysis confirms,
that the power-law dependence of the multiplicity frequency
distribution data together with knowledge of the proton-proton multiplicity
enables accurate centrality information to be obtained
in spite of these uncertainties.
Nevertheless, UPC events in principle cause the
lower end-point portion of the multiplicity frequency distribution
for minimum-bias A-A collisions to differ from the p-p limit
and these backgrounds will
contaminate the A-A data in the most-peripheral centrality bins. In this section we discuss
additional analysis methods using data available to the RHIC experiments
to minimize backgrounds and to
provide contamination level estimates for the accepted centrality bins.
Transverse particle production information via scintillators, silicon
detectors, and/or calorimetry are available in the RHIC experiments at the trigger level~\cite{startrig,phenixtrig,phobostrig,brahmstrig}.
These data record total energy deposition in the sensitive detector material for each (minimum bias) triggered
collision event including contributions
from A-A hadronic collisions, ultra-peripheral collisions, beam-gas and other
backgrounds, etc. but do not include particle track finding and primary
collision vertex finding inefficiencies. If the integrated yields from these
detectors, denoted as $SU\!M$, is approximately proportional to $N_{ch}$, then
the frequency distribution will follow an approximate power-law such that
the data can be usefully represented by $dN_{trig}/dSU\!M^{1/4}$ versus
$SU\!M^{1/4}$, where
$N_{trig}$ is the number of triggers.
MCG simulations for the integrated yields from the STAR central trigger
barrel (CTB)~\cite{startrig} plastic scintillator detector ($|\eta| \leq 1$,
$2\pi$ azimuth coverage) were done for p-p and Au-Au minimum-bias collisions
at $\sqrt{s_{NN}}$ = 200~GeV. Details of this simulation are discussed in
Appendix~C. The results for $dN_{trig}/dSU\!M^{1/4}$ versus $SU\!M^{1/4}$
are shown in Fig.~\ref{Fig4}. Both the p-p and Au-Au simulations reproduce
the general shapes of measured STAR CTB minimum-bias trigger yields~\cite{dunlop}. The
simulated Au-Au CTB yields follow the $SU\!M^{-3/4}$ power-law distribution
very well where the lower half-max point coincides with the mode of the p-p
distribution. Trigger inefficiency in Au-Au causes a depletion at low
multiplicity as indicated by the green, dashed curve. Over- or under-corrected
yields result in the blue dotted curves. From Fig.~\ref{Fig4} it is clear
how the p-p trigger yield can be used to constrain corrections for trigger inefficiency in A-A.
\begin{figure}[htb]
\begin{center}
\includegraphics[keepaspectratio,width=4.0in]{Ray_figure4}
\end{center}
\caption{\label{Fig4} (color online)
Monte Carlo simulation results for the minimum-bias trigger frequency distribution for
transverse particle production integrated trigger detector
yield quantity $SU\!M$ to the 1/4 power. Simulated yields are shown for 1M 200~GeV Au-Au (solid black curve, multiplied by 3)
and 1M proton-proton (solid red curve) collisions.
The yield for Au-Au collisions when
trigger inefficiency is large is illustrated by the lowest (right-most)
green dashed curve (hand-drawn sketch).
Over- and under-corrected yields are similarly illustrated by the upper (left-most)
and middle dotted blue lines, respectively. Background contamination contributes
at lower multiplicity as illustrated by the dashed-dotted magenta curve (hand-drawn sketch).
}
\end{figure}
Background triggers appear at lower values of $SU\!M$ where UPC events
typically produce of order two charged particles at mid-rapidity compared
with the average from p-p collisions of about $5$ in $|\eta| \leq 1$ and
therefore appear as a peak or enhancement (dashed-dotted curve in
Fig.~\ref{Fig4}) below the mode of the p-p distribution.
Comparisons between the measured p-p and Au-Au
$dN_{trig}/dSU\!M^{1/4}$ distributions and the power-law simulations for Au-Au
provide a reasonable means for defining additional event cuts to reduce
background contamination. This information can also be used to estimate the contamination level for
accepted events. Measurements of the dependence of the
lower end-point region of the $dN_{trig}/dSU\!M^{1/4}$
distribution on integrated beam current, luminosity, and ion species
would help disentangle background contributions from beam-gas collisions,
event pile-up, and UPC events. A similar power-law analysis
using reconstructed particle tracks, but without a primary collision vertex
requirement, would permit the vertex finding inefficiencies to be estimated
and corrected.
\section{Summary and Conclusions}
\label{Sec:summary}
A Monte Carlo Glauber and two-component multiplicity production model with fluctuations for high energy
heavy ion collisions
was used to describe minimum-bias multiplicity frequency distribution
data from RHIC.
Updated Woods-Saxon
radii and diffusivities for the point matter densities of $^{197}$Au and
$^{63}$Cu were determined using a combination of charge density measurements
from electron scattering and theoretical Hartree-Fock predictions.
The binary scattering parameter $x$ was estimated using a compilation
of available RHIC data from 20 to 200~GeV. The values for $x$ obtained here
display some energy dependence, although with large uncertainty.
The model was fitted
to the 130~GeV Au-Au minimum-bias data~\cite{starspec} in both the conventional
semi-log format and in the more sensitive power-law inspired format introduced
in Ref.~\cite{powerlaw}.
Quantitative agreement between the present model and data was obtained.
Systematic errors in centrality
bin averages for the geometrical quantities $\langle b \rangle$,
$\langle N_{part} \rangle$, $\langle N_{bin} \rangle$ and
$\nu = \langle N_{bin} \rangle/(\langle N_{part} \rangle /2)$ were
estimated based on
charged particle multiplicity for
$|\eta| \leq 0.5$ and full $2\pi$ azimuth acceptance.
The sources of systematic error considered here included
uncertainties associated with the model (nuclear density, nucleon-nucleon
inelastic total cross section,
multiplicity production model analytic form and parameters) and with the corrected
minimum-bias multiplicity frequency distribution data (background contamination,
trigger and collision vertex finding inefficiencies, and particle trajectory
reconstruction inefficiencies).
The TP centrality analysis method
was applied to the MCG predictions and error analysis for
six collision systems relevant to the RHIC heavy ion program.
The analysis showed that significantly reduced errors in the
collision geometry quantities result when the uncertainties in the
minimum-bias multiplicity frequency distribution data are constrained by the empirical power-law behavior and
the minimum-bias p-p collision data. The reduction in errors is in agreement
with the original TP power-law analysis;
the resulting errors are in general smaller
than previous estimates~\cite{starmc,glauber-rev,starspec}
which did not utilize the power-law and p-p constraints.
The accuracy of simple parametrizations of centrality dependent quantities
developed by Trainor and Prindle~\cite{powerlaw} was confirmed.
We also discussed how particle production data at the trigger level
({\em e.g.} scintillator hits and calorimeter energy depositions)
can be used within the power-law
and p-p constraint methodology~\cite{powerlaw} to define additional event cuts which
minimize background contamination and which
enable the detrimental effects of trigger inefficiencies to be reduced.
The systematic errors presented in this paper demonstrate the accuracy of collision geometry quantities which
can be achieved with RHIC data by exploiting the power-law behavior of the A-A
data and using constraints from minimum-bias p-p collision data. The results
presented here assume that
backgrounds from UPC events,
pileup, beam-gas collisions, etc. are not too large and that the minimum-bias
trigger and collision vertex reconstruction efficiencies are not too
small for peripheral collisions. It is intended that the MCG results
presented here will serve as a useful resource for the RHIC community
and that the analysis method developed by Trainor and Prindle~\cite{powerlaw}
and applied in this paper, together with the trigger data analysis method discussed here,
will enable accurate description of, and better
access to the heavy ion peripheral collision data from RHIC.
\ack
The authors gratefully acknowledge helpful discussions, detailed comments
and the plots in Fig.~\ref{Fig3} from Prof. T. A. Trainor (Univ. of Washington)
as well as helpful discussions and comments from Drs. J. C. Dunlop and
Zhangbu Xu (both from Brookhaven National Lab).
This research was supported in part by The U. S. Dept. of Energy Grant No.
DE-FG03-94ER40845.
\begin{table}[htb]
\begin{center}
\caption{\label{TableII}
Centrality-bin averaged collision geometry quantities and errors for Monte Carlo Glauber
Au-Au collisions at $\sqrt{s_{NN}}$ = 200~GeV using the nominal model
parameters discussed in the text.
Centrality was based on $N_{ch}$ for $|\eta|<0.5$ and full
$2\pi$ azimuthal acceptance.
Estimated positive and negative systematic errors are listed
as magnitudes and
percentages (in parentheses) of the mean values.
Errors in mean multiplicities were not
computed (see text).}
\vspace{0.1in}
\begin{tabular}{cccccc} \br
\multicolumn{6}{c}{{\bf Au-Au 200 GeV}} \\
Centrality (\%) & $\langle N_{ch} \rangle$ & $\langle b \rangle$ &
$\langle N_{part} \rangle$ & $\langle N_{bin} \rangle$ & $\nu$ \\
\mr
\vspace{0.2cm}
90-100 & 2.9 &$14.68^{+0.28(1.9)}_{-0.30(2.0)}$ &$ 2.9^{+0.3(8.6)}_{-0.2(6.5
)}$ &$ 1.8^{+ 0.2(12.4)}_{- 0.2(9.4)}$ &$1.23^{+0.04(3.0)}_{-0.03(2.4)}$\\
\vspace{0.2cm}
80-90 & 8.2 &$13.89^{+0.27(1.9)}_{-0.28(2.0)}$ &$ 6.4^{+0.3(4.2)}_{-0.2(3.9)
}$ &$ 4.9^{+ 0.3(6.5)}_{- 0.3(5.5)}$ &$1.51^{+0.03(2.3)}_{-0.03(1.9)}$\\
\vspace{0.2cm}
70-80 & 18.7 &$12.96^{+0.24(1.9)}_{-0.26(2.0)}$ &$ 14.1^{+0.6(4.2)}_{-0.5(3.3)
}$ &$ 12.7^{+ 0.9(7.2)}_{- 0.7(5.6)}$ &$1.81^{+0.05(2.9)}_{-0.04(2.4)}$\\
\vspace{0.2cm}
60-70 & 37.8 &$12.03^{+0.23(1.9)}_{-0.24(2.0)}$ &$ 27.2^{+1.1(4.0)}_{-0.9(3.2)
}$ &$ 29.5^{+ 2.3(7.9)}_{- 1.8(6.1)}$ &$2.17^{+0.08(3.8)}_{-0.07(3.0)}$\\
\vspace{0.2cm}
50-60 & 69.7 &$11.04^{+0.21(1.9)}_{-0.22(2.0)}$ &$ 47.5^{+1.4(2.9)}_{-1.1(2.4)
}$ &$ 62.5^{+ 4.6(7.3)}_{- 3.8(6.2)}$ &$2.63^{+0.11(4.2)}_{-0.10(3.8)}$\\
\vspace{0.2cm}
40-50 & 118.5 &$ 9.98^{+0.18(1.8)}_{-0.20(2.0)}$ &$ 76.3^{+1.8(2.4)}_{-1.4(1.8)
}$ &$ 120.7^{+ 8.3(6.9)}_{- 6.7(5.5)}$ &$3.16^{+0.14(4.4)}_{-0.12(3.9)}$\\
\vspace{0.2cm}
30-40 & 190.2 &$ 8.80^{+0.18(2.0)}_{-0.16(1.8)}$ &$115.5^{+1.4(1.2)}_{-1.4(1.3)
}$ &$ 216.2^{+11.8(5.5)}_{-10.7(4.9)}$ &$3.74^{+0.16(4.2)}_{-0.14(3.8)}$\\
\vspace{0.2cm}
20-30 & 291.5 &$ 7.43^{+0.14(1.9)}_{-0.13(1.7)}$ &$167.1^{+1.2(0.7)}_{-1.8(1.1)
}$ &$ 364.1^{+16.7(4.6)}_{-17.2(4.7)}$ &$4.36^{+0.17(3.9)}_{-0.16(3.7)}$\\
\vspace{0.2cm}
10-20 & 433.5 &$ 5.75^{+0.11(1.9)}_{-0.11(1.9)}$ &$234.8^{+1.2(0.5)}_{-1.6(0.7)
}$ &$ 586.9^{+24.9(4.2)}_{-26.1(4.4)}$ &$5.00^{+0.19(3.8)}_{-0.19(3.8)}$\\
\vspace{0.2cm}
5-10 & 577.6 &$ 4.05^{+0.08(1.9)}_{-0.06(1.5)}$ &$299.7^{+0.8(0.3)}_{-1.5(0.5)
}$ &$ 826.0^{+32.3(3.9)}_{-34.7(4.2)}$ &$5.51^{+0.20(3.7)}_{-0.21(3.7)}$\\
\vspace{0.2cm}
0-5 & 705.6 &$ 2.30^{+0.06(2.8)}_{-0.05(2.0)}$ &$350.6^{+1.7(0.5)}_{-2.0(0.6)}
$ &$1043.6^{+43.2(4.1)}_{-44.1(4.2)}$ &$5.95^{+0.23(3.8)}_{-0.23(3.8)}$\\
\br
\end{tabular}
\end{center}
\end{table}
\begin{table*}[t]
\begin{center}
\caption{\label{TableIII}
Same as Table~\ref{TableII} except for Au-Au collisions at
$\sqrt{s_{NN}}$ = 130~GeV.}
\vspace{0.1in}
\begin{tabular}{cccccc} \hline \hline
\multicolumn{6}{c}{{\bf Au-Au 130 GeV}} \\
Centrality (\%) & $\langle N_{ch} \rangle$ & $\langle b \rangle$ &
$\langle N_{part} \rangle$ & $\langle N_{bin} \rangle$ & $\nu$ \\
\hline
\vspace{0.2cm}
90-100 & 2.7 &$14.61^{+0.30(2.1)}_{-0.30(2.0)}$ &$ 3.0^{+0.2(7.8)}_{-0.2(6.2
)}$ &$ 1.8^{+ 0.2(11.1)}_{- 0.2(8.7)}$ &$1.24^{+0.04(2.9)}_{-0.03(2.3)}$\\
\vspace{0.2cm}
80-90 & 7.4 &$13.83^{+0.27(1.9)}_{-0.27(2.0)}$ &$ 6.4^{+0.3(4.2)}_{-0.2(3.5)
}$ &$ 4.8^{+ 0.3(6.4)}_{- 0.2(5.0)}$ &$1.50^{+0.03(2.1)}_{-0.03(1.7)}$\\
\vspace{0.2cm}
70-80 & 16.6 &$12.90^{+0.25(1.9)}_{-0.25(2.0)}$ &$ 14.0^{+0.6(4.2)}_{-0.5(3.6)
}$ &$ 12.5^{+ 0.9(7.2)}_{- 0.7(5.9)}$ &$1.78^{+0.05(2.9)}_{-0.04(2.4)}$\\
\vspace{0.2cm}
60-70 & 33.2 &$11.97^{+0.24(2.0)}_{-0.24(2.0)}$ &$ 27.0^{+1.1(4.0)}_{-0.9(3.5)
}$ &$ 28.7^{+ 2.3(7.9)}_{- 1.9(6.5)}$ &$2.13^{+0.08(3.8)}_{-0.07(3.1)}$\\
\vspace{0.2cm}
50-60 & 60.1 &$10.99^{+0.22(2.0)}_{-0.22(2.0)}$ &$ 47.0^{+1.5(3.2)}_{-1.3(2.8)
}$ &$ 60.1^{+ 4.6(7.6)}_{- 3.8(6.3)}$ &$2.56^{+0.11(4.2)}_{-0.09(3.5)}$\\
\vspace{0.2cm}
40-50 & 100.4 &$ 9.93^{+0.19(1.9)}_{-0.19(1.9)}$ &$ 75.5^{+1.7(2.2)}_{-1.5(2.0)
}$ &$ 115.1^{+ 7.3(6.4)}_{- 6.8(5.9)}$ &$3.05^{+0.13(4.1)}_{-0.12(4.0)}$\\
\vspace{0.2cm}
30-40 & 157.8 &$ 8.76^{+0.15(1.8)}_{-0.18(2.1)}$ &$114.0^{+1.8(1.6)}_{-1.2(1.0)
}$ &$ 203.9^{+12.1(5.9)}_{- 9.3(4.6)}$ &$3.58^{+0.15(4.2)}_{-0.13(3.7)}$\\
\vspace{0.2cm}
20-30 & 237.7 &$ 7.41^{+0.11(1.4)}_{-0.16(2.2)}$ &$164.9^{+2.3(1.4)}_{-0.8(0.5)
}$ &$ 341.0^{+18.7(5.5)}_{-14.0(4.1)}$ &$4.14^{+0.17(4.1)}_{-0.15(3.7)}$\\
\vspace{0.2cm}
10-20 & 348.7 &$ 5.72^{+0.10(1.8)}_{-0.12(2.0)}$ &$232.4^{+1.6(0.7)}_{-1.0(0.4)
}$ &$ 548.1^{+25.1(4.6)}_{-22.2(4.0)}$ &$4.72^{+0.18(3.9)}_{-0.17(3.6)}$\\
\vspace{0.2cm}
5-10 & 459.5 &$ 4.05^{+0.05(1.1)}_{-0.08(2.0)}$ &$296.8^{+1.2(0.4)}_{-0.6(0.2)
}$ &$ 767.9^{+32.6(4.2)}_{-29.6(3.9)}$ &$5.17^{+0.20(3.9)}_{-0.19(3.7)}$\\
\vspace{0.2cm}
0-5 & 557.6 &$ 2.29^{+0.06(2.7)}_{-0.06(2.6)}$ &$348.2^{+2.0(0.6)}_{-1.9(0.5)}
$ &$ 968.0^{+42.3(4.4)}_{-40.4(4.2)}$ &$5.56^{+0.22(3.9)}_{-0.21(3.8)}$\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t]
\begin{center}
\caption{\label{TableIV}
Same as Table~\ref{TableII} except for Au-Au collisions at
$\sqrt{s_{NN}}$ = 62~GeV.}
\vspace{0.1in}
\begin{tabular}{cccccc} \hline \hline
\multicolumn{6}{c}{{\bf Au-Au 62 GeV}} \\
Centrality (\%) & $\langle N_{ch} \rangle$ & $\langle b \rangle$ &
$\langle N_{part} \rangle$ & $\langle N_{bin} \rangle$ & $\nu$ \\
\hline
\vspace{0.2cm}
90-100 & 2.4 &$14.52^{+0.29(2.0)}_{-0.29(2.0)}$ &$ 3.1^{+0.1(4.4)}_{-0.1(4.1
)}$ &$ 1.9^{+ 0.1(6.3)}_{- 0.1(5.5)}$ &$1.24^{+0.02(1.9)}_{-0.02(1.4)}$\\
\vspace{0.2cm}
80-90 & 6.6 &$13.77^{+0.25(1.8)}_{-0.29(2.1)}$ &$ 6.4^{+0.3(4.9)}_{-0.2(2.8)
}$ &$ 4.8^{+ 0.3(7.1)}_{- 0.2(4.3)}$ &$1.49^{+0.03(2.3)}_{-0.02(1.5)}$\\
\vspace{0.2cm}
70-80 & 14.6 &$12.83^{+0.24(1.8)}_{-0.25(1.9)}$ &$ 13.9^{+0.6(4.4)}_{-0.4(3.1)
}$ &$ 12.1^{+ 0.9(7.6)}_{- 0.6(5.2)}$ &$1.75^{+0.05(3.0)}_{-0.04(2.2)}$\\
\vspace{0.2cm}
60-70 & 29.0 &$11.91^{+0.21(1.7)}_{-0.24(2.0)}$ &$ 26.6^{+1.2(4.4)}_{-0.7(2.7)
}$ &$ 27.5^{+ 2.2(8.1)}_{- 1.5(5.5)}$ &$2.07^{+0.07(3.6)}_{-0.06(3.0)}$\\
\vspace{0.2cm}
50-60 & 52.3 &$10.94^{+0.19(1.7)}_{-0.22(2.0)}$ &$ 46.2^{+1.5(3.3)}_{-1.1(2.3)
}$ &$ 56.8^{+ 4.2(7.4)}_{- 3.2(5.6)}$ &$2.46^{+0.10(4.1)}_{-0.09(3.5)}$\\
\vspace{0.2cm}
40-50 & 87.1 &$ 9.88^{+0.17(1.7)}_{-0.18(1.8)}$ &$ 74.2^{+1.9(2.6)}_{-1.1(1.5)
}$ &$ 107.8^{+ 7.4(6.9)}_{- 5.3(4.9)}$ &$2.90^{+0.12(4.3)}_{-0.10(3.6)}$\\
\vspace{0.2cm}
30-40 & 136.9 &$ 8.71^{+0.15(1.7)}_{-0.17(1.9)}$ &$112.4^{+2.0(1.8)}_{-1.3(1.2)
}$ &$ 190.2^{+11.5(6.0)}_{- 8.7(4.6)}$ &$3.38^{+0.14(4.2)}_{-0.12(3.7)}$\\
\vspace{0.2cm}
20-30 & 205.5 &$ 7.36^{+0.12(1.6)}_{-0.15(2.0)}$ &$162.6^{+2.0(1.2)}_{-0.9(0.6)
}$ &$ 315.2^{+16.6(5.3)}_{-12.3(3.9)}$ &$3.88^{+0.16(4.1)}_{-0.14(3.5)}$\\
\vspace{0.2cm}
10-20 & 300.7 &$ 5.69^{+0.09(1.6)}_{-0.12(2.1)}$ &$229.2^{+1.9(0.8)}_{-1.0(0.4)
}$ &$ 503.6^{+23.6(4.7)}_{-20.4(4.0)}$ &$4.39^{+0.17(3.9)}_{-0.17(3.8)}$\\
\vspace{0.2cm}
5-10 & 395.5 &$ 4.02^{+0.06(1.4)}_{-0.07(1.8)}$ &$293.3^{+1.3(0.5)}_{-0.7(0.2)
}$ &$ 703.5^{+30.6(4.3)}_{-28.3(4.0)}$ &$4.80^{+0.19(3.9)}_{-0.18(3.8)}$\\
\vspace{0.2cm}
0-5 & 480.8 &$ 2.28^{+0.05(2.4)}_{-0.06(2.4)}$ &$345.0^{+1.8(0.5)}_{-2.0(0.6)}
$ &$ 884.9^{+38.8(4.4)}_{-39.5(4.5)}$ &$5.13^{+0.20(4.0)}_{-0.21(4.0)}$\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t]
\begin{center}
\caption{\label{TableV}
Same as Table~\ref{TableII} except for Au-Au collisions at
$\sqrt{s_{NN}}$ = 20~GeV.}
\vspace{0.1in}
\begin{tabular}{cccccc} \hline \hline
\multicolumn{6}{c}{{\bf Au-Au 20 GeV}} \\
Centrality (\%) & $\langle N_{ch} \rangle$ & $\langle b \rangle$ &
$\langle N_{part} \rangle$ & $\langle N_{bin} \rangle$ & $\nu$ \\
\hline
\vspace{0.2cm}
90-100 & 1.9 &$14.38^{+0.27(1.9)}_{-0.30(2.1)}$ &$ 3.4^{+0.1(3.9)}_{-0.1(1.9
)}$ &$ 2.2^{+ 0.1(6.0)}_{- 0.1(3.3)}$ &$1.29^{+0.02(1.9)}_{-0.02(1.3)}$\\
\vspace{0.2cm}
80-90 & 4.8 &$13.65^{+0.27(2.0)}_{-0.27(2.0)}$ &$ 6.9^{+0.4(6.3)}_{-0.4(5.3)
}$ &$ 5.2^{+ 0.4(8.5)}_{- 0.3(6.7)}$ &$1.51^{+0.04(2.4)}_{-0.03(1.9)}$\\
\vspace{0.2cm}
70-80 & 10.5 &$12.72^{+0.24(1.9)}_{-0.27(2.1)}$ &$ 14.6^{+0.9(6.0)}_{-0.6(4.2)
}$ &$ 12.8^{+ 1.2(9.1)}_{- 0.7(5.8)}$ &$1.75^{+0.06(3.3)}_{-0.04(2.1)}$\\
\vspace{0.2cm}
60-70 & 20.5 &$11.79^{+0.22(1.9)}_{-0.26(2.2)}$ &$ 27.6^{+1.5(5.5)}_{-1.1(4.1)
}$ &$ 28.4^{+ 2.6(9.3)}_{- 1.9(6.7)}$ &$2.06^{+0.08(4.0)}_{-0.06(3.1)}$\\
\vspace{0.2cm}
50-60 & 36.2 &$10.82^{+0.20(1.8)}_{-0.23(2.1)}$ &$ 47.3^{+2.2(4.5)}_{-1.5(3.2)
}$ &$ 57.4^{+ 4.9(8.5)}_{- 3.5(6.1)}$ &$2.42^{+0.10(4.2)}_{-0.09(3.5)}$\\
\vspace{0.2cm}
40-50 & 59.2 &$ 9.78^{+0.19(1.9)}_{-0.20(2.1)}$ &$ 75.2^{+2.6(3.5)}_{-1.9(2.5)
}$ &$ 106.6^{+ 8.2(7.6)}_{- 6.0(5.6)}$ &$2.83^{+0.12(4.4)}_{-0.11(3.7)}$\\
\vspace{0.2cm}
30-40 & 91.6 &$ 8.62^{+0.16(1.9)}_{-0.18(2.1)}$ &$113.2^{+3.1(2.7)}_{-2.2(1.9)
}$ &$ 185.1^{+12.3(6.6)}_{- 9.6(5.2)}$ &$3.27^{+0.14(4.2)}_{-0.13(3.9)}$\\
\vspace{0.2cm}
20-30 & 135.9 &$ 7.27^{+0.15(2.1)}_{-0.15(2.1)}$ &$163.3^{+3.0(1.8)}_{-2.5(1.5)
}$ &$ 304.2^{+16.7(5.5)}_{-14.9(4.9)}$ &$3.72^{+0.15(4.0)}_{-0.14(3.8)}$\\
\vspace{0.2cm}
10-20 & 196.1 &$ 5.61^{+0.11(2.0)}_{-0.12(2.2)}$ &$229.1^{+2.8(1.2)}_{-2.3(1.0)
}$ &$ 479.5^{+24.6(5.1)}_{-22.1(4.6)}$ &$4.19^{+0.17(4.1)}_{-0.16(3.9)}$\\
\vspace{0.2cm}
5-10 & 255.4 &$ 3.96^{+0.07(1.8)}_{-0.08(2.1)}$ &$292.2^{+2.2(0.8)}_{-2.1(0.7)
}$ &$ 664.2^{+31.6(4.8)}_{-30.9(4.7)}$ &$4.55^{+0.19(4.1)}_{-0.18(4.0)}$\\
\vspace{0.2cm}
0-5 & 309.2 &$ 2.29^{+0.07(3.2)}_{-0.07(3.1)}$ &$342.2^{+2.5(0.7)}_{-2.7(0.8)}
$ &$ 825.9^{+38.2(4.6)}_{-38.5(4.7)}$ &$4.83^{+0.19(4.0)}_{-0.20(4.1)}$\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t]
\begin{center}
\caption{\label{TableVI}
Same as Table~\ref{TableII} except for Cu-Cu collisions at
$\sqrt{s_{NN}}$ = 200~GeV.}
\vspace{0.1in}
\begin{tabular}{cccccc} \hline \hline
\multicolumn{6}{c}{{\bf Cu-Cu 200 GeV}} \\
Centrality (\%) & $\langle N_{ch} \rangle$ & $\langle b \rangle$ &
$\langle N_{part} \rangle$ & $\langle N_{bin} \rangle$ & $\nu$ \\
\hline
\vspace{0.2cm}
90-100 & 2.2 &$10.18^{+0.22(2.1)}_{-0.22(2.1)}$ &$ 2.6^{+0.2(7.0)}_{-0.2(7.1
)}$ &$ 1.5^{+ 0.2(10.8)}_{- 0.2(10.6)}$ &$1.18^{+0.04(3.1)}_{-0.04(3.0)}$\\
\vspace{0.2cm}
80-90 & 4.7 &$ 9.79^{+0.22(2.2)}_{-0.20(2.0)}$ &$ 3.8^{+0.2(4.5)}_{-0.2(4.7)
}$ &$ 2.6^{+ 0.2(6.6)}_{- 0.2(6.7)}$ &$1.35^{+0.02(1.8)}_{-0.02(1.8)}$\\
\vspace{0.2cm}
70-80 & 8.4 &$ 9.10^{+0.20(2.2)}_{-0.19(2.1)}$ &$ 6.6^{+0.2(2.3)}_{-0.2(2.8)
}$ &$ 5.0^{+ 0.2(4.1)}_{- 0.2(4.4)}$ &$1.53^{+0.03(1.7)}_{-0.03(1.8)}$\\
\vspace{0.2cm}
60-70 & 14.3 &$ 8.40^{+0.19(2.2)}_{-0.18(2.1)}$ &$ 10.9^{+0.3(2.6)}_{-0.3(2.9)
}$ &$ 9.3^{+ 0.4(4.5)}_{- 0.5(4.9)}$ &$1.71^{+0.03(2.0)}_{-0.04(2.2)}$\\
\vspace{0.2cm}
50-60 & 23.3 &$ 7.68^{+0.17(2.2)}_{-0.14(1.8)}$ &$ 17.2^{+0.4(2.1)}_{-0.5(2.6)
}$ &$ 16.7^{+ 0.7(4.3)}_{- 0.8(5.1)}$ &$1.94^{+0.05(2.4)}_{-0.05(2.5)}$\\
\vspace{0.2cm}
40-50 & 36.6 &$ 6.92^{+0.14(2.0)}_{-0.14(2.0)}$ &$ 26.1^{+0.5(1.8)}_{-0.6(2.4)
}$ &$ 28.9^{+ 1.3(4.5)}_{- 1.5(5.2)}$ &$2.21^{+0.06(2.8)}_{-0.06(2.8)}$\\
\vspace{0.2cm}
30-40 & 55.4 &$ 6.10^{+0.13(2.1)}_{-0.13(2.2)}$ &$ 38.1^{+0.6(1.7)}_{-0.8(2.0)
}$ &$ 48.2^{+ 2.3(4.8)}_{- 2.5(5.1)}$ &$2.53^{+0.08(3.1)}_{-0.08(3.1)}$\\
\vspace{0.2cm}
20-30 & 81.5 &$ 5.15^{+0.10(2.0)}_{-0.11(2.2)}$ &$ 53.9^{+0.8(1.5)}_{-0.9(1.7)
}$ &$ 77.9^{+ 3.9(5.0)}_{- 3.9(5.0)}$ &$2.89^{+0.10(3.4)}_{-0.10(3.4)}$\\
\vspace{0.2cm}
10-20 & 117.4 &$ 3.97^{+0.09(2.2)}_{-0.07(1.9)}$ &$ 74.4^{+0.8(1.1)}_{-1.1(1.5)
}$ &$ 123.0^{+ 5.6(4.6)}_{- 6.1(4.9)}$ &$3.31^{+0.11(3.4)}_{-0.12(3.5)}$\\
\vspace{0.2cm}
5-10 & 152.2 &$ 2.80^{+0.05(1.9)}_{-0.05(1.9)}$ &$ 93.2^{+0.8(0.8)}_{-1.0(1.1)
}$ &$ 169.9^{+ 7.4(4.3)}_{- 7.9(4.7)}$ &$3.65^{+0.13(3.6)}_{-0.13(3.6)}$\\
\vspace{0.2cm}
0-5 & 184.8 &$ 1.75^{+0.11(6.5)}_{-0.10(5.9)}$ &$106.5^{+1.5(1.4)}_{-1.6(1.5)}
$ &$ 211.4^{+10.1(4.8)}_{-10.5(5.0)}$ &$3.97^{+0.15(3.8)}_{-0.15(3.8)}$\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t]
\begin{center}
\caption{\label{TableVII}
Same as Table~\ref{TableII} except for Cu-Cu collisions at
$\sqrt{s_{NN}}$ = 62~GeV.}
\vspace{0.1in}
\begin{tabular}{cccccc} \hline \hline
\multicolumn{6}{c}{{\bf Cu-Cu 62 GeV}} \\
Centrality (\%) & $\langle N_{ch} \rangle$ & $\langle b \rangle$ &
$\langle N_{part} \rangle$ & $\langle N_{bin} \rangle$ & $\nu$ \\
\hline
\vspace{0.2cm}
90-100 & 1.8 &$10.00^{+0.21(2.1)}_{-0.21(2.1)}$ &$ 2.8^{+0.1(3.4)}_{-0.1(2.7
)}$ &$ 1.7^{+ 0.1(5.1)}_{- 0.1(4.4)}$ &$1.20^{+0.02(1.4)}_{-0.02(1.3)}$\\
\vspace{0.2cm}
80-90 & 3.8 &$ 9.65^{+0.20(2.1)}_{-0.20(2.1)}$ &$ 3.8^{+0.1(2.9)}_{-0.1(3.3)
}$ &$ 2.6^{+ 0.1(4.2)}_{- 0.1(4.1)}$ &$1.34^{+0.02(1.1)}_{-0.02(1.3)}$\\
\vspace{0.2cm}
70-80 & 6.7 &$ 8.97^{+0.21(2.3)}_{-0.17(1.9)}$ &$ 6.5^{+0.1(2.2)}_{-0.2(3.2)
}$ &$ 4.9^{+ 0.2(3.3)}_{- 0.2(4.7)}$ &$1.50^{+0.02(1.3)}_{-0.02(1.6)}$\\
\vspace{0.2cm}
60-70 & 11.2 &$ 8.27^{+0.17(2.1)}_{-0.16(2.0)}$ &$ 10.6^{+0.3(2.7)}_{-0.3(2.8)
}$ &$ 8.8^{+ 0.4(4.6)}_{- 0.4(4.5)}$ &$1.66^{+0.03(2.0)}_{-0.03(1.8)}$\\
\vspace{0.2cm}
50-60 & 17.9 &$ 7.57^{+0.15(1.9)}_{-0.15(2.0)}$ &$ 16.7^{+0.4(2.2)}_{-0.5(2.9)
}$ &$ 15.5^{+ 0.7(4.5)}_{- 0.8(5.2)}$ &$1.85^{+0.04(2.3)}_{-0.04(2.4)}$\\
\vspace{0.2cm}
40-50 & 27.7 &$ 6.82^{+0.14(2.0)}_{-0.13(1.9)}$ &$ 25.3^{+0.4(1.8)}_{-0.7(2.6)
}$ &$ 26.3^{+ 1.1(4.3)}_{- 1.4(5.2)}$ &$2.08^{+0.05(2.6)}_{-0.06(2.7)}$\\
\vspace{0.2cm}
30-40 & 41.3 &$ 6.00^{+0.13(2.1)}_{-0.11(1.8)}$ &$ 36.8^{+0.5(1.5)}_{-0.8(2.2)
}$ &$ 43.2^{+ 1.9(4.3)}_{- 2.2(5.1)}$ &$2.35^{+0.07(3.0)}_{-0.07(3.1)}$\\
\vspace{0.2cm}
20-30 & 60.0 &$ 5.06^{+0.11(2.1)}_{-0.08(1.6)}$ &$ 52.1^{+0.5(1.0)}_{-1.0(1.9)
}$ &$ 69.0^{+ 2.5(3.6)}_{- 3.5(5.1)}$ &$2.65^{+0.07(2.7)}_{-0.09(3.3)}$\\
\vspace{0.2cm}
10-20 & 84.9 &$ 3.91^{+0.07(1.8)}_{-0.06(1.6)}$ &$ 71.8^{+0.7(1.0)}_{-1.0(1.4)
}$ &$ 106.8^{+ 4.5(4.2)}_{- 5.3(5.0)}$ &$2.97^{+0.09(3.1)}_{-0.11(3.6)}$\\
\vspace{0.2cm}
5-10 & 108.9 &$ 2.76^{+0.05(1.7)}_{-0.03(1.3)}$ &$ 90.1^{+0.8(0.9)}_{-1.1(1.2)
}$ &$ 145.7^{+ 6.5(4.5)}_{- 6.9(4.7)}$ &$3.23^{+0.12(3.6)}_{-0.12(3.6)}$\\
\vspace{0.2cm}
0-5 & 132.2 &$ 1.77^{+0.10(5.6)}_{-0.08(4.7)}$ &$103.2^{+1.4(1.3)}_{-1.5(1.5)}
$ &$ 178.4^{+ 9.0(5.0)}_{- 9.1(5.1)}$ &$3.46^{+0.14(4.0)}_{-0.14(3.9)}$\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}
\begin{table}[htb]
\caption{\label{TableVIII}
Absolute values of individual error contributions (in percent) to collision geometry bin average
quantities for Au-Au collisions at $\sqrt{s_{NN}}$ = 200~GeV
relative to the reference values in Table~\ref{TableII} as explained in the text. Errors are
denoted as $\Delta \langle b \rangle$, etc. The left-most column lists the
sources of uncertainty. Average errors for centrality bins 60-100\%, 20-60\%
and 0-20\% are listed from left to right, respectively, for each instance. }
\begin{indented}
\vspace{0.1in}
\item[]\begin{tabular}{@{}ccccc} \br
Error & \multicolumn{4}{c}{{\bf Au-Au 200 GeV Errors}} \\
Source & $\Delta \langle b \rangle$(\%) & $\Delta \langle N_{part} \rangle$(\%) & $\Delta \langle N_{bin} \rangle$(\%) & $\Delta \nu$(\%) \\
\mr
$c_{pt,m}$ & 0.9, 1.0, 0.8 \hspace{0.1in} & 0.1, 0.1, 0.2 \hspace{0.1in} &
0.1, 0.9, 2.2 \hspace{0.1in} & 0.0, 0.9, 2.0 \\
$z_{pt,m}$ & 1.7, 1.6, 1.5 \hspace{0.1in} & 3.2, 1.6, 0.3 \hspace{0.1in} &
5.2, 5.1, 2.6 \hspace{0.1in} & 2.1, 3.5, 2.3 \\
$\sigma_{inel}$ & 0.1, 0.1, 0.0 \hspace{0.1in} & 0.4, 0.5, 0.3 \hspace{0.1in} &
1.0, 2.0, 2.5 \hspace{0.1in} & 0.6, 1.5, 2.2 \\
$x$ & 0.0, 0.0, 0.1 \hspace{0.1in} & 0.2, 0.0, 0.0 \hspace{0.1in} &
0.3, 0.0, 0.0 \hspace{0.1in} & 0.2, 0.1, 0.0 \\
$a$ & 0.0, 0.0, 0.2 \hspace{0.1in} & 0.3, 0.1, 0.0 \hspace{0.1in} &
0.4, 0.2, 0.1 \hspace{0.1in} & 0.1, 0.1, 0.0 \\
$\cal P$-form & 0.1, 0.0, 0.7 \hspace{0.1in} & 2.2, 0.1, 0.2 \hspace{0.1in} &
3.5, 0.3, 0.3 \hspace{0.1in} & 1.2, 0.2, 0.1 \\
trigger
& 0.0, 0.0, 0.0 \hspace{0.1in} & 0.1, 0.0, 0.0 \hspace{0.1in} &
0.1, 0.0, 0.0 \hspace{0.1in} & 0.0, 0.0, 0.0 \\
vertex & 0.0, 0.0, 0.1 \hspace{0.1in} & 0.1, 0.0, 0.0 \hspace{0.1in} &
0.1, 0.0, 0.0 \hspace{0.1in} & 0.0, 0.0, 0.0 \\
tracking
& 0.0, 0.0, 0.1 \hspace{0.1in} & 0.0, 0.0, 0.0 \hspace{0.1in} &
0.0, 0.0, 0.0 \hspace{0.1in} & 0.0, 0.0, 0.0 \\
\br
\end{tabular}
\end{indented}
\end{table}
\begin{table}[htb]
\caption{\label{TableIX}
Same as Table~\ref{TableVIII} except absolute values of errors (in percent) for Cu-Cu collisions at
$\sqrt{s_{NN}}$ = 200~GeV relative to the reference values in Table~\ref{TableVI} as explained in the text.}
\begin{indented}
\vspace{0.1in}
\item[]\begin{tabular}{@{}ccccc} \br
Error & \multicolumn{4}{c}{{\bf Cu-Cu 200 GeV Errors}} \\
Source & $\Delta \langle b \rangle$(\%) & $\Delta \langle N_{part} \rangle$(\%) & $\Delta \langle N_{bin} \rangle$(\%) & $\Delta \nu$(\%) \\
\mr
$c_{pt,m}$ & 1.0, 1.0, 1.0 \hspace{0.1in} & 0.2, 0.4, 0.5 \hspace{0.1in} &
0.2, 1.1, 2.6 \hspace{0.1in} & 0.2, 0.8, 2.0 \\
$z_{pt,m}$ & 1.9, 1.8, 1.7 \hspace{0.1in} & 2.1, 1.8, 0.8 \hspace{0.1in} &
3.2, 4.3, 3.2 \hspace{0.1in} & 1.2, 2.5, 2.3 \\
$\sigma_{inel}$ & 0.2, 0.2, 0.2 \hspace{0.1in} & 0.2, 0.4, 0.4 \hspace{0.1in} &
0.5, 1.5, 2.2 \hspace{0.1in} & 0.3, 1.1, 1.8 \\
$x$ & 0.0, 0.0, 0.0 \hspace{0.1in} & 0.0, 0.0, 0.0 \hspace{0.1in} &
0.2, 0.0, 0.1 \hspace{0.1in} & 0.2, 0.1, 0.1 \\
$a$ & 0.1, 0.1, 0.3 \hspace{0.1in} & 0.8, 0.2, 0.1 \hspace{0.1in} &
1.2, 0.4, 0.2 \hspace{0.1in} & 0.4, 0.2, 0.1 \\
$\cal P$-form & 0.3, 0.1, 1.9 \hspace{0.1in} & 2.8, 0.1, 0.5 \hspace{0.1in} &
4.7, 0.5, 0.6 \hspace{0.1in} & 1.5, 0.5, 0.3 \\
trigger
& 0.0, 0.0, 0.1 \hspace{0.1in} & 0.2, 0.1, 0.0 \hspace{0.1in} &
0.2, 0.1, 0.0 \hspace{0.1in} & 0.1, 0.0, 0.0 \\
vertex & 0.0, 0.0, 0.0 \hspace{0.1in} & 0.0, 0.0, 0.0 \hspace{0.1in} &
0.2, 0.0, 0.0 \hspace{0.1in} & 0.0, 0.0, 0.0 \\
tracking
& 0.0, 0.0, 0.0 \hspace{0.1in} & 0.0, 0.0, 0.0 \hspace{0.1in} &
0.0, 0.0, 0.0 \hspace{0.1in} & 0.0, 0.0, 0.0 \\
\br
\end{tabular}
\end{indented}
\end{table}
\begin{table}[htb]
\caption{\label{TableX}
Percent changes in collision geometry bin average quantities relative to the
reference values in Tables~\ref{TableII} - \ref{TableVII}
due to background contamination
of the peripheral collision events as explained in Sec.~\ref{Sec:results}.
Background contamination errors
for centrality bins not listed were negligible.}
\begin{indented}
\vspace{0.1in}
\item[]\begin{tabular}{@{}cccccc} \br
& \multicolumn{5}{c}{{\bf Background Contamination Errors}} \\
System & Centrality & $\Delta \langle b \rangle$ &
$\Delta \langle N_{part} \rangle$ & $\Delta \langle N_{bin} \rangle$
& $\Delta \nu$ \\
& (percent) & (\%) & (\%) & (\%) & (\%) \\
\mr
Au-Au 200~GeV & 90-100 & -0.3 & 5.6 & 7.9 & 1.8 \\
23\% contamination & 80-90 & 0.2 & -1.6 & -1.7 & -0.1 \\
\vspace{0.2cm}
in 88-100\% bin & & & & & \\
Au-Au 130~GeV & 90-100 & -0.3 & 4.8 & 6.7 & 1.6 \\
22\% contamination & 80-90 & 0.2 & -1.5 & -1.5 & 0.0 \\
\vspace{0.2cm}
in 91-100\% bin & & & & & \\
Au-Au 62~GeV & 90-100 & -0.1 & 1.3 & 1.6 & 0.3 \\
18\% contamination & 10-50$^{\rm a}$
& -0.1 & 0.2 & 0.4 & 0.1 \\
\vspace{0.2cm}
in 96-100\% bin & & & & & \\
\hline
Au-Au 20~GeV$^{\rm b}$
& 90-100 & -0.1 & 2.7 & 3.5 & 0.8 \\
& 80-90 & 0.0 & 0.0 & 0.1 & 0.0 \\
\vspace{0.2cm}
& 40-70$^{\rm a}$ & -0.1 & 0.5 & 0.7 & 0.2 \\
\vspace{0.2cm}
Cu-Cu 200~GeV$^{\rm b}$ & 90-100 & -0.2 & 1.7 & 2.3 & 0.8 \\
\vspace{0.2cm}
Cu-Cu 62~GeV$^{\rm b}$ & 90-100 & -0.1 & 1.5 & 2.0 & 0.6 \\
\hline \hline
\end{tabular}
\item[] $^{\rm a}$ Average percent changes within the combined centrality bin.
\item[] $^{\rm b}$ Reference uncertainties assuming 10\% background contamination in the number of collisions in the
nominal 90-100\% centrality bin, equivalent to 1\% overall background.
\end{indented}
\end{table}
\begin{table}[h]
\caption{\label{TableXI}
Upper half-max end-point positions $N_{part,max}$ and $N_{bin,max}$ from the
power-law distributions $d\sigma/d(N_{part}/2)^{1/4}$ and
$d\sigma/dN_{bin}^{1/6}$, respectively, for the six collision systems studied here.}
\begin{indented}
\item[]\begin{tabular}{@{}ccc} \br
System & $N_{part,max}$ & $N_{bin,max}$ \\
\mr
Au-Au 200 GeV & 379.3 & 1166 \\
Au-Au 130 GeV & 377.5 & 1082 \\
Au-Au 62 GeV & 374.9 & 988 \\
Au-Au 20 GeV & 373.4 & 926 \\
Cu-Cu 200 GeV & 115.8 & 230 \\
Cu-Cu 62 GeV & 112.7 & 196 \\
\br
\end{tabular}
\end{indented}
\end{table}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,207 |
\section{Introduction}
Let $R$ be a complete discrete valuation ring, $K$ its field of fractions, and $k$ its residue field. We assume moreover that $k$ is algebraically closed and $\Char k\ne2$. Let $\nu:K\to\Z\cup\{\infty\}$ be the normalized valuation of $K$.
Let $C$ be a smooth projective geometrically connected curve of genus $g>0$ over $K$. There exists a regular $R$-curve $X$ whose generic fiber is isomorphic to $C$ and such that if $X'$ is another regular $R$-curve whose generic fiber is isomorphic to $C$ then $X'$ dominates $X$. $X$ is said to be the minimal regular model of $C$. The type of the minimal regular model is determined by the structure of its special fiber $X_k$ which we are going to call the reduction type of $X$.
If $C$ is an elliptic curve, then given a Weierstrass equation describing $C$, Tate's algorithm produces the minimal regular model of $C$ by analyzing the $a$- and $b$-invariants of $C$, see \cite[Chapter IV, \S 9]{Silverman}. If $C$ has genus $2$, then the complete classification of $X_k$ can be found in \cite{NamikawaUeno}. The reduction type of $X$ takes one out of more than $120$ possibilities.
Let $C$ be a curve of genus $2$. Let $L/K$ be the smallest field extension over which $C$ admits a stable model. In \cite{Liumodelesminimaux}, using the stable model of $C$ together with a careful analysis of the Igusa and affine invariants attached to $C$, Liu reproduced the reduction type of the minimal regular model of $C$.
If $K(\sqrt{D})$ is a quadratic extension of $K$ with associated character $\chi$, we denote the corresponding quadratic twist of the curve $C$ by $C^{\chi}$. If $C$ is an elliptic curve, given the reduction type of the minimal regular model $X$ of $C$, one can find the reduction type of the minimal regular model $X^{\chi}$ of $C^{\chi}$. Indeed, if $\nu(D)$ is even, then the reduction type of $X^{\chi}$ is the same as the one of $X$. If $\nu(D)$ is odd, then the complete list of reduction types of $X^{\chi}$ can be found in \cite{Comalada}. For example, if $\Char k\ne 2$, then
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
type of $X_k$ & ${\rm I}_0$ & ${\rm I}_n$ & ${\rm II}$ & ${\rm III}$ & ${\rm IV}$ & ${\rm I}_0^*$ & ${\rm I}_n^*$ & ${\rm IV}^*$ &${\rm III}^*$&${\rm II}^*$\\
\hline
type of $X^{\chi}_k$ & ${\rm I}_0^*$ & ${\rm I}_n^*$ & ${\rm IV}^*$ & ${\rm III}^*$ & ${\rm II}^*$ & ${\rm I}_0$ & ${\rm I}_n$ & ${\rm II}$&${\rm III}$&${\rm IV}$ \\
\hline
\end{tabular}
\end{center}
\vskip10pt
If $C$ has genus $g>1$, then one may pose the same question on the reduction type of $X^{\chi}$. In \cite{Sadek}, if $C$ has genus $g>0$, the description of the special fiber of $X^{\chi}$ is given if $X$ is smooth. In this article, given the reduction type of $X$, we display the reduction type of $X^{\chi}$ if the genus of $C$ is $g=2$.
We first investigate the Igusa and affine invariants attached to $C^{\chi}$ and link these invariants to the ones attached to $C$. This enables us to describe the stable model $\CC^{\chi}$ of $C^{\chi}$ and compute the degrees of singularity of its ordinary double points. Then one will be able to construct the minimal regular model $X^{\chi}$ from $\CC^{\chi}$.
\section{Invariants of genus two curves}
Let $C$ be a smooth projective geometrically connected genus two curve defined over a field $K$. If $\Char K\ne 2$, then $C$ is defined by a hyperelliptic equation of the form $y^2=P(x)$ where $P(x)$ is a polynomial in $K[x]$ of degree $5$ or $6$ with no repeated roots. If $K$ is the fraction field of some ring $R$, one may assume that $P(x)\in R[x]$. Let $f:C\to\PP_K^1$ be a finite separable morphism of degree $2$. Let $\sigma$ be the hyperelliptic involution of $C$. A point $x\in C(K)$ is a ramification point of $f$ if $\sigma(x)=x$. In particular, the ramification points of $f$ are the zeros of $P(x)$, plus the point at infinity if $\deg P(x)=5$. Assuming that $z^2=Q(u)$ is another hyperelliptic equation defining $C$, there exists
\[\left(
\begin{array}{cc}
a & b \\
c & d \\
\end{array}
\right)\in\GL_2(K), e\in K^{\times}\] such that $\displaystyle x=\frac{a u+b}{c u+d}$, $\displaystyle y=\frac{ez}{(cu+d)^3}$.
Assuming that $P(x)=a_0x^6+a_1x^5+a_2x^4+a_3x^3+a_4x^2+a_5x+a_6\in K[x]$, one may define the Igusa invariants (projective invariants), $J_{2i},1\le i\le 5$, associated to $P(x)$, see \cite{Liumodelesminimaux} for an explicit description of these invariants. One knows that $J_{2i}\in\Z[a_0,\ldots,a_6][1/2]$ is a homogeneous polynomial of degree $2i$ in the $a_i$'s. Moreover, one may define the invariants $I_4 = J_2^2 - 24J_4$ and $I_{12} = 2^{-2}(J_2^2J_4^2- 32J_4^3 - J_2^3J_6 + 36J_2J_4J_6 - 108J_6^2)$ which are homogeneous polynomials in the $a_i$'s of degree $4$ and $12$, respectively.
If $J_{2i}',1\le i\le 5$, are the Igusa invariants associated to another hyperelliptic equation describing $C$, then there is an $a\in K^{\times}$ such that $J_{2i}'=a^{2i}J_{2i}$. Furthermore, if $C$ and $C'$ are two genus $2$ curves with Igusa invariants $J_{2i}$ and $J_{2i}'$ satisfying the latter equality, then $C$ and $C'$ are isomorphic over the algebraic closure of $K$.
In \cite[\S2]{Liumodelesminimaux}, Liu introduced the following invariants (affine invariants) attached to $P(x)$:
\begin{eqnarray*}
A_2 &=& -5a_1^2 + 12a_0a_2 \\
A_3 &=& 5a_1^3 + 9a_0(-2a_2a_1 + 3a_0a_3) \\
A_4 &=& -5a_1^4+ 24a_0(a_2a_1^2 - 3a_3a_0a_1 + 6a_4a_0^2)\\
A_5 &=& a_1^5+ 3a_0(-2a_2a_1^3 + 9a_0a_3a_1^2 - 36a_0^2a_4a_1 + 108a_0^3a_5)\\
B_2 &=& 2a_2^2 - 5a_1a_3 + 10a_0a_4.
\end{eqnarray*}
One observes that $A_i,B_2\in\Z[a_0,\ldots,a_6]$.
The invariants $A_i$ and $B_2$ are homogeneous polynomials in the $a_i$'s of degree $i$ and $2$, respectively.
\section{Models of genus two curves}
Throughout this note $R$ is a complete discrete valuation ring, $K$ its field of fractions, $\mathfrak{m}$ its maximal ideal generated by $t$, and $k$ its residue field. We assume moreover that $k$ is algebraically closed and $\Char k\ne2$. Let $\nu:K\to\Z\cup\{\infty\}$ be the normalized valuation of $K$, i.e., $\nu(t)=1$. The map $\alpha\mapsto\overline{\alpha}$ is the canonical homomorphism from $R$ to $k$.
For a scheme $X$ over $R$, $X_K$ and $X_k$ will denote the generic fiber and the special fiber of $X$, respectively.
\begin{Definition}
Let $C$ be a smooth projective curve over $K$, $R'$ a discrete valuation ring dominating $R$. We say that $\CC$ is a {\em stable model} for $C$ over $R'$ if $\CC$ is a stable curve over $R'$ whose generic fiber is isomorphic to $C\times_K K'$, where $K'$ is the fraction field of $R'$.
\end{Definition}
Therefore, one knows that the singular points of a stable model $\CC$ of $C$ over $R'$ are ordinary double points. In particular, if $P$ is a singular point of $\CC_k$, then the $\mathfrak{m}_P$-adic completion of $\mathcal{O}_{\CC,P}$ satisfies
\[ \widehat{\mathcal{O}}_{\CC,P}\simeq \widehat{R}[[u,v]]/(uv-\pi),\,\pi\in\mathfrak{m}_{K'}\setminus\{0\}.\] The {\em degree of singularity} of $P$ in $\CC$ is the integer $\nu_{K'}(\pi)$, where $\nu_{K'}$ is the normalized valuation of $K'$.
The following proposition, \cite[Proposition 4]{LiuCourbesstablesdegenre2}, shows that if the genus of $C$ is positive, then $C$ admits a stable model over a Galois extension of $K$.
\begin{Proposition}
\label{prop:fieldextension}
Suppose that $C$ is a smooth projective geometrically connected curve over $K$ of genus $g\ge 1$. There exists a Galois extension $L$ of $K$ such that for every finite extension $F$ of $K$, $C\times_K F$ admits a stable model if and only if $L\subseteq F$.
\end{Proposition}
\begin{Definition}
Let $C$ be a smooth projective geometrically connected curve of genus $g\ge 1$ over $K$. Let $L/K$ be the smallest Galois field extension over which $C$ admits a stable model $\CC$, see Proposition \ref{prop:fieldextension}. Then we say that $\CC$ is the {\em stable model} of $C$. Furthermore, the stable model $\CC$ of $C$ is unique up to isomorphism.
\end{Definition}
Let $\Gamma$ be the set of curves defined over $k$ by $y^2=x^6+x^4+x^2+a,a\in k^{\times}$ when $\Char k=3$. Indeed, $\CC_k\in\Gamma$ if and only if
\[J_2\ne 0,\,J_4J_2^{-2}\in\mathfrak{m},\;J_{10}J_2^{-5}\in R^{\times},\,\textrm{ and } J_6J_2^{-3}-J_{10}J_{2}^{-5}\in\mathfrak{m}.\]
Let $C_0$ be the curve defined by $y^2=x^5-x$ if $\Char k=5$. In fact, $\CC_k\simeq C_0$ if and only if $J_{2i}^5J_{10}^{-i}\in\mathfrak{m}$. Now, one has the following result.
\begin{Proposition}
\label{prop:tamelyramified}\cite[Proposition 4.1.2]{Liumodelesminimaux}
Let $C$ be a smooth projective geometrically connected curve of genus $2$ over $K$. If one of the following conditions holds, then $L$ is a tamely ramified extension of $K$:
\begin{itemize}
\item[(1)] $\CC_k\ne 3,5$;
\item[(2)] $f:C\to \PP^1_K$ is ramified above two rational points of $\PP^1_K$;
\item[(3)] $\Char k=3$, $\overline{\omega}$ is ramified or $\CC_k\not\in\Gamma$;
\item[(4)] $\Char k=5$, $\overline{\omega}$ is non-ramified or $\CC_k\not\simeq C_0$.
\end{itemize}
\end{Proposition}
The special fiber $\CC_k$ of the stable model $\CC$ of $C$ is either smooth, irreducible with one or two double points, the union of two rational curves intersecting transversally in three points, or the union of two irreducible components intersecting in one and only one point. In the latter case, the two irreducible components are either smooth, singular, or one is smooth and the other component is singular. The following theorem, \cite[Th\'{e}or\`{e}me 1]{LiuCourbesstablesdegenre2}, gives explicit criteria for each of these possibilities of $\CC_k$ in terms of the invariants of $C$.
We set $\epsilon=1$ if $\Char k\ne 2,3$; $\epsilon =3$ if $\Char k=3$; and $\epsilon =4$ if $\Char k=2$. We define $I_2:=12^{-1}J_2,\,I_6:=J_6,\,I_8:=J_8$.
\begin{Theorem}
\label{Thm:singularity}
Let $C$ be a hyperelliptic curve defined over $K$ of genus $2$. Then
\begin{itemize}
\item[(I)] $\CC_k$ is smooth if and only if $J_{2i}^5J_{10}^{-i}\in R$ for every $i\le 5$;
\item[(II)] $\CC_k$ is irreducible with one double point if and only if $J_{2i}^6I_{12}^{-i}\in R$ for every $i\le 5$ and $J_{10}^6I_{12}^{-5}\in\mathfrak{m}$. The normalisation of $\CC_k$ is an elliptic curve with $j$-invariant $j=\overline{(I_4^3I_{12}^{-1})}$;
\item[(III)] $C_k$ is irreducible with two double points if and only if $J_{2i}^2I_4^{-i}\in R$ for $i\le 5$, $J_{10}^2I_4^{-5}\in\mathfrak{m}$, $I_{12}I_4^{-3}\in\mathfrak{m}$, and either $J_4I_4^{-1}$ or $J_6^2I_4^{-3}$ invertible in $R$;
\item[(IV)] $\CC_k$ consists of two rational curves intersecting transversally in three points if and only if $J_{2i}^2I_4^{-i}\in\mathfrak{m}$ for $2\le i\le 5$;
\item[(V*)] $\CC_k$ is the union of two irreducible components intersecting in one point if and only if \begin{eqnarray}
I_4^{\epsilon}I_{2\epsilon}^{-2}\in\mathfrak{m},\; J_{10}^{\epsilon}I_{2\epsilon}^{-5}\in\mathfrak{m},\; I_{12}^{\epsilon}I_{2\epsilon}^{-6}\in\mathfrak{m}
\end{eqnarray} (which implies that $J_{2i}^{\epsilon}I_{2\epsilon}^{-i}\in R$ for $i\le5$). Furthermore,
\item[(V)] Both components of $\CC_k$ are smooth if and only if: in addition to (1), $I_4^{3\epsilon}J_{10}^{-\epsilon}I_{2\epsilon}^{-1}\in R$, $I_{12}^{\epsilon}J_{10}^{-\epsilon}I_{2\epsilon}^{-1}\in R$. If $j_1$ and $j_2$ are the modular invariants of the components of $\CC_k$, then
\[(j_1j_2)^{\epsilon}=\overline{(I_4^{3\epsilon}J_{10}^{-\epsilon}I_{2\epsilon}^{-1})},\;(j_1+j_2)^{\epsilon}=2^6.3^3+\overline{(I_{12}^{\epsilon}J_{10}^{-\epsilon}I_{2\epsilon}^{-1})};\]
\item[(VI)] Only one of the two components of $\CC_k$ is smooth if and only if: in addition to (1), $I_4^3I_{12}^{-1}\in R, J_{10}^{\epsilon}I_{2\epsilon}I_{12}^{-\epsilon}\in\mathfrak{m}$. The modular invariant of the smooth component of $\CC_k$ is $j=\overline{(I_4^3I_{12}^{-1})}$;
\item[(VII)] Both components of $\CC_k$ are singular if and only if: in addition to (1), $I_{12}I_4^{-3}\in\mathfrak{m}$, and $J_{10}^{\epsilon}I_{2\epsilon}I_4^{-3\epsilon}\in\mathfrak{m}$.
\end{itemize}
\end{Theorem}
The following proposition provides us with the degrees of singularity of the singular points of $\CC_k$.
\begin{Proposition}\cite[Proposition 2]{LiuCourbesstablesdegenre2}
\label{prop:degreeofsingularity}
Let $C$ be a smooth projective geometrically connected curve of genus $2$ over $K$. Let $L/K$ be the smallest Galois field extension over which $C$ has its stable model $\CC$, and $\nu_L$ its normalized valuation.
The following statements hold.
\begin{itemize}
\item[(I)] If $\CC_k$ is smooth, then the minimal regular model of $C$ is $\CC$.
\item[(II)] If $\CC_k$ is irreducible with one double point, the degree of singularity of its singular point is $e=\nu_L(J_{10}^6I_{12}^{-5})/6$.
\item[(III)] If $\CC_k$ is irreducible with two double points, the degrees of singularity, $e_1\le e_2$, of the singular points are given by
\[e_1=\inf\{\nu_L(I_{12}I_4^{-3},\nu_L(J_{10}^2I_4^{-5})\},\;e_2=\frac{1}{2}\nu_L(J_{10}^2I_4^{-5})-e_1.\]
\item[(IV)] If $\CC_k$ consists of two rational curves intersecting in three points, we assume $e_1\le e_2\le e_3$ are the degrees of singularity of the singular points of $\CC_k$. Set $l=\nu_L(J_{10}J_2^{-5}),n=\nu_L(I_{12}J_2^{-6})$ and $m=\nu_L(J_4J_2^{-2}).$ Then
\[e_1=\inf\{l/3,n/2,m\},\;e_2=\inf\{(l-e_1)/2,n-e_1\},\;e_3=l-e_1-e_2.\]
\item[(V)] If $\CC_k$ is the union of two smooth irreducible components intersecting in one point, then the degree of singularity of the singular point of $\CC_k$ is $e=\nu_L(J_{10}^{\epsilon}I_{2\epsilon}^{-5})/12\epsilon$.
\item[(VI)] If $\CC_k$ is the union of one smooth and one singular irreducible component intersecting in one point, we set $e_0$ to be the degree of singularity of the point of intersection of the components of $\CC_k$, and $e_1$ the degree of singularity of the other singular point. Then \[e_0=\nu_L(I_{12}^{\epsilon}I_{2\epsilon}^{-6})/12\epsilon,\;e_1=\nu_L(J_{10}^{\epsilon}I_{2\epsilon}I_{12}^{-\epsilon})/\epsilon.\]
\item[(VII)] If $\CC_k$ is the union of two singular irreducible components intersecting in one point, we set $e_0$ to be the degree of singularity of the point of intersection of the two components of $\CC_k$ and $e_1\le e_2$ the degrees of singularity of the other singular points. Then $e_0=\nu_L(I_4^{\epsilon}I_{2\epsilon}^{-2})$,
\[e_1=\inf\{\nu_L(I_{12}I_4^{-3}),\nu_L(J_{10}^{\epsilon}I_{2\epsilon}I_4^{-3\epsilon})/2\epsilon\},\;e_2=\frac{1}{\epsilon}\nu_L(J_{10}^{\epsilon}I_{2\epsilon}I_4^{-3\epsilon})-e_1.\]
\end{itemize}
\end{Proposition}
\section{Stable models of quadratic twists}
Let $C$ be a smooth projective geometrically connected curve of genus $2$ defined over $K$. Let $y^2=P(x)$ be a defining polynomial of $C$. Let $K(\sqrt{D})$ be a quadratic extension of $K$ with associated character $\chi$. Let $C^{\chi}$ be the quadratic twist of $C$ by the character $\chi$. One knows that $C^{\chi}$ is defined by the hyperelliptic equation $y^2=DP(x)$. One may and will assume without loss of generality that $\nu(D)=0$ or $1$.
\begin{Proposition}
Let $C$ be a smooth projective geometrically connected curve of genus $2$ defined over $K$. Let $K(\sqrt{D})$ be a quadratic extension of $K$ with associated character $\chi$. If $\nu(D)=0$, then the minimal regular model $ X^{\chi}$ of $C^{\chi}$ is isomorphic to the minimal regular model $ X$ of $C$ over $R$.
\end{Proposition}
\begin{Proof}
Since $\nu(D)=0$, one knows that $K(\sqrt D)$ is an unramified extension of $K$. Now, the proof follows from \cite[Proposition 10.1.17]{Liubook}.
\end{Proof}
The proposition above allows us to assume from now on that $\nu(D)=1$.
\begin{Lemma}
\label{Lem:IgusaofTwists}
Let $J_{2i},1\le i\le 5$, $I_4$, $I_{12}$, $A_i,2\le i\le 5$, $B_2$ be the invariants attached to a hyperelliptic equation $y^2=P(x)$ defining a smooth projective geometrically connected genus $2$ curve over $K$, and $J_{2i}',1\le i\le 5$, $I_4'$, $I_{12}'$, $A_i',2\le i\le 5$, $B_2'$ the invariants attached to the hyperelliptic equation $y^2=DP(x)$.
Then one has $J_{2i}'=D^{2i}J_{2i}$, $I_4'=D^4I_4$, $I_{12}'=D^{12}I_{12}$, $A_{i}'=D^iA_i$, and $B_2'=D^2B_2$.
\end{Lemma}
\begin{Proof}
One may assume that $P(x)=a_0x^6+a_1x^5+\ldots+a_6\in R[x]$. Now, $y^2=DP(x)=a_0'x^6+a_1'x^5+\ldots+a_6'$ where $a_i'=Da_i$. The result holds using the fact that the invariants $J_{i}$, $I_i$, $A_i$, and $B_i$ are homogeneous of degree $i$ in the $a_j$'s, see \S 2.
\end{Proof}
\begin{Lemma}
\label{Lem:StableModelofTwist}
Let $C$ and $C^{\chi}$ be as above. Let $\CC$ and $\CC^{\chi}$ be the stable models of $C$ and $C^{\chi}$ respectively. Then the following statements are true.
\begin{itemize}
\item[(I)] $\CC_k$ is smooth if and only if $\CC^{\chi}_k$ is smooth;
\item[(II)] $\CC_k$ is irreducible with one double point if and only if $\CC^{\chi}_k$ is irreducible with one double point;
\item[(III)] $C_k$ is irreducible with two double points if and only if $C^{\chi}_k$ is irreducible with two double points;
\item[(IV)] $\CC_k$ consists of two rational curves intersecting transversally in three points if and only if $\CC^{\chi}_k$ consists of two rational curves intersecting transversally in three points;
\item[(V*)] $\CC_k$ is the union of two irreducible components intersecting in one point if and only if $\CC^{\chi}_k$ is the union of two irreducible components intersecting in one point. Furthermore,
\item[(V)] Both components of $\CC_k$ are smooth if and only if both components of $\CC^{\chi}_k$ are smooth;
\item[(VI)] Only one of the two components of $\CC_k$ is smooth if and only if only only one of the two components of $\CC^{\chi}_k$ is smooth;
\item[(VII)] Both components of $\CC_k$ are singular if and only if both components of $\CC^{\chi}_k$ are singular.
\end{itemize}
\end{Lemma}
\begin{Proof}
Let $C$ be defined by the hyperelliptic equation $y^2=P(x)$ and $C^{\chi}$ defined by $y^2=DP(x)$. If $J_{2i},\;1\le i\le 5,$ is an Igusa invariant attached to the hyperelliptic equation describing $C$, then $J_{2i}'$ will be an Igusa invariant of the hyperelliptic equation describing $C^{\chi}$. Similarly, $I_{2i}$ and $I_{2i}'$ are invariants of $C$ and $C^{\chi}$, respectively.
According to Theorem \ref{Thm:singularity}, one only needs to study a quotient of products of Igusa invariants powers. We prove (I) and (II) and the other cases are similar.
(I) One has that $\CC_k$ is smooth if and only if $J_{2i}^5J_{10}^{-i}\in R$ for every $i\le 5$. Since $J_{2i}$ is homogeneous of degree $2i$, one has $J_{2i}'^5J_{10}'^{-i}=J_{2i}^5J_{10}^{-i}\in R$. (II) $\CC_k$ is irreducible with one double point if and only if $J_{2i}^6I_{12}^{-i}\in R$ for every $i\le 5$ and $J_{10}^6I_{12}^{-5}\in\mathfrak{m}$, see Theorem \ref{Thm:singularity} (II). Since $J_{2i}^6I_{12}^{-i}$ and $J_{10}^6I_{12}^{-5}$ are quotients of invariants of the same degree, it follows that $J_{2i}'^6I_{12}'^{-i}=J_{2i}^6I_{12}^{-i}\in R$ for every $i\le 5$, and $J_{10}'^6I_{12}'^{-5}=J_{10}^6I_{12}^{-5}\in\mathfrak{m}$.
\end{Proof}
\section{Minimal regular models of quadratic twists}
Let $C$ be a smooth projective geometrically connected curve of genus $2$ defined over $K$. We let $\sigma$ be the hyperelliptic involution of $C$. It extends to an involution of the stable model $\CC$, which we will denote by $\sigma$ again. We set $\mathcal Z=\CC/\langle\sigma\rangle$ and $L$ the field extension of $K$ over which $C$ attains its stable model, Proposition \ref{prop:fieldextension}. Then $\mathcal Z$ is a semi-stable curve over $R_L$, the ring of integers of $L$, whose generic fibre is isomorphic to $\PP_L^1$ and its special fibre is $\mathcal{Z}_k=\CC_k/\langle\sigma\rangle$. Let $f:\CC\to\mathcal Z$ be the canonical morphism, $\omega\in\mathcal{Z}_L$ the point corresponding to $x=\infty$, $\overline{\omega}\in\mathcal{Z}_k$ its specialization. We say $\overline{\omega}$ is ramified if $f$ is ramified above $\overline{\omega}$.
In \cite{Liumodelesminimaux}, Liu displayed the possible types for the special fiber of the minimal regular model of $C$ when $\mathcal{Z}_k$ is either smooth, irreducible and singular, or not irreducible; and when the field extension $L/K$ is tamely ramified. Moreover, he presented the possible reduction types for the minimal regular model of $C$ that appear when $L/K$ is a wildly ramified extension.
In this work, we follow the notation of Liu. In particular, the reduction type of the minimal regular model of $C$ will be given the same symbol as in \cite{Liumodelesminimaux}. Moreover, according to Lemma \ref{Lem:StableModelofTwist}, one knows that the special fibers of the stable models of the hyperelliptic curve $C$ and its quadratic twist have the same number of irreducible components and singular points. This reduces the number of possibilities that one needs to investigate in order to find the reduction type of the minimal regular model of the quadratic twist.
Furthermore, in this section we are going to deal with the tamely ramified extensions $L/K$ that occur when one of the conditions of Proposition \ref{prop:tamelyramified} holds. It turns out that when this happens then the field extension $L'/K$ over which the quadratic twist attains its stable model is also tamely ramified. Finally, we treat the case of widely ramified extensions in Corollary \ref{cor:widelyramified}.
Throughout this section, any table will contain the reduction type of the minimal regular model of $C$, the positive integer $n$ which represents the degree of the field extension $L/K$ over which $C$ attains its stable model, the congruence classes of two positive integers $r$ and $q$ mod $n$, where these integers associated to $C$ are introduced in \cite{Liumodelesminimaux}, and finally the reduction type of the minimal regular model of the quadratic twist of $C$.
\subsection{$\CC_k$ is smooth and $L/K$ is tamely ramified}
The following result can be found in \cite{Liumodelesminimaux}. It describes the ramification of the field extension $L/K$, see Proposition \ref{prop:fieldextension}, and its degree if the special fiber of the stable model is smooth.
\begin{Proposition}
\label{prop:smooth}
Assume that $\CC_k$ is smooth. The point $\overline{\omega}$ is ramified if and only if $A_5\ne 0$ and $a_0^{20}J_{10}A_5^{-6}\in\mathfrak{m}$. Moreover, the morphism $C\to \PP_K^1$ is ramified above some rational point $x_0\in \PP_K^1$.
Furthermore, the field extension $L/K$ is tamely ramified in the following situations:
\begin{itemize}
\item[(a)] $\Char k\ne 3,5$;
\item[(b)] $f:C\to \PP_K^1$ is ramified above two rational points in $\PP_K^1$;
\item[(c)] $\Char k =3$, $\overline{\omega}$ is ramified or $\CC_k\not\in \Gamma$ where $\Gamma$ is the set of isomorphis classes of the smooth proper curves over $k$ defined by $z^2=v^6+v^4+v^2+a,\;a\in k^{\times}$.
\item[(d)] $\Char k\ne 5$, $\overline{\omega}$ is non-ramified or $J_{2i}^5J_{10}^{-i}\not\in\mathfrak{m}$ for some $i\le 3$.
\end{itemize}
We assume that $L/K$ is tamely ramified. We define $n,r,q$ as follows:
\begin{itemize}
\item[(a)] If $\overline{\omega}$ is non-ramified, $n$ is the least common denominator of $\nu(a_0^{10}J_{10}^{-1})/30$ and $\nu(a_0^5J_{10}^{-1})/10$, $r=n\nu(a_0^{10}J_{10}^{-1})/30$ and $q=n\nu(a_0^5J_{10}^{-1})/10$;
\item[(b)] If $\overline{\omega}$ is ramified, $n$ is the least common denominator of $\nu(A_5^{-2}J_{10})/20$ and $\nu(A_5^{-6}J_{10}^{5})/40$, $r=n\nu(A_5^{-2}J_{10})/20$ and $q=n\nu(A_5^{-6}J_{10}^{5})/40$.
\end{itemize}
Then $[L:K]=n$.
\end{Proposition}
The integers $r$ and $q$ in the proposition above are used in the description of the action of $\Gal(L/K)$ on $\CC_k$, and will be used to determine the reduction type of the minimal regular model of the quadratic twist of $C$.
The field extension $L'$ is the field over which $C^{\chi}$ attains its stable model, and the degree of the extension extension is $n'$.
\begin{Lemma}
\label{lem:n-r-qsmooth}
Let $K(\sqrt{D})$, where $\nu(D)=1$, be a quadratic extension of $K$ with associated character $\chi$. Assume that $C$ is a smooth projective curve of genus two defined by $y^2=P(x)$ and $C^{\chi}$ is the quadratic twist of $C$ by $\chi$ defined by $y^2=DP(x)$. Let $n,q,r$ be the integers attached to $C$ defined in Proposition \ref{prop:smooth}, and $n',q',r'$ be the ones attached to $C^{\chi}$. We assume moreover that the special fiber of the stable model of $C$ is smooth. If $L/K$ is tamely ramified, then the following statements hold:
\begin{itemize}
\item[(a)] If $\overline{\omega}$ is non-ramified, then $\overline{\omega}'$ is non-ramified. Furthermore, $L'$ is a tamely ramified extension of $K$. The integer $n'$ is the least common denominator of $\nu(a_0^{10}J_{10}^{-1})/30$ and $\nu(a_0^5J_{10}^{-1})/10-1/2$, $nr'=n'r$ and $q'=n'(q/n-1/2)$;
\item[(b)] If $\overline{\omega}$ is ramified, then $\overline{\omega}'$ is ramified. Furthermore, $L'$ is a tamely ramified extension of $K$. The integer $n'$ is the least common denominator of $\nu(A_5^{-2}J_{10})/20$ and $\nu(A_5^{-6}J_{10}^{5})/40+1/2$, $nr'=n'r$ and $q'=n'(q/n+1/2)$.
\end{itemize}
\end{Lemma}
\begin{Proof}
In view of Proposition \ref{prop:smooth}, $\overline{\omega}'$ is ramified if and only if $A'_5\ne 0$ and $a_0^{20}J_{10}A_5^{-6}\in\mathfrak{m}$. The latter statement is equivalent to $\overline{\omega}$ being ramified since $A_5=D^{-5}A_5'\ne 0$ and $a_0^{20}J_{10}A_5^{-6}=a_0'^{20}J_{10}'A_5'^{-6}\in\mathfrak{m}$.
Again, we use the fact that the invariants of $C$ and $C^{\chi}$ are homogeneous in the $a_i$'s to evaluate $n',r'$ and $q'$. For (a), one has $n'$ is the least common denominator of $\nu(a_0'^{10}J_{10}'^{-1})/30$ and $\nu(a_0'^5J_{10}'^{-1})/10-1/2$, see Proposition \ref{prop:smooth}. Now $a_0'^{10}J_{10}'^{-1}=a_0^{10}J_{10}^{-1}$ and $a_0'^5J_{10}'^{-1}=a_0^5J_{10}^{-1}d^{-5}$. Now, the value for $n'$ follows from the fact that $\nu(ab)=\nu(a)+\nu(b)$ for $a,b\in K$ and $\nu(D)=1$. One has that $r'=n'\nu(a_0'^{10}J_{10}'^{-1})/30=n'\nu(a_0^{10}J_{10}^{-1})/30=n'r/n$, and $q'=n'\nu(a_0'^5J_{10}'^{-1})/10=n'\left(\nu(a_0^5J_{10}^{-1})-5\right)/10=n'(q/n-1/2)$.
The argument is similar for (b).
\end{Proof}
We set $u_1=\nu(a_0^5J_{10}^{-1})$, $u_2=\nu(a_0^5J_{10}^{-1})$; and $v_1=\nu(A_5^{-2}J_{10})$, $v_2=\nu(A_5^{-6}J_{10}^{5})$, whereas $u_1',u_2',v_1',v_2'$ are the corresponding values for $C^{\chi}$.
\begin{Theorem}
Let $C$ be a hyperelliptic curve defined over $K$. Assume that $L/K$ is tamely ramified and $\CC_k$ is smooth. Let $K(\sqrt{D})/K$ be a quadratic extension whose associated character is $\chi$, and $\nu(D)=1$. If $X$ and $X^{\chi}$ are the minimal regular models of the curves $C$ and its quadratic twist by $\chi$, $C^{\chi}$, then the reduction type of $X^{\chi}$ is given in the following table.
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
type($X_k$) & n&$r$ mod $n$ & $q$ mod $n$ &type($X^{\chi}_k$)\\
\hline
$[{\rm I}_{0-0-0}]$ &1&&& $[{\rm I}^*_{0-0-0}]$\\
\hline
$[{\rm I}^*_{0-0-0}]$ &2&0&&$[{\rm I}_{0-0-0}]$ \\
\hline
$[\rm{II}]$ &2& 1&&$[{\rm II}]$\\
\hline
$[\rm{III}]$ &3& && $[\rm{IV}]$\\
\hline
$[\rm{VI}]$ &4&&& $[\rm{VI}]$ \\
\hline
$[\rm{IX}-3]$ &5&1&& $[\rm{VIII}-1]$\\
\hline
$[\rm{IX}-1]$ &5&2&&$[\rm{VIII}-3]$ \\
\hline
$[\rm{IX}-4]$ &5&3&& $[\rm{VIII}-2]$\\
\hline
$[\rm{IX}-2]$ &5&4&& $[\rm{VIII}-4]$\\
\hline
$[\rm{V}]$ &6&1&0& $[\rm{V}^*]$\\
&6&5&3&\\
\hline
$[\rm{V}^*]$ &6&1&3& $[\rm{V}]$\\
&6&5&0&\\
\hline
$[\rm{IV}]$ &6& 2 or 4&& $[\rm{III}]$\\
\hline
$[\rm{VII}^*]$ &8&&1 or 3&$[\rm{VII}]$ \\
\hline
$[\rm{VII}]$ &8&&5 or 7&$[\rm{VII}^*]$ \\
\hline
$[\rm{VIII}-1]$ &10&2&& $[\rm{IX}-3]$\\
\hline
$[\rm{VIII}-3]$ &10&4&&$[\rm{IX}-1]$ \\
\hline
$[\rm{VIII}-2]$ &10&6&&$[\rm{IX}-4]$ \\
\hline
$[\rm{VIII}-4]$ &10&8&& $[\rm{IX}-2]$\\
\hline
\end{tabular}
\end{center}
\end{Theorem}
\begin{Proof}
We assume that $C$ is given by the hyperelliptic equation $y^2=P(x)$, $\deg P(x)=5$ or $6$ and $C^{\chi}$ is given by $y^2=DP(x)$. Unless otherwise stated we will assume throughout the proof that $\overline{\omega}$ is non-ramified, since the proof for $\overline{\omega}$ being ramified will be similar. We recall that if $\CC_k$ is smooth, then the special fiber $\CC^{\chi}_k$ of the stable model of $C^{\chi}$ is smooth, see Lemma \ref{Lem:StableModelofTwist}.
We assume that the reduction type of $X_k$ is $[{\rm I}_{0-0-0}]$. Since $n=1$, $30\mid u_1$ and $10\mid u_2$, see Proposition \ref{prop:smooth}. Since $u'_1/30=u_1/30$ and $u'_2/10:=u_2/10-1/2$, see Lemma \ref{lem:n-r-qsmooth}, it follows that $n'=2$. Moreover, $r'=n'r/n=2r\in2\Z$, i.e., $r'=0$ mod $2$. One has that the type of $X^{\chi}_k$ is $[{\rm I}^*_{0-0-0}]$.
If $X_k$ has reduction type $[{\rm I}_{0-0-0}^*]$, then $30\mid u_1$ since $r=2u_1/30=0$ mod $(n=2)$. It follows that $5\mid u_2$ and $2\nmid u_2$ since otherwise $n=1$. This implies that $n'=1$ as $u_1/30,u_2/10-1/2\in\Z$, see Lemma \ref{lem:n-r-qsmooth}. Thus, the type of $X^{\chi}_k$ is $[{\rm I}_{0-0-0}]$.
If $X_k$ has reduction type $[{\rm II}]$, then $15\mid u_1$ and $2\nmid u_1$ since $r=2u_1/30=1$ mod $2$. Therefore, $n'=2$ as $\displaystyle u_1/30\in\frac{1}{2}\Z$ and the least denominator of $u_2/10-1/2$ is either $1$ or $2$. Furthermore, $r'=n'r/n=r=1$ mod $2$. One obtains that the reduction type of $X^{\chi}_k$ is $[{\rm II}]$.
If $X$ has reduction type $[{\rm III}]$, then $10\mid u_1,\;3\nmid u_1$ and $10\mid u_2$ since $n=3$. Now, $\displaystyle \frac{u_1}{30}\in\frac{1}{3}\Z$ and $\displaystyle \frac{u_2}{10}-\frac{1}{2}\in\frac{1}{2}\Z$ which implies that $n'=6$, see Lemma \ref{lem:n-r-qsmooth}. One moreover has $\displaystyle r'=6r/3=2r\in 2\Z$, it follows that $r'= 2$ or $4$ mod $6$. Thus, the type of $X^{\chi}_k$ is $[{\rm IV}]$. One observes that if $\overline{\omega}$ is ramified, then the reduction type of $X$ cannot be $[{\rm III}]$ since $3$ does not divide the denominator of $v_1/20$ and $v_2/40$.
If $X$ has reduction type $[{\rm VI}]$, then $n=4$. If $\overline{\omega}$ is non-ramified, then the least common denominator of $u_1/30$ and $u_2/10$ cannot be $4$. Therefore, $\overline{\omega}$ is ramified. One has $n'$ is the least common denominator of $\displaystyle \nu(v_1)/20$ and $\nu(v_2)/40+1/2$, Lemma \ref{lem:n-r-qsmooth}. If the least denominator of $\nu(v_1)/40$ is $4$, then $n'=4$; otherwise the least denominator of $\nu(v_2)/40$ is $4$ and so $\nu(v_2)/10=1$ or $3$ mod $4$, thus $\nu(v_2)/10+2= 1$ or $3$ mod $4$, which implies that $n'=4$. Therefore, the type of $X^{\chi}_k$ is $[{\rm VI}]$.
Assuming that $n=5$, and $\overline{\omega}$ is non-ramified, one has $5$ is the least common denominator of $ u_1/30$ and $u_2/10$. Now, $n'$ is the least common denominator of $\displaystyle u_1/30$ $u_2/10-1/2$. This yields that $n'=10$. Moreover, $r'=2r$. Therefore, if the reduction type of $X$ is $[\textrm{IX}-i]$, $i=3, 1, 4,2$, then $r=1,2,3,4$ mod $5$, respectively, and so $r'=2, 4, 6, 8$ mod $10$, respectively, i.e., the reduction type of $X^{\chi}$ is $[\textrm{VIII}-j]$, $j=1,3,2,4$, respectively.
Assuming that $n=6$, $\overline{\omega}$ must be non-ramified. One has $5\mid u_1$ and $5\mid u_2$. If $r:=6u_1/30=1$ or $5$ mod $6$, then $r=5u_1$ mod $6$, and $u_1=5$ or $1$ mod $6$, respectively. In particular, $2\nmid u_1$, hence $\displaystyle u_1/30\in\frac{1}{6}\Z$. In this case, the least common denominator, $n'$, of $u_1/30$ and $u_2/10-1/2$ is $6$, moreover, $r'=r$ and $\displaystyle q'= q-3$, Lemma \ref{lem:n-r-qsmooth}. This implies that if $X$ has reduction type $[{\rm V}]$ or $[{\rm V}^*]$, then the reduction type of $X^{\chi}$ is $[{\rm V}^*]$ or $[{\rm V}]$, respectively. If $r=2$ or $4$ mod $6$, hence the reduction type of $X$ is $[\rm{IV}]$, then $ u_1=4$ or $2$ mod $6$, respectively, which forces $2\nmid u_2$, therefore, the least common denominator of $\displaystyle u_1/30\in\frac{1}{3}\Z$ and $\displaystyle \frac{u_2}{10}-\frac{1}{2}\in \Z$ is $3$. This implies that $n'=3$ and the reduction type of $X^{\chi}$ is $[\rm{III}]$.
If $n=8$, then $\overline{\omega}$ must be ramified. Now, the least denominator of $v_2/40$ is $8$, in particular, $v_2/5=1,3,5$ or $7$ mod $8$. It follows that $\displaystyle v_2/40+1/2\in\frac{1}{8}\Z$. Therefore, $n'=8$, $r'=r$, and $q'=q+4$, Lemma \ref{lem:n-r-qsmooth}. If $q=1$ or $3$ mod $8$ ($5$ or $7$ mod $8$), then $q'=5$ or $7$ mod $8$ ($1$ or $3$ mod $8$). More specifically, if the reduction type of $X$ is $[\rm{VII}^*]$ or $[\rm{VII}]$, then the reduction type of $X^{\chi}$ is $[\rm{VII}]$ or $[\rm{VII}^*]$, respectively.
If $n=10$, one has $r$ is even. This implies that $u_1$ is even and $u_2$ is odd. We recall that we assume that $\overline{\omega}$ is non-ramified. The integer $n'$ is the least common denominator of $u_1/30$ and $\displaystyle u_2/10-1/2$ where the denominator of the latter is either $1$ or $5$. Thus, $n'=5$ and $r=2r'$, the same holds when $\overline{\omega}$ is ramified.
Now when $r=2, 4, 6,$ or $8$ mod $10$, i.e., the reduction type of $X$ is $[{\rm VIII}-i]$, $i=1,3, 2,$ or $4$, it follows that $r'=1, 2,3,$ or $4$ mod $5$
and the reduction type of $X^{\chi}$ is $[{\rm IX}-j]$, $j=3,1,4,$ or $2$, respectively.
\end{Proof}
\subsection{$\CC_k$ is singular, $\CC_k/\langle\sigma\rangle$ is irreducible, and $L/K$ is tamely ramified}
In this section we assume that $\CC_k/\langle\sigma\rangle$ is irreducible and $\CC_k$ is singular. In particular, $\CC_k$ is irreducible with one or two double point; or $\CC_k$ consists of two rational curves intersecting transversally in three ordinary double points.
Let $C_{000}$ be the stable curve over $k$ consisting of two rational curves intersecting in three points. We set:
\begin{align*}J_{12}=\left\{\begin{array}{ll}
I_{12} & \textrm{if $\CC_k$ has a singular point} \\
I_4^3 & \textrm{if $\CC_k$ is irreducible and rational}\\
J_2^6 & \textrm{if $\CC_k=C_{000}$} \end{array}\right.\end{align*}
The following proposition, see \cite{Liumodelesminimaux}, describes the ramification of $f:\CC\to \mathcal Z$ over $\overline{\omega}$, the cases when $L/K$ is tamely ramified, and the field extension $L/K$ over which $C$ attains its stable model.
\begin{Proposition}
\label{prop:irreduciblesingular}
Assume that $\CC_k$ is singular and $\CC_k/\langle\sigma\rangle$ is irreducible.
\begin{itemize}
\item[(a)] The point $\overline{\omega}$ is non-ramified if and only if $a_0^{-6}B_2^9J_{12}^{-1},a_0^{-120}A_5^{36}J_{12}^{-5}\in R$;
\item[(b)] The point $\overline{\omega}$ is ramified and $f^{-1}(\overline{\omega})$ is a regular point if and only if $a_0^{120}A_5^{-36}J_{12}^{5}\in \mathfrak{m},\;B_2^{60}A_5^{-12}J_{12}^{-5}\in R$;
\item[(c)] The point $\overline{\omega}$ is ramified and $f^{-1}(\overline{\omega})$ is a singular point if and only if $a_0^6B_2^{-9}J_{12}\in\mathfrak{m}$, and $B_2^{-60}A_5^{12}J_{12}^5\in\mathfrak{m}$.
\end{itemize}
Furthermore, the field extension $L/K$ is tamely ramified in the following situations:
\begin{itemize}
\item[(a)] $\Char k\ne 3$ or $\CC_k\ne C_{000}$;
\item[(b)] $\Char k=3$, $\CC_k=C_{000}$ and $\overline{\omega}$ is ramified.
\end{itemize}
We assume that $L/K$ is tamely ramified. We define $n,r,q$ as follows:
\begin{itemize}
\item[(a)] If $\overline{\omega}$ is non-ramified, $n$ is the least common denominator of $\nu(a_0^{12}J_{12}^{-1})/36$ and $\nu(a_0^6J_{12}^{-1})/12$, $r=n\nu(a_0^{12}J_{12}^{-1})/36$ and $q=n\nu(a_0^6J_{12}^{-1})/12$;
\item[(b)] If $\overline{\omega}$ is ramified and $f^{-1}(\overline{\omega})$ is a regular point, $n$ is the least denominator of $\nu(A_5^{36}J_{12}^{-25})/240$, $q=n\nu(A_5^{36}J_{12}^{-25})/240$ and $r=-2q$;
\item[(c)] If $\overline{\omega}$ is ramified and $f^{-1}(\overline{\omega})$ is singular, $n$ is the least common denominator of $\nu(B_2^{-6}J_{12})/12$ and $\nu(B_2^{-9}J_{12})/12$, $r=n\nu(B_2^{-6}J_{12})/12$ and $q=n\nu(B_2^{-9}J_{12})/12$.
\end{itemize}
Then $[L:K]=n$.
\end{Proposition}
The following lemma introduces the degree of the field extension, $n'$, over which $C^{\chi}$ admits a stable model, and the integers $r'$, $q'$ attached to $C^{\chi}$.
\begin{Lemma}
\label{lem:n-r-qirrdeuciblesingular}
Assume that $\CC_k$ is singular and $\CC_k/\langle\sigma\rangle$ is irreducible. Assume that $C$ is defined by $y^2=P(x)$ and $C^{\chi}$ is defined by $y^2=DP(x)$ where $\nu(D)=1$. We assume moreover that $L/K$ is tamely ramified. The following statements hold:
\begin{itemize}
\item[(a)] If $\overline{\omega}$ is non-ramified, then $\overline{\omega}'$ is non-ramified. Furthermore, $L'$ is a tamely ramified extension of $K$. The integer $n'$ is the least common denominator of $\nu(a_0^{12}J_{12}^{-1})/36$ and $\nu(a_0^6J_{12}^{-1})/12-1/2$, $nr'=n'r$ and $q'=n'(q/n-1/2)$;
\item[(b)] If $\overline{\omega}$ is ramified and $f^{-1}(\overline{\omega})$ is a regular point, then $\overline{\omega}'$ is ramified and $f'^{-1}(\overline{\omega'})$ is regular. Furthermore, $L'$ is a tamely ramified extension of $K$. The integer $n'$ is the least denominator of $\nu(A_5^{36}J_{12}^{-25})/240-1/2$, $q'=n'(q/n-1/2)$ and $r'=-2q'$;
\item[(c)] If $\overline{\omega}$ is ramified and $f^{-1}(\overline{\omega})$ is singular, then $\overline{\omega}'$ is ramified and $f'^{-1}(\overline{\omega}')$ is singular. Furthermore, $L'$ is a tamely ramified extension of $K$. The integer $n'$ is the least common denominator of $\nu(B_2^{-6}J_{12})/12$ and $\nu(B_2^{-9}J_{12})/12-1/2$, $nr'=n'r$ and $q'=n'(q/n-1/2)$.
\end{itemize}
\end{Lemma}
\begin{Proof}
This follows from Proposition \ref{prop:irreduciblesingular} and the fact that $a_0,B_2, A_5,J_{12}$ are homogeneous polynomials in the $a_i$'s of degrees $1,2,5,12$, respectively.
\end{Proof}
We set $u_1=\nu(a_0^{12}J_{12}^{-1})$, $u_2=\nu(a_0^6J_{12}^{-1})$.
\begin{Theorem}
Let $C$ be a hyperelliptic curve defined over $K$. Assume that $L/K$ is tamely ramified and $\CC_k$ consists of one irreducible component with a unique double point. Let $K(\sqrt{D})/K$ be a quadratic extension whose associated character is $\chi$, and $\nu(D)=1$. If $X$ and $X^{\chi}$ are the minimal regular models of the curves $C$ and its quadratic twist by $\chi$, $C^{\chi}$, then the reduction type of $X^{\chi}$ is given in the following table.
\vskip10pt
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
type($X_k$) &$n$& $r$ mod $n$ & $q$ mod $n$ &type($X^{\chi}_k$)\\
\hline
$[{\rm I}_{d-0-0}]$ & 1& & & $[{\rm I}^*_{d-0-0}]$\\
\hline
$[{\rm I}^*_{d/2-0-0}]$ &2&0&&$[{\rm I}_{d/2-0-0}]$ \\
\hline
$[{\rm II}^*_{d/2-0}]$ &2&1&0& $[{\rm II}_{d/2-0}]$\\
\hline
$[{\rm II}_{d/2-0}]$ & 2&1&1&$[{\rm II}^*_{d/2-0}]$\\
\hline
$[{\rm IV}-{\rm II}_{(d-2)/3}]$ &3&1&& $[{\rm II}^*-{\rm II}^*_{(d-2)/3}]$\\
\hline
$[{\rm IV}^*-{\rm II}_{(d-1)/3}]$ &3&2&&$[{\rm II}-{\rm II}^*_{(d-1)/3}]$ \\
\hline
$[{\rm III}-{\rm II}_{(d-2)/4}]$ &4&1&1& $[{\rm III}^*-{\rm II}^*_{(d-2)/4}]$\\
\hline
$[{\rm III}^*-{\rm II}^*_{(d-2)/4}]$ &4&1&3&$[{\rm III}-{\rm II}_{(d-2)/4}]$ \\
\hline
$[{\rm III}-{\rm II}^*_{(d-2)/4}]$ &4&3&1 &$[{\rm III}^*-{\rm II}_{(d-2)/4}]$ \\
\hline
$[{\rm III}^*-{\rm II}_{(d-2)/4}]$ &4&3&3& $[{\rm III}-{\rm II}^*_{(d-2)/4}]$\\
\hline
$[{\rm II}^*-{\rm II}^*_{(d-4)/6}]$ &6&2&& $[{\rm IV}-{\rm II}_{(d-4)/6}]$\\
\hline
$[{\rm II}-{\rm II}^*_{(d-2)/6}]$ &6&4&& $[{\rm IV}^*-{\rm II}_{(d-2)/6}]$\\
\hline
\end{tabular}
\end{center}
\end{Theorem}
\begin{Proof}
According to Lemma \ref{lem:n-r-qirrdeuciblesingular}, we have three subcases to consider: $\overline{\omega}$ is non-ramified, $f^{-1}(\overline{\omega})$ is regular, or $\overline{\omega}$ is ramified and $f^{-1}(\overline{\omega})$ is singular. Unless otherwise stated, we will assume that $\overline{\omega}$ is non-ramified since the proofs for the other two subcases will be similar. We recall that the degree of singularity of the unique double point in $\CC_k$ is given by $\nu_L(J_{10}^6I_{12}^{-5})/6$, see Proposition \ref{prop:degreeofsingularity}. Moreover, $\CC^{\chi}_k$ consists of one irreducible component with a unique double point.
If $X$ has reduction type $[{\rm I}_{d-0-0}]$, then this implies that the degree of singularity of the ordinary double point in $\CC_k$ is $d$. In view of Proposition \ref{prop:irreduciblesingular}, since $n=1$, one has $\displaystyle 36\mid u_1$ and $12 \mid u_2$. Now, $n'$ is the least common denominator of $u_1/36$ and $u_2/12-1/2$, Lemma \ref{lem:n-r-qirrdeuciblesingular}. This yields $n'=2$, $r'=2r= 0$ mod $2$. Therefore, the reduction type of $X^{\chi}$ is $[{\rm I}^*_{d'/2-0-0}]$ where $d'$ is the degree of singularity of the double point of $\CC^{\chi}_k$. One observes that $d'=\nu_{L'}(J_{10}'^6I_{12}'^{-5})/6=2\nu(J_{10}^6I_{12}^{-5})/6=2d$.
Assume that the reduction type of $X$ is $[{\rm I}^*_{d/2-0-0}]$. Since $\overline{r}=0$, it follows that $36\mid u_1$. One has $\displaystyle \frac{u_1}{36}\in\Z,\frac{u_2}{12}\in\frac{\Z}{2}$. Now, $n'$ is the least common denominator of $u_1/36$ and $u_2/12-1/2\in\Z$, i.e., $n'=1$. The reduction type of $X^{\chi}$ is $[{\rm I}_{d'-0-0}]$ where $d'$ is the degree of singularity of the double point of $\CC^{\chi}_k$, and is given by $\displaystyle \nu(J_{10}'^6I_{12}'^{-5})/6=\nu(J_{10}^6I_{12}^{-5})/6=d/2$. Thus, the reduction type of $X^{\chi}$ is $[{\rm I}_{d/2-0-0}]$.
When $n=2$ and $r=1$ mod $2$, one has $\displaystyle u_1/36\in\frac{1}{2}\Z$. Since $n'$ is the least common denominator of $u_1/36$ and $u_2/12-1/2$, it follows that $n'=2$, $r'=r$, and the degree of singularity of the double point in $\CC^{\chi}_k$ is $d'=d$. Moreover, one has $q'=q-1$. Thus, If $q=0$ mod $2$, then the reduction type of $X$ is $[{\rm II}^*_{d/2-0}]$ and $q'=1$ mod $2$, hence the reduction type of $X^{\chi}$ is $[{\rm II}_{d/2-0}]$. If $q=1$ mod $2$, then the reduction type of $X$ is $[{\rm II}_{d/2-0}]$ and $q'=0$ mod $2$, hence the reduction type of $X^{\chi}$ is $[{\rm II}^*_{d/2-0}]$. One observes that when $r=1$ mod $2$, if $\overline{\omega}$ ir ramified, then $f^{-1}(\overline{\omega})$ cannot be regular as according to Proposition \ref{prop:irreduciblesingular}, $r$ must be even.
If $n=3$, then the least common denominator of $u_1/36$ and $u_2/12$ is $3$. Since $n'$ is the least common denominator of $u_1/36$ and $u_2/12-1/2$, it follows that $n'=6$ and $r'=2r$. If $r=1$ mod $3$ (the reduction type of $X$ is $[{\rm IV}-{\rm II}_{(d-2)/3}]$), then $r'=2$ mod $6$ and the reduction type of $X^{\chi}$ is $[{\rm II}^*-{\rm II}^*_{(d'-4)/6}]$ where $d'=2d$. If $r=2$ mod $3$ (the reduction type of $X$ is $[{\rm IV}^*-{\rm II}_{(d-1)/3}]$), then $r'= 4$ mod $6$ and the reduction type of $X^{\chi}$ is $[{\rm II}-{\rm II}^*_{(d'-2)/6}]$ where $d'=2d$.
If $n=4$, then the least common denominator of $u_1/36$ and $u_2/12-1/2$ is $n'=4$. Moreover, $r'=r$, $q'=q-2$, and the degree of singularity $d'=d$.
If $n=6$, one has that either $r:=u_1/6=2$ or $4$ mod $6$. In particular, the least denominator of $u_1/36$ is either $1$ or $3$, which implies that the least denominator of $u_2/12$ is either $6$ or $2$. It follows that the least common denominator of $u_1/36$ and $u_2/12-1/2$ is $n'=3$, $r=2r'$, and the degree of singularity $d'=d/2$. Therefore, if the reduction type of $X$ is $[{\rm II}^*-{\rm II}^*_{(d-4)/6}]$, then the reduction type of $X^{\chi}$ is $[{\rm IV}-{\rm II}_{(d-4)/6}]$, whereas if the reduction type of $X$ is $[{\rm II}-{\rm II}^*_{(d-2)/6}]$, then the reduction type of $X^{\chi}$ is $[{\rm IV}^*-{\rm II}_{(d-2)/6}]$
\end{Proof}
In the following theorem, we find the reduction type of the minimal regular model of $C^{\chi}$ when the special fiber of the stable model is irreducible with two double points.
\begin{Theorem}
Let $C$ be a hyperelliptic curve defined over $K$. Assume that $L/K$ is tamely ramified and $\CC_k$ consists of one irreducible component with exactly two ordinary double points. Let $K(\sqrt{D})/K$ be a quadratic extension whose associated character is $\chi$, and $\nu(D)=1$. If $X$ and $X^{\chi}$ are the minimal regular models of the curves $C$ and its quadratic twist by $\chi$, $C^{\chi}$, then the reduction type of $X^{\chi}$ is given in the following table.
\vskip10pt
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
type($X_k$)& $n$ &$r$ mod $n$ & $f^{-1}(\omega)$ &type($X^{\chi}_k$)\\
\hline
$[{\rm I}_{d_1-d_2-0}]$ & 1& && $[{\rm I}^*_{d_1-d_2-0}]$\\
\hline
$[{\rm I}^*_{d_1/2-d_2/2-0}]$&2 &0& &$[{\rm I}_{d_1/2-d_2/2-0}]$\\
\hline
$[2{\rm I}_{d_1}-0]$ &2&1®ular &$[2{\rm I}_{d_1}-0]$ \\
\hline
$[{\rm II}_{d_1/2-d_2/2}]$ &2&1 & singular&$[{\rm II}_{d_1/2-d_2/2}]$ \\
\hline
$[{\rm III}_{d_1/2}]$ & 4&& &$[{\rm III}_{d_1/2}]$\\
\hline
\end{tabular}
\end{center}
\end{Theorem}
\begin{Proof}
Unless otherwise stated, we will assume that $\overline{\omega}$ is non-ramified since the proofs for the other two subcases are similar. The integers $d_1$ and $d_2$ are the degrees of singularity of the two ordinary double points in $\CC_k$. Recall that $\CC^{\chi}_k$ is irreducible with exactly two ordinary double points, Lemma \ref{Lem:StableModelofTwist}. The degrees of singularity of the double points is given in Proposition \ref{prop:degreeofsingularity}.
Assume that the reduction type of $X$ is $[{\rm I}_{d_1-d_2-0}]$. Since $n'$ is the least common denominator of $u_1/36$ nd $u_2/12-1/2$, see Lemma \ref{lem:n-r-qirrdeuciblesingular}, one has $n'=2$, $r'=2r=0$ mod $2$. Therefore, the reduction type of $X^{\chi}$ is $[{\rm I}^*_{d'_1/2-d'_2/2-0}]$ where $d'_1$, $d_2'$ are the degrees of singularity of the two ordinary double points in $\CC^{\chi}_k$. In fact, since $n'=2$, one obtains $d_i'=2d_i$, $i=1,2$.
If the reduction type of $X$ is $[{\rm I}^*_{d_1/2-d_2/2-0}]$, then $r:=2u_1/36=0$ mod $2$ which implies that $u_1/36\in\Z$ and the least denominator of $u_2/12$ is $2$. Thus, $n'=1$ and the reduction type of $X^{\chi}$ is $[{\rm I}_{d_1'-d_2'-0}]$ where $d_i'=d_i/2$.
We assume that the reduction type of $X$ is $[2{\rm I}_{d_1}-0]$. One has that the degrees of the double points in $\CC_k$ are the same, $d_1$. One has $r:=2u_1/36=1$ mod $2$ which implies that the least denominator of $u_1/36$ is $2$. Therefore, $n'=2$, $r'=r$. This yields that reduction type of $X^{\chi}$ is $[2{\rm I}_{d_1}-0]$. One observes that subcase (b) in Proposition \ref{prop:irreduciblesingular} does not yield the reduction type $[2{\rm I}_{d_1}-0]$ since otherwise $r$ is even, moreover, subcase (c) does not yield this reduction type since $f^{-1}(\overline{\omega})$ is regular.
We assume that the reduction type of $X^{\chi}$ is $[{\rm II}_{d_1/2-d_2/2}]$. One has $n=2$, $r=1$, $f^{-1}(\overline{\omega})$ is singular. If $\overline{\omega}$ is non-ramified, then $n'=2,r'=r$ and $f^{-1}(\overline{\omega})$ is singular. If $\overline{\omega}$ is non ramified, then $6\mid u_1:=\nu(B_2^{-6}J_{12})$ and $12\nmid u_1$. Therefore, $n'=2$, and $r'=r$, see Lemma \ref{lem:n-r-qirrdeuciblesingular}. Thus, the reduction type of $X^{\chi}$ is the same as that of $X$.
We assume that the reduction type of $X$ is $[{\rm III}_{d_1/2}]$ and $\overline{\omega}$ is non-ramified. Since $n=4$, one has $n'=4$, see Lemma \ref{lem:n-r-qirrdeuciblesingular}. Thus, the reduction type of $X^{\chi}$ is the same as that of $X$. The same holds when $\overline{\omega}$ is ramified.
\end{Proof}
One remarks that the degrees of singularity of the two double points in $\CC_k$ when the reduction type of $X$ is either $[2{\rm I}_{d_1}-0]$ or $[{\rm III}_{d_1/2}]$ are both equal to $d_1$.
\begin{Theorem}
\label{thm:threerationalcurves}
Let $C$ be a hyperelliptic curve defined over $K$. Assume that $L/K$ is tamely ramified and $\CC_k$ is the union of two rational curves intersecting transversally in three ordinary double points. Let $K(\sqrt{D})/K$ be a quadratic extension whose associated character is $\chi$, and $\nu(D)=1$. If $X$ and $X^{\chi}$ are the minimal regular models of the curves $C$ and its quadratic twist by $\chi$, $C^{\chi}$, then the reduction type of $X^{\chi}$ is given in the following table.
\vskip10pt
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
type($X_k$) &$n$& $r$&$q$&type($X^{\chi}_k$)\\
\hline
$[{\rm I}_{d_1-d_2-d_3}]$ &1&&&$[{\rm I}^*_{d_1-d_2-d_3}]$ \\
\hline
$[{\rm I}^*_{d_1/2-d_2/2-d_3/2}]$&2&0&&$[{\rm I}_{d_1/2-d_2/2-d_3/2}]$\\
\hline
$[{\rm II}^*_{e_1/2-e_2}]$& 2&1&0&$[{\rm II}_{e_1/2-e_2}]$\\
\hline
$[{\rm II}_{e_1/2-e_2}]$ &2&1&1& $[{\rm II}^*_{e_1/2-e_2}]$\\
\hline
$[{\rm III}_{d_1}]$& 3&&&$[{\rm III}^*_{d_1}]$\\
\hline
$[{\rm III}^*_{d_1/2}]$ &6&&& $[{\rm III}_{d_1/2}]$\\
\hline
\end{tabular}
\end{center}
\end{Theorem}
\begin{Proof}
We will assume that $\overline{\omega}$ is non-ramified, unless otherwise stated, since the argument for the ramified case will be similar. The integers $d_1,d_2,d_3$ are the degrees of singularity of the three ordinary double points, and can be evaluated using Proposition \ref{prop:degreeofsingularity}. According to Lemma \ref{Lem:StableModelofTwist}, the special fiber $\CC^{\chi}_k$ of the stable model $\CC^{\chi}$ is the union of two rational curves intersecting transversally in three double points.
We assume the reduction type of $X$ is $[{\rm I}_{d_1-d_2-d_3}]$. One knows that if $n=1$, then $n'=2$, $r'=2r$, see Lemma \ref{lem:n-r-qirrdeuciblesingular}. Therefore, the reduction type of $X^{\chi}$ is $[{\rm I}^*_{d'_1/2-d'_2/2-d'_3/2}]$ where $d_i'=2d_i$.
If the reduction type of $X$ is $[{\rm I}^*_{d_1/2-d_2-d_3}]$, one has $n'=1$ and $d_i'=d_i/2$. Thus, the reduction type of $X^{\chi}$ is $[{\rm I}_{d_1/2-d_2/2-d_3/2}]$.
When $n=2$ and $r=1$ mod $2$, one obtains that $n'=2$, $r'=r$ and $q'=q-1$. Moreover, the degrees of singularity of the three double points remain the same. Thus, if the reduction type of $X$ is $[{\rm II}^*_{e_1/2-e_2}]$ ($[{\rm II}_{e_1/2-e_2}]$), then the reduction type of $X^{\chi}$ is $[{\rm II}_{e_1/2-e_2}]$ ($[{\rm II}^*_{e_1/2-e_2}]$). One observes that if $\overline{\omega}$ is ramified and $f^{-1}(\overline{\omega})$ is regular, then the reduction type of $X$ can neither be $[{\rm II}^*_{e_1/2-e_2}]$ nor $[{\rm II}_{e_1/2-e_2}]$ since otherwise $r$ is even.
We assume that the reduction type of $X$ is $[{\rm III}_{d_1}]$. According to Lemma \ref{lem:n-r-qirrdeuciblesingular}, one has $n'=6$. In particular, the reduction type of $X^{\chi}$ is $[{\rm III}^*_{d'_1/2}]$ where $d'_1=2d_1$.
We assume that the reduction type of $X$ is $[{\rm III}^*_{d_1/2}]$. Since $r$ is even, in view of Lemma \ref{lem:n-r-qirrdeuciblesingular}, one has $n'=3$. In particular, the reduction type of $X^{\chi}$ is $[{\rm III}_{d'_1}]$ where $d'_1=d_1/2$.
\end{Proof}
In Theorem \ref{thm:threerationalcurves}, when the reduction type of $X$ is either $[{\rm II}^*_{e_1/2-e_2}]$ or $[{\rm II}_{e_1/2-e_2}]$, then exactly two of the ordinary double points in $\CC_k$ have the same degree of singularity $e_1$, and the degree of singularity of the third ordinary double point is $e_2$. If the reduction type of $X$ is either $[{\rm III}_{d_1}]$ or $[{\rm III}^*_{d_1/2}]$, then the three ordinary double points of $\CC_k$ have the same degree of singularity $d_1$.
\subsection{$\CC_k/\langle\sigma\rangle$ is not irreducible and $L/K$ is tamely ramified}
Now, we assume that $\CC_k/\langle\sigma\rangle$ is not irreducible where the field $L$ over which $C$ admits a stable model is a tamely ramified extension.
Assuming that $\Char k\ne 3$ and $\mathcal{Z}_k$ is the union of two projective curves intersecting in one point. The possible divisors of $[L:K]$ are $2$ and $3$. It follows that $L/K$ is tamely ramified. Letting $E_1$ and $E_2$ be the irreducible components of $\CC_k$, with $\overline{\omega}\in f(E_1)$, we set:
\begin{align*}d_K=\left\{\begin{array}{ll}
\nu(J_{10}J_2^{-5})/12 & \textrm{if $E_1,E_2$ are smooth} \\
\nu(I_{12}J_2^{-6})/12 & \textrm{if $\CC_k$ has a unique smooth component}\\
\nu(I_4J_2^{-2})/4& \textrm{if $E_1,E_2$ are singular} \end{array}\right.\end{align*}
The degree of singularity of the point of intersection $E_1\cap E_2$ in $\CC$ is $d=[L:K]d_K$. It follows that the degree of singularity of $f(E_1\cap E_2)$ in $\mathcal{Z}$ is $2d$.
The following proposition summarizes the behavior of the stable model in that case.
\begin{Proposition}
\label{prop:notirreducible}
Assume $\Char k\ne 3$. Assume that $\CC_k/\langle\sigma\rangle$ is not irreducible.
\begin{itemize}
\item[(a)] The point $\overline{\omega}$ is non-ramified if and only if $a_0^{-2}B_2^3J_{2}^{-2}\in R,\;,a_0^{-4}A_3^{2}J_{2}^{-1},a_0^{-20}A_5^6J_2^{-5}\in R$ and at least one of the two latter elements are invertible in $R$;
\item[(b)] $f^{-1}(\overline{\omega})$ is a regular point if and only if $a_0^{20}A_5^{-6}J_{2}^{5}\in \mathfrak{m}$ and $B_2^{10}A_5^{-2}J_{2}^{-5}\in R$;
\item[(c)] $\overline{\omega}$ is regular and $f^{-1}(\overline{\omega})$ is a singular point if and only if $a_0^2B_2^{-3}J_{2}^2\in\mathfrak{m}$ and $B_2^{-10}A_5^{2}J_{2}^5\in\mathfrak{m}$;
\item[(d)] $\overline{\omega}$ is singular if and only if $a_0^{-2}B_2^{3}J_{2}^{-2}\in R$ and $a_0^{-4}A_3^2J_2^{-1},a_0^{-20}A_5^6J_2^{-5}\in\mathfrak{m}$.
\end{itemize}
Assume that $2\mid \nu(J_2)$, then one has
\begin{itemize}
\item[(a)] If $\overline{\omega}$ is non-ramified, $n$ is the smallest common denominator of $d_K$ and $\nu(a_0J_{2})/6$, $r=n\nu(a_0J_{2})/6$;
\item[(b)] If $\overline{\omega}$ is regular and ramified, and $f^{-1}(\overline{\omega})$ is a regular point, then $n$ is the least common denominator of $d_K$ and $\nu(A_5^{2}J_{2})/8$, $r=n\nu(A_5^2J_2)/8$;
\item[(c)] If $\overline{\omega}$ is regular such that $f^{-1}(\overline{\omega})$ is singular, then $n$ is the least common denominator of $d_K$ and $\nu(B_2)/4$, $r=n\nu(B_2)/4$;
\item[(d)] If $\overline{\omega}$ is singular, $n$ is the least common denominator of $d_K$ and $r_K$, $r=nr_K$, where
\[r_K=\nu(a_0)/2+\min\{d_K/2,\nu(A_2^{-3}A_3^2)/8,\nu(A_2^{-5}(A_2A_3-3A_5)^2)/12\}\in\Q.\]
\end{itemize}
Then $[L:K]=n$ and $d=nd_K$.
If $2\nmid \nu(J_2)$, then $d_K+\nu(a_0)=2r_K$ and $[L:K]=2m$, where $m$ is the least denominator of $d_K$, and $r=md_K$.
\end{Proposition}
\begin{Lemma}
\label{lem:n-r-qnotirreducible}
Assume $\Char k\ne 3$. Assume that $\CC_k/\langle\sigma\rangle$ is not irreducible. Then $L'/K$ is tamely ramified. Moreover,
\begin{itemize}
\item[(a)] If the point $\overline{\omega}$ is non-ramified, then $\overline{\omega}'$ is non-ramified;
\item[(b)] If $f^{-1}(\overline{\omega})$ is a regular point, then $f'^{-1}(\overline{\omega}')$ is regular;
\item[(c)] If $\overline{\omega}$ is regular and $f^{-1}(\overline{\omega})$ is a singular point, then $\overline{\omega}'$ is regular and $f'^{-1}(\overline{\omega}')$ is singular;
\item[(d)] If $\overline{\omega}$ is singular, then $\overline{\omega}'$ is singular.
\end{itemize}
Moreover, if $2\mid \nu(J_2)$, then $2\mid \nu(J_2')$, and the following statements hold:
\begin{itemize}
\item[(a)] If $\overline{\omega}$ is non-ramified, $n'$ is the least common denominator of $d_K$ and $\nu(a_0J_{2})/6+1/2$, $nr'=n'(r+n/2)$;
\item[(b)] If $\overline{\omega}$ is regular and ramified, and $f^{-1}(\overline{\omega})$ is a regular point, then $n'$ is the least common denominator of $d_K$ and $\nu(A_5^{2}J_{2})/8+3/2$, $nr'=n'(r+3n/2)$;
\item[(c)] If $\overline{\omega}$ is regular such that $f^{-1}(\overline{\omega})$ is singular, then $n'$ is the least common denominator of $d_K$ and $\nu(B_2)/4+1/2$, $nr'=n'(r+n/2)$;
\item[(d)] If $\overline{\omega}$ is singular, $n$ is the least common denominator of $d_K$ and $r_K+1/2$, $nr'=n'(r+n/2)$.
\end{itemize}
Moreover, $nd'=n'd$.
If $2\nmid \nu(J_2)$, then $2\nmid \nu(J_2')$. Furthermore, $[L':K]=[L:K]=2m$, where $m$ is the least denominator of $d_K$, and $r'=r$.
\end{Lemma}
\begin{Proof}
We assume that $C$ is defined by $y^2=P(x)=a_0x^6+\ldots+a_6\in R[x]$, and $C'$ is defined by $y^2=DP(x)$ and $\nu(D)=1$.
According to Proposition \ref{prop:notirreducible}, $\overline{\omega}'$ is non-ramified if $a_0'^{-2}B_2'^3J_{2}'^{-2}\in R$, $a_0'^{-4}A_3'^{2}J_{2}'^{-1},$ $a_0'^{-20}A_5'^6J_2'^{-5}\in R$ and at least one of the two latter elements are invertible in $R$. These elements are quotients of homogeneous invariants where the numerator is of the same degree as the denominator, and so each of these quotients is homogeneous of degree $0$. Therefore, this is equivalent to saying that $a_0^{-2}B_2^3J_{2}^{-2}\in R,\;,a_0^{-4}A_3^{2}J_{2}^{-1},a_0^{-20}A_5^6J_2^{-5}\in R$ and at least one of the two latter elements are invertible in $R$. This implies that $\overline{\omega}'$ is non-ramified if $\overline{\omega}$ is non-ramified. The proofs of (b), (c), and (d) are similar.
Since $J_2'=d^2J_2$, one has $2\mid(\nu(d^2)+\nu(J_2))=2+\nu(J_2)$ if and only if $2\mid \nu(J_2)$.
If $\overline{\omega}$ is ramified, then $\overline{\omega}'$ is ramified and $n'$ is the least common denominator of $d'_K=d_K$ and $\nu(a_0'J_2')/6=\nu(a_0J_2)/6+1/2$. Moreover, $r'=n'\nu(a_0'J_2')/6=n'(\nu(a_0J_2)+3)/6$. The same argument holds for the other subcases.
\end{Proof}
\begin{Theorem}
\label{thm:twoellipticcurves}
Let $C$ be a hyperelliptic curve defined over $K$. Assume that $L/K$ is tamely ramified and $\CC_k$ is the union of two elliptic curves intersecting in one point. Let $K(\sqrt{D})/K$ be a quadratic extension whose associated character is $\chi$, and $\nu(D)=1$. Let $X$ and $X^{\chi}$ be the minimal regular models of the curves $C$ and its quadratic twist by $\chi$, $C^{\chi}$.
If $2\mid\nu(J_2)$, then the reduction type of $X^{\chi}$ is given in the following table.
\vskip10pt
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
type($X_k$) &$n$ & $d$ mod $n$ & $r$ mod $n$ &type($X^{\chi}_k$)\\
\hline
$[{\rm I}_0-{\rm I}_0-d]$ & 1 & & & $[{\rm I}^*_0-{\rm I}^*_0-(d-1)]$\\
\hline
$[{\rm I}^*_0-{\rm I}^*_0-(d-2)/2]$ & 2& 0& &$[{\rm I}_0-{\rm I}_0-d/2]$\\
\hline
$[{\rm I}_0-{\rm I}^*_0-(d-1)/2]$ &2& 1 & &$[{\rm I}_0-{\rm I}^*_0-(d-1)/2]$ \\
\hline
$[{\rm IV}-{\rm IV}^*-(d-3)/3]$& 3 & 0& &$[{\rm II}-{\rm II}^*-(d-3)/3]$\\
\hline
$[{\rm I}_0-{\rm IV}-(d-1)/3]$ & 3 & 1& 0 or 1&$[{\rm I}^*_0-{\rm II}^*-(d-4)/3]$\\
\hline
$[{\rm IV}^*-{\rm IV}^*-(d-4)/3]$ &3 & 1 & 2& $[{\rm II}-{\rm II}-(d-1)/3]$\\
\hline
$[{\rm I}_0-{\rm IV}^*-(d-2)/3]$ & 3 & 2& 0 or 2& $[{\rm I}_0^*-{\rm II}-(d-2)/3]$\\
\hline
$[{\rm IV}-{\rm IV}-(d-2)/3]$ &3& 2& 1&$[{\rm II}^*-{\rm II}^*-(d-5)/3]$\\
\hline
$[{\rm III}-{\rm III}^*-(d-4)/4]$ &4&0&&$[{\rm III}-{\rm III}^*-(d-4)/4]$\\
\hline
$[{\rm I}_0-{\rm III}-(d-1)/4]$ &4&1& 0 or 1& $[{\rm I}^*_0-{\rm III}^*-(d-5)/4]$\\
\hline
$[{\rm I}^*_0-{\rm III}^*-(d-5)/4]$ & 4 &1 & 2 or 3& $[{\rm I}_0-{\rm III}-(d-1)/4]$\\
\hline
$[{\rm III}-{\rm III}-(d-2)/4]$ &4& 2& 1& $[{\rm III}^*-{\rm III}^*-(d-6)/4]$\\
\hline
$[{\rm III}^*-{\rm III}^*-(d-6)/4]$ &4&2&3&$[{\rm III}-{\rm III}-(d-2)/4]$ \\
\hline
$[{\rm I}_0-{\rm III}^*-(d-3)/4]$ &4&3& 0 or 3& $[{\rm I}^*_0-{\rm III}-(d-3)/4]$\\
\hline
$[{\rm I}^*_0-{\rm III}-(d-3)/4]$ &4&3&1 or 2& $[{\rm I}_0-{\rm III}^*-(d-3)/4]$ \\
\hline
$[{\rm II}-{\rm II}^*-(d-6)/6]$ &6& 0& &$[{\rm IV}-{\rm IV}^*-(d-6)/6]$\\
\hline
$[{\rm I}_0-{\rm II}-(d-1)/6]$ &6&1&0 or 1 & $[{\rm I}^*_0-{\rm IV}^*-(d-7)/6]$\\
\hline
$[{\rm II}^*-{\rm IV}-(d-7)/6]$ &6&1& 2 or 5 &$[{\rm II}^*-{\rm IV}-(d-7)/6]$\\
\hline
$[{\rm I}^*_0-{\rm IV}^*-(d-7)/6]$ &6& 1& 3 or 4&$[{\rm I}_0-{\rm II}-(d-1)/6]$ \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
type($X_k$) & $n$ & $d$ mod $n$ & $r$ mod $n$ &type($X^{\chi}_k$)\\
\hline
$[{\rm II}-{\rm II}-(d-2)/6]$&6&2&1 &$[{\rm IV}^*-{\rm IV}^*-(d-8)/6]$\\
\hline
$[{\rm I}^*_0-{\rm II}^*-(d-8)/6]$&6&2& 3 or 5 &$[{\rm I}_0-{\rm IV}-(d-2)/6]$\\
\hline
$[{\rm II}-{\rm IV}-(d-3)/6]$ &6& 3& 1 or 2&$[{\rm II}^*-{\rm IV}^*-(d-9)/6]$\\
\hline
$[{\rm II}^*-{\rm IV}^*-(d-9)/6]$ &6&3&4 or 5&$[{\rm II}-{\rm IV}-(d-3)/6]$\\
\hline
$[{\rm I}^*_0-{\rm II}-(d-4)/6]$ &6&4&1 or 3&$[{\rm I}_0-{\rm IV}^*-(d-4)/6]$\\
\hline
$[{\rm II}^*-{\rm II}^*-(d-10)/6]$ &6&4&5&$[{\rm IV}-{\rm IV}-(d-4)/6]$\\
\hline
$[{\rm I}_0-{\rm II}^*-(d-5)/6]$ &6&5&0 or 5&$[{\rm I}^*_0-{\rm IV}-(d-5)/6]$\\
\hline
$[{\rm II}-{\rm IV}^*-(d-5)/6]$ &6& 5& 1 or 4& $[{\rm II}-{\rm IV}^*-(d-5)/6]$\\
\hline
$[{\rm I}^*_0-{\rm IV}-(d-5)/6]$ & 6& 5& 2 or 3&$[{\rm I}_0-{\rm II}^*-(d-5)/6]$\\
\hline
$[{\rm II}^*-{\rm III}-(d-13)/12]$ &12&1& 3 or 10&$[{\rm IV}-{\rm III}^*-(d-13)/12]$\\
\hline
$[{\rm IV}-{\rm III}^*-(d-13)/12]$ &12 & 1& 4 or 9& $[{\rm II}^*-{\rm III}-(d-13)/12]$\\
\hline
$[{\rm II}-{\rm III}-(d-5)/12]$ & 12 & 5& 2 or 3 & $[{\rm IV}^*-{\rm III}^*-(d-17)/12]$\\
\hline
$[{\rm IV}^*-{\rm III}^*-(d-17)/12]$ &12 & 5 & 8 or 9 &$[{\rm II}-{\rm III}-(d-5)/12]$\\
\hline
$[{\rm IV}-{\rm III}-(d-7)/12]$ &12 & 7& 3 or 4 & $[{\rm II}^*-{\rm III}^*-(d-19)/12]$\\
\hline
$[{\rm II}^*-{\rm III}^*-(d-19)/12]$ & 12 & 7 & 9 or 10& $[{\rm IV}-{\rm III}-(d-7)/12]$\\
\hline
$[{\rm IV}^*-{\rm III}-(d-11)/12]$ & 12 & 11& 3 or 8&$[{\rm II}-{\rm III}^*-(d-11)/12]$\\
\hline
$[{\rm II}-{\rm III}^*-(d-11)/12]$ & 12 & 11& 2 or 9&$[{\rm IV}^*-{\rm III}-(d-11)/12]$\\
\hline
\end{tabular}
\end{center}
If $2\nmid\nu(J_2)$, then the reduction type of $X^{\chi}$ is given in the following table.
\begin{center}
\begin{tabular}{|c|c|c||c|}
\hline
type($X_k$) & $n$&$r$ mod $n/2$ &type($X^{\chi}_k$)\\
\hline
$[2{\rm I}_0-r]$ & 2&&$[2{\rm I}_0-r]$ \\
\hline
$[2{\rm I}^*_0-(r-1)/2]$ &4&&$[2{\rm I}^*_0-(r-1)/2]$\\
\hline
$[2{\rm IV}-(r-1)/3]$ &6&1&$[2{\rm IV}-(r-1)/3]$\\
\hline
$[2{\rm IV}^*-(r-2)/3]$ &6&2&$[2{\rm IV}^*-(r-2)/3]$\\
\hline
$[2{\rm III}-(r-1)/4]$ &8&1&$[2{\rm III}-(r-1)/4]$\\
\hline
$[2{\rm III}^*-(r-3)/4]$&8&3 &$[2{\rm III}^*-(r-3)/4]$\\
\hline
$[2{\rm II}-(r-1)/6]$ &12&1&$[2{\rm II}-(r-1)/6]$\\
\hline
$[2{\rm II}^*-(r-5)/6]$ &12&5&$[2{\rm II}^*-(r-5)/6]$\\
\hline
\end{tabular}
\end{center}
\end{Theorem}
\begin{Proof}
One knows that if $\CC_k$ consists of two elliptic curves, then so does $\CC^{\chi}_k$, see Proposition \ref{Lem:StableModelofTwist}. We assume for now that $2\mid\nu(J_2)$. Unless otherwise stated, we are assuming that $\overline{\omega}$ is non-ramified since the proof for the other subcases (b), (c), (d) of Lemma \ref{lem:n-r-qnotirreducible} is similar. Recall that $d$ is the degree of singularity of the intersection point of the irreducible components of $\CC_k$, see Proposition \ref{prop:degreeofsingularity}. Since $\CC_k$ consists of two elliptic curves, one has $d_K=\nu(J_{10}J_2^{-5})/12$. We set $u=\nu(a_0J_2)$.
When the type of $X_k$ is $[{\rm I}_0-{\rm I}_0-d]$, one has $n=1$. In view of Lemma \ref{lem:n-r-qnotirreducible}, $n'=2$, $r'=2r+1$, and $d'=2d=0$ mod $2$. Therefore, the reduction type of $X^{\chi}$ is $[{\rm I}^*_0-{\rm I}^*_0-(d'-2)/2]$.
When $n=2$ and $d=2d_K=0$ mod $2$, one sees that the smallest denominator of $d_K$ is $1$. It follows that $\displaystyle u/6\in\frac{1}{2}\Z$, see Proposition \ref{prop:notirreducible}. Thus, $n'=1$, see Lemma \ref{lem:n-r-qnotirreducible}. It follows that the reduction type of $X^{\chi}$ is $[{\rm I}_0-{\rm I}_0-d']$ where $d=2d'$. When $n=2$ and $d=1$ mod $2$, i.e., the reduction type of $X$ is $[{\rm I}_0-{\rm I}^*_0-(d-1)/2]$, one has that $\displaystyle d_K\in\frac{1}{2}\Z$. Thus, $n'=2$ and $d'=d$. Therefore, the reduction type of $X^{\chi}$ is $[{\rm I}_0-{\rm I}^*_0-(d'-1)/2]$.
When $n=3$ and $d=3d_K=0$ mod $3$ which implies that the least denominator of $d_K$ is not divisible by $3$, one has $\displaystyle u\in \frac{1}{3}\Z$ and so $n'=6$, $d'=2d=0$ mod $6$, see Lemma \ref{lem:n-r-qnotirreducible}. It follows that the reduction type of $X^{\chi}$ is $[{\rm II}-{\rm II}^*-(d'-6)/6]$. When $n=3$ and $d=1$ mod $3$, one has that $\displaystyle d_K\in\frac{1}{3}\Z$. Thus, $n'=6$, $r'=2r+3$, and $d'=2d$ is $2$ mod $6$. If $r=0$ or $1$ mod $3$, then $r'=3$ or $5$ mod $6$. In other words, the reduction type of $X^{\chi}$ is $[{\rm I}^*_0-{\rm II}^*-(d'-8)/6]$. If $r=2$ mod $3$, then $r'=1$ mod $6$ and the reduction type of $X^{\chi}$ is $[{\rm II}-{\rm II}-(d'-2)/6]$. When $n=3$ and $d=2$ mod $3$, one has $\displaystyle d_K\in\frac{1}{3}\Z$. Thus, $n'=6$ and $d'=2d=4$ mod $6$. If, moreover, $r=0$ or $2$ mod $3$, then $r'=3$ or $1$ mod $6$, and the reduction type of $X^{\chi}$ is then $[{\rm I}_0^*-{\rm II}-(d'-4)/6]$, whereas if $r=1$ mod $3$, then $r'=5$ mod $6$ and the reduction type of $X^{\chi}$ is then $[{\rm II}^*-{\rm II}^*-(d'-10)/6]$.
When $n=4$ and $d=0$ mod $4$, one has that the denominator of $d_K$ is $1$ which implies that when $\overline{\omega}$ is non-ramified, the reduction type of $X$ cannot be $[{\rm III}-{\rm III}^*-(d-4)/4]$ since $4$ does not divide the denominator of $u/6$. Therefore, we assume that $\overline{\omega}$ is regular and ramified and $f^{-1}(\overline{\omega})$ is regular. Now, according to Proposition \ref{prop:notirreducible} the least denominator of $\nu(A_5^2J_2)/8$ is $4$, which yields that $n'=4$, $d'=d$ and $r'=r+6$, see Lemma \ref{lem:n-r-qnotirreducible}, hence the reduction type of $X^{\chi}$ is the same as this of $X$, namely $[{\rm III}-{\rm III}^*-(d-4)/4]$. The same argument holds if $\overline{\omega}$ satisfies the hypotheses of (c) or (d) in Lemma \ref{lem:n-r-qnotirreducible}. Now, we assume again that $\overline{\omega}$ is non-ramified. If $d\ne 0\mod 4$, then the denominator of $d_K$ is either $2$ or $4$. But since $n=4$ is the least common denominator of $d_K$ and $u/6$, it follows that $\displaystyle d_K\in\frac{1}{4}\Z$. This yields that $n'=4$, $d'=d$ and $r'=r+6$. Hence, the reduction type of $X^{\chi}$ follows.
When $n=6$ and $d=6d_K=0$ mod $6$, one has $d_K\in\Z$. It follows that $\displaystyle u/6\in\frac{1}{6}\Z$. One then has $n'=3$ and $d=2d'$. Thus, the reduction type of $X^{\chi}$ is $[{\rm IV}-{\rm IV}^*-(d'-3)/3]$. If $d=1$ or $5$ mod $6$, then $\displaystyle d_K\in\frac{1}{6}\Z$. This yields that $n'=6$, $d'=d$ and $r'=r+3$. If $d=2$ or $4$ mod $6$, then $\displaystyle d_K\in\frac{1}{3}\Z$. One has that $2$ divides the least denominator of $u/6$. If follows that $n'=3$, $d=2d'$, and $r=2r'-3$, see Lemma \ref{lem:n-r-qnotirreducible}. Thus, if $d=2$ ($d=4$) mod $6$, then $d'=1$ ($d'=2$) mod $3$. When $d=3$ mod $6$, one sees that $2$ is the least denominator of $d_K$, i.e., $\displaystyle d_K\in\frac{1}{2}\Z$. Therefore, $3$ is a divisor of the least denominator of $u/6$. It follows that $n'=6$, $d'=d$ and $r'=r+3$.
When $n=12$ and $d=1,5,7,11$ mod $12$, one recalls $\displaystyle d_K=\nu(J_{10}J_2^{-5})/12$ and hence $12$ is the least denominator of $d_K$. This implies that $n'=12$, $d'=d$ and $r'=r+6$.
Now, we assume that $2\nmid \nu(J_2)$. According to Lemma \ref{lem:n-r-qnotirreducible}, one has $[L':K]=[L:K]$, $r'=r$, and $d'=2r'=d$. It follows that the reduction type of $X^{\chi}$ is the same as the reduction type of $X$.
\end{Proof}
\begin{Theorem}
\label{thm:tworationalcurves}
Let $C$ be a hyperelliptic curve defined over $K$. Assume that $L/K$ is tamely ramified and $\CC_k$ is the union of two rational curves intersecting in a unique point. Let $K(\sqrt{D})/K$ be a quadratic extension whose associated character is $\chi$, and $\nu(D)=1$. Let $X$ and $X^{\chi}$ be the minimal regular models of the curves $C$ and its quadratic twist by $\chi$, $C^{\chi}$. Then the reduction type of $X^{\chi}$ is given in the following table.
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
type($X_k$) &$n$&$\nu(J_2)$&$d$ mod $n$&type($X^{\chi}_k$)\\
\hline
$[{\rm I}_{d_1}-{\rm I}_{d_2}-d]$ & 1&&&$[{\rm I}^*_{d_1}-{\rm I}^*_{d_2}-(d-1)]$ \\
\hline
$[{\rm I}^*_{d_1/2}-{\rm I}^*_{d_2/2}-(d-2)/2]$ & 2&even&0&$[{\rm I}_{d_1/2}-{\rm I}_{d_2/2}-d/2]$\\
\hline
$[{\rm I}_{e_1/2}-{\rm I}^*_{e_2/2}-(d-1)/2]$ &2&even&1&$[{\rm I}_{e_1/2}-{\rm I}^*_{e_2/2}-(d-1)/2]$ \\
\hline
$[2{\rm I}_{d_1}-d/2]$ &2&odd && $[2{\rm I}_{d_1}-d/2]$\\
\hline
$[2{\rm I}^*_{d_1/2}-(d-2)/4]$ &4& odd&& $[2{\rm I}^*_{d_1/2}-(d-2)/4]$\\
\hline
\end{tabular}
\end{center}
\end{Theorem}
\begin{Proof}
Lemma \ref{Lem:StableModelofTwist} implies that the special fiber $\CC^{\chi}_k$ of the stable model of $C^{\chi}$ is the union of an elliptic curve and a rational curve intersecting in exactly one point. The integer $d$ is the degree of singularity of the intersection point of the components whereas $d_1$ and $d_2$ are the degrees of singularity of the other two ordinary double points. Furthermore, $\CC^{\chi}_k$ is the union of an elliptic curve and a rational curve intersecting in exactly one point, see Lemma \ref{Lem:StableModelofTwist}.
When $n=1$, we assume that $\overline{\omega}$ is non-ramified. One has $2\mid\nu(J_2)$ since $n$ is odd, hence $2\mid\nu(J_2')$. According to Lemma \ref{lem:n-r-qnotirreducible}, one has $n'=2$, $d'=2d=0$ mod $2$. It follows that the reduction type of $X^{\chi}$ is given by $[{\rm I}^*_{d'_1/2}-{\rm I}^*_{d'_2/2}-(d'-2)/2]$ where the degrees of singularity $d_1'$, $d_2'$ are $2d_1$ and $2d_2$, respectively. The proof when $\overline{\omega}$ satisfies the hypothesis of subcases (b), (c) or (d) of Lemma \ref{lem:n-r-qnotirreducible} is similar.
If $n=2$ and $2\mid\nu(J_2)$, we will assume that $\overline{\omega}$ is non-ramified. One has $2\mid \nu(J_2)$. Moreover, if $d$ is even, then the denominator of $d_K$ is $1$. Lemma \ref{lem:n-r-qnotirreducible} implies that $n'=1$ and $d=2d'$. Hence, the reduction type of $X^{\chi}$ is given by $[{\rm I}_{d_1'}-{\rm I}_{d_2'}-d'/2]$ where $d_1'=d_1/2,d_2'=d_2/2$. Now, if $d$ is odd, one sees that the denominator of $d_K$ is $2$ which implies that $n'=2$ and $d'=d$. Therefore, the reduction type of $X^{\chi}$ is $[{\rm I}_{e_1/2}-{\rm I}^*_{e_2/2}-(d-1)/2]$. The proof for the subcases (b), (c), (d) of Lemma \ref{lem:n-r-qnotirreducible} is similar.
If $2\nmid\nu(J_2)$, then $2\nmid\nu(J_2')$ and $n'=n$, see Lemma \ref{lem:n-r-qnotirreducible}.
\end{Proof}
For the reduction types $[2{\rm I}_{d_1}-d/2]$ and $[2{\rm I}^*_{d_1/2}-(d-2)/4]$ in the table above, the degrees of singularity of the two points that are not intersection points are both equal to $d_1$.
\begin{Remark}\label{rem:reductiontype}
Let the stable model of $C$ be the union of two irreducible components $E_1$ and $E_2$ where $\overline{\omega}\in f(E_1)$. Assume that one component is a rational curve and the other component is an elliptic curve such that $[L:K]=2$ and $d=1$ mod $2$. The reduction type of the minimal regular model $X$ of $C$ is either $[{\rm I}_{0}-{\rm I}_{q}^*-(d-1)/2]$, where $q=\nu(J_2J_{10}I_{12}^{-1})$, if $E_1$ is smooth and $r$ is even or $E_1$ is singular and $r$ is odd. Otherwise, the reduction type of $X$ is $[{\rm I}_{q}-{\rm I}_{0}^*-(d-1)/2]$, see \cite[Remarque 4.4]{Liumodelesminimaux}.
\end{Remark}
\begin{Theorem}
\label{thm:oneellipticonerationalcurves}
Let $C$ be a hyperelliptic curve defined over $K$. Assume that $L/K$ is tamely ramified and $\CC_k$ is the union of an elliptic curve and a rational curve intersecting in a unique point. Let $K(\sqrt{D})/K$ be a quadratic extension whose associated character is $\chi$, and $\nu(D)=1$. Let $X$ and $X^{\chi}$ be the minimal regular models of the curves $C$ and its quadratic twist by $\chi$, $C^{\chi}$. Then the reduction type of $X^{\chi}$ is given in the following table.
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
type($X_k$) &$n$& $d$ mod $n$& $r$ mod $n$ &type($X^{\chi}_k$)\\
\hline
$[{\rm I}_{d_1}-{\rm I}_{0}-d]$ & 1&&&$[{\rm I}_{0}^*-{\rm I}_{d_1}^*-(d-1)]$ \\
\hline
$[{\rm I}_{0}^*-{\rm I}_{d_1/2}^*-(d-2)/2]$ &2&0&&$[{\rm I}_{d_1/2}-{\rm I}_{0}-d/2]$ \\
\hline
$[{\rm I}_{0}-{\rm I}_{q}^*-(d-1)/2]$ &2&1& Remark \ref{rem:reductiontype}&$[{\rm I}_{q}-{\rm I}_{0}^*-(d-1)/2]$ \\
\hline
$[{\rm I}_{q}-{\rm I}_{0}^*-(d-1)/2]$ &2&1& Remark \ref{rem:reductiontype}&$[{\rm I}_{0}-{\rm I}_{q}^*-(d-1)/2]$ \\
\hline
$[{\rm IV}-{\rm I}_{d_1/3}-(d-1)/3]$ &3&1&&$[{\rm II}^*-{\rm I}^*_{d_1/3}-(d-4)/3]$ \\
\hline
$[{\rm IV}^*-{\rm I}_{d_1/3}-(d-2)/3]$ &3&2&&$[{\rm II}-{\rm I}^*_{d_1/3}-(d-2)/3]$ \\
\hline
$[{\rm III}-{\rm I}_{d_1/4}-(d-1)/4]$ &4&1&0 or 1& $[{\rm III}^*-{\rm I}^*_{d_1/4}-(d-5)/4]$ \\
\hline
$[{\rm III}^*-{\rm I}^*_{d_1/4}-(d-5)/4]$ &4&1&2 or 3& $[{\rm III}-{\rm I}_{d_1/4}-(d-1)/4]$\\
\hline
$[{\rm III}^*-{\rm I}_{d_1/4}-(d-3)/4]$ &4&3&0 or 3&$[{\rm III}-{\rm I}^*_{d_1/4}-(d-3)/4]$ \\
\hline
$[{\rm III}-{\rm I}^*_{d_1/4}-(d-3)/4]$ &4& 3& 1 or 2&$[{\rm III}^*-{\rm I}_{d_1/4}-(d-3)/4]$ \\
\hline
$[{\rm II}-{\rm I}_{d_1/6}-(d-1)/6]$ & 6& 1& 0 or 1&$[{\rm IV}^*-{\rm I}^*_{d_1/6}-(d-7)/6]$ \\
\hline
$[{\rm IV}^*-{\rm I}^*_{d_1/6}-(d-7)/6]$ &6&1&3 or 4&$[{\rm II}-{\rm I}_{d_1/6}-(d-1)/6]$ \\
\hline
$[{\rm II}^*-{\rm I}^*_{d_1/6}-(d-8)/6]$ & 6& 2& &$[{\rm IV}-{\rm I}_{d_1/6}-(d-2)/6]$\\
\hline
$[{\rm II}-{\rm I}^*_{d_1/6}-(d-4)/6]$ &6& 4&&$[{\rm IV}^*-{\rm I}_{d_1/6}-(d-4)/6]$ \\
\hline
$[{\rm II}^*-{\rm I}_{d_1/6}-(d-5)/6]$ &6&5&0 or 5& $[{\rm IV}-{\rm I}^*_{d_1/6}-(d-5)/6]$ \\
\hline
$[{\rm IV}-{\rm I}^*_{d_1/6}-(d-5)/6]$ & 6&5&2 or 3&$[{\rm II}^*-{\rm I}_{d_1/6}-(d-5)/6]$\\
\hline
\end{tabular}
\end{center}
\end{Theorem}
\begin{Proof}
Since one irreducible component of $\CC_k$ is singular whereas the other component is smooth, it follows that each of the irreducible components is globally fixed under the action of $\Gal(L/K)$. This implies that $2\mid \nu(J_2)$, see \cite[Proposition 4.3.2]{Liumodelesminimaux}. Moreover, $d$ is the degree of singularity of the intersection point while $d_1$ is the degree of singularity on the rational curve. Lemma \ref{Lem:StableModelofTwist} indicates that the special fiber of the stable model $C^{\chi}$ is the union of a rational curve and an elliptic curve intersecting in one point. We assume that $\overline{\omega}$ is non-ramified, unless otherwise stated, since the proof for the other subcases is similar.
If $n=1$, then $n'=2$ and $d'=2d_K=0$ mod $2$. Moreover, $d'=2d$ and $d_1'=2d_1$. If $n=2$ and $d=0$ mod $2$, then $d_K\in\Z$, and so $n'=1$, see Lemma \ref{lem:n-r-qnotirreducible}. Moreover, $d'=d/2$ and $d_1'=d_1/2$.
If $n=2$ and $d=1$ mod $2$, then $\displaystyle d_K\in\frac{1}{2}\Z$, therefore, $n'=2$, $d'=d$, $q'=q$. Lemma \ref{lem:n-r-qnotirreducible} implies that if $f^{-1}(\overline{\omega})$ is regular (singular) then so is $f'^{-1}(\overline{\omega'})$. Because $r'=r+1$, Remark \ref{rem:reductiontype} implies that if the reduction type of $X$ is either $[{\rm I}_{0}-{\rm I}_{q}^*-(d-1)/2]$ or $[{\rm I}_{q}-{\rm I}_{0}^*-(d-1)/2]$), then the reduction type of $X^{\chi}$ is either $[{\rm I}_{q}-{\rm I}_{0}^*-(d-1)/2]$ or $[{\rm I}_{0}-{\rm I}_{q}^*-(d-1)/2]$, respectively.
If $n=3$ and $d=1$ or $2$ mod $3$, then $\displaystyle d_K\in\frac{1}{3}\Z$. Thus, one has $n'=6$ and $d'=2d$ mod $6$, see Lemma \ref{lem:n-r-qnotirreducible}. Moreover, $d_1'=2d_1$.
If $n=4$ and $d=1$ or $3$ mod $4$, then $\displaystyle d_K\in\frac{1}{4}\Z$. It follows that $n'=4$, $d'=d$, $d_1'=d_1$ and $r'=r+2$.
If $n=6$ and $d=1$ or $5$ mod $6$, then $\displaystyle d_K\in\frac{1}{6}\Z$. This yields that $n'=6$, $d'=d$, $d_1'=d_1$ and $r'=r+3$. If $n=6$ and $d=2$ or $4$ mod $6$, then $\displaystyle d_K\in\frac{1}{3}\Z$. It follows that $2$ divides the least denominator of $\nu(a_0J_2)/6$. Now, Lemma \ref{lem:n-r-qnotirreducible} implies that $n'=3$ and $d=2d'$. Moreover, $d_1'=d_1/2$.
\end{Proof}
\subsection{$L/K$ is wildly ramified}
In this section we assume that $C$ admits its stable model after a wildly ramified base extension. We determine the minimal regular model of a quadratic twist of $C$ by a character $\chi$ associated with the quadratic field $K(\sqrt{D})$, $\nu(D)=1$.
The following propositions can be found in \cite[\S 5]{Liumodelesminimaux}.
\begin{Proposition}
\label{prop:char3}
Suppose $3\mid[L:K]$. One has the following properties:
\begin{itemize}
\item[a)] $C$ is described by an equation of the form $z^2=a_0Q(u)$ with $Q(u)=(u^3+c_1u^2+c_2u+c_3)^2+c_4u^2+c_5u+c_6\in R[u]$, with $\nu(c_3)=1$ or $2$;
\item[b)] We set $N=\min\{3\nu(c_i)-i\nu(c_3)|4\le i\le 6\}$. The reduction type of $X$ is $[{\rm III}_N]$ if $2\mid \nu(a_0)$ or $[{\rm III}^*_N]$ otherwise.
\end{itemize}
\end{Proposition}
\begin{Proposition}
\label{prop:char5}
Suppose $5\mid[L:K]$. One has the following properties:
\begin{itemize}
\item[a)] $C$ is described by an equation of the form $z^2=b_0u^6+b_1u^5+b_2u^4+\ldots+b_6\in R[u]$, with $b_0\in\mathfrak{m}, b_1\in R^{\times}, 1\le \nu(b_6)\le 9$ and $\nu(b_6)\ne 5$;
\item[b)] If $\nu(b_6)=2m$, the reduction type of $X$ is $[{\rm IX}-m]$. If $\nu(b_6)=2m-1$, the reduction type is $[{\rm VIII}-m]$ if $m\le 2$, or $[{\rm VIII}-(m-1)]$ if $m\ge 4$.
\end{itemize}
\end{Proposition}
\begin{Corollary}
\label{cor:widelyramified}
Let $C$ be a hyperelliptic curve defined over $K$. Let $K(\sqrt{D})/K$ be a quadratic extension whose associated character is $\chi$, and $\nu(D)=1$. Let $X$ and $X^{\chi}$ be the minimal regular models of the curves $C$ and its quadratic twist by $\chi$, $C^{\chi}$.
\begin{itemize}
\item[(a)] If $\Char k=3$ and the reduction type of $X$ is either $[{\rm III}_N]$ or $[{\rm III}^*_N]$, then the reduction type of $X^{\chi}$ is $[{\rm III}^*_N]$ or $[{\rm III}_N]$, respectively.
\item[(b)] If $\Char k=5$ and the reduction type of $X^{\chi}$ is given in the following table:
\end{itemize}
{\footnotesize\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
type($X_k$) & $[{\rm IX}-1]$&$[{\rm IX}-2]$& $[{\rm IX}-3]$& $[{\rm IX}-4]$& $[{\rm VIII}-1]$&$[{\rm VIII}-2]$&$[{\rm VIII}-3]$ &$[{\rm VIII}-4]$ \\
\hline
type($X^{\chi}_k$)& $[{\rm VIII}-3]$&$[{\rm VIII}-4]$&$[{\rm VIII}-1]$ &$[{\rm VIII}-2]$&$[{\rm IX}-3]$&$[{\rm IX}-4]$& $[{\rm IX}-1]$& $[{\rm IX}-2]$ \\
\hline
\end{tabular}
\end{center}}
\end{Corollary}
\begin{Proof}
(a) follows from the fact that if is $C$ is defined by $z^2=a_0Q(u)$ with $Q(u)=(u^3+c_1u^2+c_2u+c_3)^2+c_4u^2+c_5u+c_6\in R[u]$, with $\nu(c_3)=1$ or $2$, then $C^{\chi}$ is defined by $z^2=Da_0Q(u)$. Now, since $\nu(a_0d)=1+\nu(a_0)$, the result follows using Proposition \ref{prop:char3} (b).
(b) Proposition \ref{prop:char5} indicated that $C$ is defined by an equation of the form $z^2=P(x)=b_0u^6+b_1u^5+b_2u^4+\ldots+b_6\in R[u]$, with $b_0\in\mathfrak{m}, b_1\in R^{\times}, 1\le \nu(b_6)\le 9$ and $\nu(b_6)\ne 5$. Since $C^{\chi}$ is defined by $z^2=DP(u)$, one may replace $u$ with $t^{-1}u$ and $z$ with $t^{-2} z$ to obtain the equation $t^{-4}z^2=D(b_0t^{-6}u^6+b_1t^{-5}u^5+b_2t^{-4}u^4+\ldots+b_6)$; or equivalently $z^2=P'(u)=b_0'u^6+b_1'u^5+b_2'u^4+\ldots+b'_6t^4$ where
\[b_0'=Db_0/t^2,b_1'=Db_1/t,b_2'=Db_2,b_3'=Dtb_3,b_4'=Dt^2b_4,b_5'=Dt^3b_5,b_6'=Dt^4b_6.\]
Therefore, $P'(u)\in R[u]$ since $b_0\in\mathfrak{m}$ and $\nu(D)=1$. Furthermore, $b_1'\in R^{\times}$. In fact, one may, and will, assume that $\nu(b_0')>0$, since otherwise one replaces $u$ with $\displaystyle\frac{u}{1-(b_0/b_1)u}$ and $z$ with $\displaystyle \frac{z}{\left(1-(b_0/b_1)u\right)^3}$ in order to obtain the equation $z^2=b_0''u^6+b_1''u^5+b_2''u^4+\ldots+b''_6t^4$ where $\nu(b_1'')=\nu(b_1')=0$, $\nu(b_i)>0$ for $i\ne 1$, and $b_6''=b_6'=Dt^4b_6$.
If $\nu(b_6)=1,2,3,4$, then $\nu(b_6')=6,7,8,9$, respectively. In other words, if the reduction type of $X$ is either $[{\rm VIII}-1]$, $[{\rm IX}-1]$, $[{\rm VIII}-2]$, or $[{\rm IX}-2]$, then the reduction type of $X^{\chi}$ is $[{\rm IX}-3]$, $[{\rm VIII}-3]$, $[{\rm IX}-4]$, or $[{\rm VIII}-4]$, respectively.
If $\nu(b_6)=6,7,8,9$, then $\nu(b_6')=11,12,13,14$, respectively. Since $\nu(b_6')>9$, one then may replace $u$ with $t^2u$ and $z$ with $t^5z$. Therefore, $\nu(b_6')=1,2,3,4$, respectively. In particular, if the reduction type of $X$ is either $[{\rm IX}-3]$, $[{\rm VIII}-3]$, $[{\rm IX}-4]$, or $[{\rm VIII}-4]$, then the reduction type of $X^{\chi}$ is
$[{\rm VIII}-1]$, $[{\rm IX}-1]$, $[{\rm VIII}-2]$, or $[{\rm IX}-2]$, respectively.
\end{Proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,129 |
Florida State vs. Houston — Everything You Need to Know for the Peach Bowl
Wes Glinsmann
AAC champ Houston crashes the major bowl party against a Florida State team playing in its fourth straight New Year's Six bowl.
December 31, 12 p.m. ET (ESPN)
No. 18 Houston (12-1) vs. No. 9 Florida State (10-2)
How They Got Here
Houston received an at-large bid after winning the American Athletic Conference title and finishing as the highest ranked "Group of Five" champion. Florida State fell short of its goal of a fourth straight ACC championship but still earned an at-large bid after finishing ninth in the final CFP rankings.
When Florida State Has the Ball
Florida State running back Dalvin Cook led the ACC in rushing yards, rushing touchdowns and yards per carry despite playing through nagging ankle and hamstring injuries. With a month to rest and heal, he should be ready to go against a challenging Cougar defense that ranks 12th nationally against the run.
If Houston has a weak spot, it's a pass defense that ranks 111th nationally, giving up more than 265 yards per game. That's good news for junior Sean Maguire, who will be making his fifth start of the season (1,128 yards and nine touchdowns) after taking the reins from Everett Golson (who did not make the trip to the bowl game for "personal reasons"). The Seminoles are 3-1 in Maguire's starts and he helped rally Florida State to another victory after coming off the bench against North Carolina State.
When Houston Has the Ball
The Cougars feature one of the nation's most balanced offenses, averaging 247 passing yards and 239 yards on the ground per game. But while the offense is not one-dimensional, it is focused around the talents of dual-threat quarterback Greg Ward, Jr. The junior has thrown for nearly 2,600 yards this year while rushing for over 1,000 more. He has scored 35 touchdowns and, under his leadership, the Cougars are averaging nearly 41 points per game (11th best nationally). However, he's never faced a unit like the Seminoles, who boast several future NFLers and rank fifth nationally in scoring defense (15.8 points per game).
Key to the Game
The Cougars boast one of the nation's better rush defenses, but they have never faced a runner of Dalvin Cook's caliber. If Cook can run effectively and help open up the field for Maguire, Houston could be in for a long night. While Houston has other talented players, the Cougars will only go as far as Greg Ward can carry them. Although he is a solid passer, he is at his best when the run game can spread the field. For Houston to have a chance to pull off the upset, Ward and running back Kenneth Farrow (949 yards and 12 touchdowns) will have to make some plays on the ground to open up the passing game.
But motivation could be a factor--this is one of the biggest games in Houston's history, so the Cougars will be fired up for this one. But after falling short of their goals of a fourth straight ACC title and a playoff spot, can the Seminoles say the same?
Filed Under: College Football | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 388 |
{"url":"https:\/\/explog.in\/notes\/walk.html","text":"# [wip] A Random Walk\n\n## Simulations\n\nHamming famously wrote about a random walk in The Art of Science and Engineering, and that's the cover of the new edition of the book. I thought it would be fun to simulate a random walk in Python.\n\nA one-dimensional walk can be expressed elegantly as the sum over a generator;\n\nimport random\n\ndef walk(steps):\nstep = [-1, 1]\nreturn sum(random.choice(step) for _ in range(steps))\n\nfor i in range(1, 6):\nprint(f\"{i} steps: {walk(i)}\")\n\n1 steps: 1\n2 steps: -2\n3 steps: 1\n4 steps: -4\n5 steps: 1\n\n\nThe next step to see how far a random walk actually gets would be to run it several times, and record the distance covered. I'm also going to look at the root mean squared distance, which is easier to calculate with a direct solution.\n\nimport math\n\ndef average_distance(steps, iterations=10):\ntotal = sum(abs(walk(steps)) for _ in range(iterations))\n\ndef rms(steps, iterations=10):\ntotal = sum(walk(steps) ** 2 for _ in range(iterations))\nreturn math.sqrt(total \/ iterations)\n\ndef all_stats(steps, iterations=10):\ntotal_dist = 0\ntotal_disp = 0\ntotal_sq_disp = 0\nfor _ in range(iterations):\ndisp = walk(steps)\ntotal_disp += disp\ntotal_dist += abs(disp)\ntotal_sq_disp += disp ** 2\n\nreturn (total_dist \/ iterations, math.sqrt(total_sq_disp \/ iterations), total_disp \/ iterations)\n\nfor _ in range(3):\nprint(all_stats(9))\n\n(2.8, 3.255764119219941, 0.6)\n(1.8, 2.23606797749979, 0.2)\n(3.0, 3.605551275463989, -1.4)\n\n\nThere are two ways I'm interested in looking at this data: to see how it converges for a given number of steps as the number of iterations increase, and to see the average value across different numbers of steps.\n\nimport matplotlib.pyplot as plt\nplt.style.use('ggplot')\n\ndef converge_steps(steps, max_iterations=10):\ndistances = []\nrms_distances = []\ndisplacement = []\niterations = []\nfor i in range(1, max_iterations):\ndist, rms, disp = all_stats(steps, i)\ndistances.append(dist)\ndisplacement.append(disp)\nrms_distances.append(rms)\niterations.append(i)\n\nplt.figure(figsize=(7,4))\nplt.plot(iterations, distances, linewidth=.5)\nplt.plot(iterations, displacement, linewidth=.5)\nplt.plot(iterations, rms_distances, linewidth=.5)\nplt.savefig(\"..\/static\/images\/converge.png\")\nplt.close()\n\nconverge_steps(16, 1000)\n\n\/tmp\/babel-1rbHGK\/python-R7MvGt:16: RuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (matplotlib.pyplot.figure) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam figure.max_open_warning).\nplt.figure(figsize=(7,4))\n\n\nLooking at the distance traversed after taking a thousand iterations:\n\ndef steps(max_steps):\ndistances = []\nrms_distances = []\ndisplacements = []\nsteps = []\n\nfor i in range(1, max_steps + 1):\ndist, rms, disp = all_stats(i, 1000)\ndistances.append(dist)\nrms_distances.append(rms)\ndisplacements.append(disp)\nsteps.append(i)\n\nplt.figure(figsize=(7,4))\nplt.plot(steps, distances, linewidth=.5)\nplt.plot(steps, rms_distances, linewidth=.5)\nplt.plot(steps, displacements, linewidth=.5)\nplt.savefig(\"..\/static\/images\/steps.png\")\n\nsteps(100)\n\n\/tmp\/babel-1rbHGK\/python-6Z9dnp:14: RuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (matplotlib.pyplot.figure) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam figure.max_open_warning).\nplt.figure(figsize=(7,4))\n\n\n## Analytical Solutions\n\nIt's fairly easy to derive the average root mean square distance after N steps:\n\n\u2022 Represent a walk as the series of steps $$x_1$$, $$x_2$$, $$x_3$$, \u2026 , $$x_n$$. $$x_i$$ can only be +1 or -1.\n\u2022 The total distance at the end of the walk is $$x_1 + x_2 + x_3 + ... + x_n$$.\n\u2022 Let $$N$$ be the total possible walks, which is $$2^n$$ because at each of the $$n$$ steps we have $$2$$ options.\n\u2022 Then, the square of the root mean square distance (just to allow simpler typing):\n\\begin{align*} rms^2 & = \\frac{1}{N} * ((x_{1,1} + x_{2,1} + ... + x_{n,1})^2 + (x_{1,2} + x_{2,2} + ... + x_{n,2})^2 + ... + (x_{1,N} + x_{2,N} + ... + x_{n,N})^2) \\\\ & = \\frac{1}{N} * (x_{1,1}^2 + x_{2,1}^2 + ... + 2x_{1,1}x_{1,2} + ... + 2x_{1,2}x_{2,2} + ...) \\\\ & = \\frac{1}{N} * ((1 + 1 + ... + 1) * N + 2(x_{1,1}x{2, 1} + x_{1,2}x_{2,2} + ...)) \\\\ & = \\frac{1}{N} * (N * n + 2 (0)) \\\\ & = n \\\\ rms & = \\sqrt n \\end{align*}\n\u2022 $$(x_{1,1}x_{2, 1} + x_{1,2}x_{2,2} + ...)$$ evaluates to zero because it covers all possible combinations of pairs of values being added together: $$x_{i,k}x_{j,k}$$ must be either $$1$$ or $$-1$$ depending on the individual values, with equal chance. Once we cover all possibilities, all the values cancel out.","date":"2020-09-27 13:19:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9626286029815674, \"perplexity\": 2733.2229687523018}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600400279782.77\/warc\/CC-MAIN-20200927121105-20200927151105-00399.warc.gz\"}"} | null | null |
Union has multiple legal concerns regarding new conduct policy
Posted by Mike Florio on December 14, 2014, 7:09 PM EDT
The NFL Players Association intended to scrutinize immediately the new personal conduct policy for terms that permit potential legal challenges, either through arbitration or a claim with the National Labor Relations Board.
Per a source with knowledge of the situation, the NFLPA has identified multiple specific areas of concern and communicated those concerns to the Executive Committee and board of player representatives. The three biggest issues are summarized below.
First, the union disputes the league's belief that Fifth Amendment rights have no relevance to the new policy. "We disagree and will vigorously protect ALL players' 5th Amendment rights; we will ensure that an NFLPA attorney and criminal attorney protect a player at every step of any NFL investigation," NFLPA management informed the Executive Committee and player representatives in a memo, a copy of which PFT has obtained. "This is crucial for players' protection because any information gathered in an employer's investigation is not privileged, and law enforcement could obtain and make prosecutorial decisions based on such information." The NFLPA also will "aggressively address" the possibility that a player who invokes his Fifth Amendment right will be disciplined for failing to cooperate with the league's investigation.
Second, the NFLPA disagrees with the plan to put players on paid leave when charged with a crime of violence. "[W]e strongly object to such unilateral action by the Commissioner/Owners, in no small part because many players have contracts that include significant roster bonuses, etc.," the NFLPA said. "Moreover, the removal of a player
from the field is a form of discipline regardless of whether he is paid his paragraph 5 salary. We do not believe that unilaterally placing players on the Commissioner Exempt list with pay before issuance of discipline in form of fine or suspension is permitted by the CBA or NFL Constitution."
Third, the NFLPA contends that the new approach to discipline under the Personal Conduct Policy, with the Commissioner delegating the initial decision to a to-be-hired "Special Counsel for Investigations and Conduct," violates the plain language of the Collective Bargaining Agreement.
"This new disciplinary structure violates the CBA," the NFLPA explained to the players. "Article 46 of the CBA, which was obviously collectively bargained, contains the specific agreement that the Commissioner issues the initial
discipline, and the parties agreed that he can delegate appeal decision rights; the CBA language allowing for delegation is specific and its absence for the Commissioner to delegate to another NFL paid position is clear."
The NFLPA may challenge these and other provisions by pursuing a "system arbitration" under the labor deal or filing a complaint with the National Labor Relations Board.
13 responses to "Union has multiple legal concerns regarding new conduct policy"
whatevnfl says:
I don't like the idea of the NFL doing its own investigation and jeopardizing a legal case. Who do they think they are to be enforcing the law. Then again, considering how inept ALL of their investigations are, including investigating their own Commissioner, or how difficult they claim it was to obtain a video from a CASINO with 18,000 cameras, maybe players have nothing to fear.
fanasaurus says:
I was under the impression that the 5th Amendment was for criminal law protection against the government, not labor law protecting employees against their employers. I'm not positive, as I'm not a lawyer. I would think with all the lawyers the NFL has that it would've been covered in the CBA and amendments they could add.
bobsnygiants says:
Players go on strike just before the playoffs , dom' let the NFL be like our goverment.
melikefootball says:
Why are we on the players side when we have witnessed the last two big events that clearly aren't excepted in society. Why shouldn't the head of the corp pan out the penalty. I blame the players association that seems to not care how players act on or off the field. if players come into a game that is the entertainment world
they should be expected to act inside these rules not as they please. When they don't they should answer the price. As we have seen in Rice and Peterson they clearly by photo and admission were bad characters yet they want to cry foul now.
NFL IS LIKE OUR GOVERNMENT LIKES CONTROL , GO ON STRIKE BEFORE THE PLAYOFFS.
dartwick says:
Points one and three are fairly obvious and the union is completely right about them.
The union should just make a flat policy of non-cooperation if point one stands.
Point obviously contradicts the CBA.
Point 2 – the union is might be right on this but it will depend on the judge.
cwwgk says:
I understand the first two concerns, but the third is laughable. The NFLPA has a constant drumbeat against Goodell, but when he seeks to delegate authority in the disciplinary process they object? I guess they're concerned that people will start to realize that Goodell isn't the problem after all.
cinvis says:
The real challenge will be after the Panthers last game and Hardy is no longer being paid. He will then be a free agent and the 'exempt list' will no longer apply (He'll be exempt from what?). What then NFL? Will you stand in his way from signing with a team?
rajbais says:
If you can prove that the CBA is being violated, Sue them. Mr. Smith, you know that you are an attorney, right?
Please file a lawsuit immediately.
Please file a lawsuit immediately!
FinFan68 says:
How does the 5th ammendment apply here? The investigation in question initiates after the legal case has gone through the court system. As for the delegation argument, it depends on the wording. The NFLPA has a history of misinterpreting the language but even if they are correct Goodell can simply use the decision of the special counsel as a recommendation and then always follow it as a matter of course. It still would be coming from Goodell but he would not be the "bad guy" initially.
. This Union has never taken the initiative to hold their players accountable and I don't remember a single time when it said the player's actions warranted any discipline at all.
andreasx1 says:
The only way to get the owners attention is to strike, but do the players have the resolve to stand up to this "bully"?
What happens if the player's team makes the playoffs and he's suspended? Can those be considered games suspended? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,954 |
Grey Power (or Greypower) was an Australian political party and lobby group, first registered in 1983. At the federal elections of 1984 and 1987 it ran candidates, but on both occasions these candidates (who included former Liberal cabinet minister Bill Wentworth) did poorly. The group was designed to represent the elderly vote, advocating issues dealing with aged care and a mature perspective on national policy; hence the name "grey power".
The party's state president in New South Wales was Robert Clark, an anti-immigration campaigner and the founder of the Immigration Control Association, which advocated for a return to the White Australia policy. In 1989 the party's deputy president resigned in protest at the "abhorrent literature ... anti-Asian, anti-Jews, anti-everyone" that had been circulated at party meetings.
Grey Power ran in the 1989 Western Australian state election, garnering 5.2% of the total lower house vote. However after the election a "bitter power struggle" emerged which resulted in the "virtual collapse" of the party in Western Australia. Police were called to the annual meeting of the party at a suburban hall in Dalkeith.
The Canberra Times observed in 1989 that the party "remains fractured, with, as yet, little unity of purpose among state organisations" and noted that the movement had been "dogged by allegations of links with right wing or racist groups".
The last election which Grey Power contested was the 1997 South Australian state election, but then it only managed to receive 1.6% of the South Australian Legislative Council vote. Their preferences however significantly contributed to the election of Nick Xenophon.
The best result Grey Power ever achieved was at the 1994 Taylor state by-election in South Australia. Without a Liberal candidate in the running on this occasion, Grey Power took 13 percent of the primary vote and finished second after preferences had been distributed with a 27 percent two-candidate preferred vote.
See also
List of pensioners' parties
References
Defunct political parties in Australia
Pensioners' parties
Pensioners' parties in Australia
Political parties established in 1983
Retirement in Australia
1983 establishments in Australia
Political parties with year of disestablishment missing | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,259 |
// Copyright (c) Microsoft Corporation. All rights reserved.
// Licensed under the MIT License.
package com.azure.spring.cloud.autoconfigure.context;
import com.azure.core.credential.TokenCredential;
import com.azure.core.management.AzureEnvironment;
import com.azure.identity.ClientCertificateCredentialBuilder;
import com.azure.identity.ClientSecretCredentialBuilder;
import com.azure.identity.DefaultAzureCredentialBuilder;
import com.azure.identity.ManagedIdentityCredentialBuilder;
import com.azure.identity.UsernamePasswordCredentialBuilder;
import com.azure.spring.cloud.autoconfigure.AzureServiceConfigurationBase;
import com.azure.spring.cloud.autoconfigure.implementation.properties.core.AbstractAzureHttpConfigurationProperties;
import com.azure.spring.cloud.core.customizer.AzureServiceClientBuilderCustomizer;
import com.azure.spring.cloud.core.implementation.credential.resolver.AzureTokenCredentialResolver;
import com.azure.spring.cloud.core.implementation.factory.AbstractAzureServiceClientBuilderFactory;
import com.azure.spring.cloud.core.implementation.factory.credential.AbstractAzureCredentialBuilderFactory;
import com.azure.spring.cloud.core.implementation.factory.credential.ClientCertificateCredentialBuilderFactory;
import com.azure.spring.cloud.core.implementation.factory.credential.ClientSecretCredentialBuilderFactory;
import com.azure.spring.cloud.core.implementation.factory.credential.DefaultAzureCredentialBuilderFactory;
import com.azure.spring.cloud.core.implementation.factory.credential.ManagedIdentityCredentialBuilderFactory;
import com.azure.spring.cloud.core.implementation.factory.credential.UsernamePasswordCredentialBuilderFactory;
import com.azure.spring.cloud.core.provider.authentication.TokenCredentialOptionsProvider;
import org.springframework.beans.BeansException;
import org.springframework.beans.factory.BeanFactory;
import org.springframework.beans.factory.BeanFactoryAware;
import org.springframework.beans.factory.ObjectProvider;
import org.springframework.beans.factory.annotation.Qualifier;
import org.springframework.beans.factory.config.BeanPostProcessor;
import org.springframework.boot.autoconfigure.AutoConfigureAfter;
import org.springframework.boot.autoconfigure.EnableAutoConfiguration;
import org.springframework.boot.autoconfigure.condition.ConditionalOnMissingBean;
import org.springframework.boot.autoconfigure.task.TaskExecutionAutoConfiguration;
import org.springframework.boot.task.TaskExecutorBuilder;
import org.springframework.context.annotation.Bean;
import org.springframework.context.annotation.Configuration;
import org.springframework.core.annotation.Order;
import org.springframework.scheduling.concurrent.ThreadPoolTaskExecutor;
import org.springframework.util.StringUtils;
import static com.azure.spring.cloud.autoconfigure.context.AzureContextUtils.DEFAULT_CREDENTIAL_TASK_EXECUTOR_BEAN_NAME;
import static com.azure.spring.cloud.autoconfigure.context.AzureContextUtils.DEFAULT_CREDENTIAL_THREAD_NAME_PREFIX;
import static com.azure.spring.cloud.autoconfigure.context.AzureContextUtils.DEFAULT_TOKEN_CREDENTIAL_BEAN_NAME;
/**
* {@link EnableAutoConfiguration Auto-configuration} for Spring Cloud Azure default {@link TokenCredential}.
*
* @since 4.0.0
*/
@Configuration(proxyBeanMethods = false)
@AutoConfigureAfter(TaskExecutionAutoConfiguration.class)
public class AzureTokenCredentialAutoConfiguration extends AzureServiceConfigurationBase {
private final IdentityClientProperties identityClientProperties;
AzureTokenCredentialAutoConfiguration(AzureGlobalProperties azureGlobalProperties) {
super(azureGlobalProperties);
this.identityClientProperties = loadProperties(azureGlobalProperties, new IdentityClientProperties());
}
@ConditionalOnMissingBean(name = DEFAULT_TOKEN_CREDENTIAL_BEAN_NAME)
@Bean(name = DEFAULT_TOKEN_CREDENTIAL_BEAN_NAME)
@Order
TokenCredential tokenCredential(DefaultAzureCredentialBuilderFactory factory,
AzureTokenCredentialResolver resolver) {
TokenCredential globalTokenCredential = resolver.resolve(this.identityClientProperties);
if (globalTokenCredential != null) {
return globalTokenCredential;
} else {
return factory.build().build();
}
}
@Bean
@ConditionalOnMissingBean
DefaultAzureCredentialBuilderFactory azureCredentialBuilderFactory(
ObjectProvider<AzureServiceClientBuilderCustomizer<DefaultAzureCredentialBuilder>> customizers,
@Qualifier(DEFAULT_CREDENTIAL_TASK_EXECUTOR_BEAN_NAME) ThreadPoolTaskExecutor threadPoolTaskExecutor) {
DefaultAzureCredentialBuilderFactory factory = new DefaultAzureCredentialBuilderFactory(identityClientProperties);
factory.setExecutorService(threadPoolTaskExecutor.getThreadPoolExecutor());
customizers.orderedStream().forEach(factory::addBuilderCustomizer);
return factory;
}
@Bean
@ConditionalOnMissingBean
AzureTokenCredentialResolver azureTokenCredentialResolver(
ClientSecretCredentialBuilderFactory clientSecretCredentialBuilderFactory,
ClientCertificateCredentialBuilderFactory clientCertificateCredentialBuilderFactory,
UsernamePasswordCredentialBuilderFactory usernamePasswordCredentialBuilderFactory,
ManagedIdentityCredentialBuilderFactory managedIdentityCredentialBuilderFactory) {
return new AzureTokenCredentialResolver(azureProperties -> {
if (azureProperties.getCredential() == null) {
return null;
}
final TokenCredentialOptionsProvider.TokenCredentialOptions properties = azureProperties.getCredential();
final String tenantId = azureProperties.getProfile().getTenantId();
final String clientId = properties.getClientId();
final boolean isClientIdSet = StringUtils.hasText(clientId);
String authorityHost = azureProperties.getProfile().getEnvironment().getActiveDirectoryEndpoint();
if (authorityHost == null) {
authorityHost = AzureEnvironment.AZURE.getActiveDirectoryEndpoint();
}
if (StringUtils.hasText(tenantId)) {
if (isClientIdSet && StringUtils.hasText(properties.getClientSecret())) {
return clientSecretCredentialBuilderFactory.build()
.authorityHost(authorityHost)
.clientId(clientId)
.clientSecret(properties.getClientSecret())
.tenantId(tenantId)
.build();
}
String clientCertificatePath = properties.getClientCertificatePath();
if (StringUtils.hasText(clientCertificatePath)) {
ClientCertificateCredentialBuilder builder = clientCertificateCredentialBuilderFactory
.build()
.authorityHost(authorityHost)
.tenantId(tenantId)
.clientId(clientId);
if (StringUtils.hasText(properties.getClientCertificatePassword())) {
builder.pfxCertificate(clientCertificatePath, properties.getClientCertificatePassword());
} else {
builder.pemCertificate(clientCertificatePath);
}
return builder.build();
}
}
if (isClientIdSet && StringUtils.hasText(properties.getUsername())
&& StringUtils.hasText(properties.getPassword())) {
return usernamePasswordCredentialBuilderFactory.build()
.authorityHost(authorityHost)
.username(properties.getUsername())
.password(properties.getPassword())
.clientId(clientId)
.tenantId(tenantId)
.build();
}
if (properties.isManagedIdentityEnabled()) {
ManagedIdentityCredentialBuilder builder = managedIdentityCredentialBuilderFactory.build();
if (isClientIdSet) {
builder.clientId(clientId);
}
return builder.build();
}
return null;
});
}
@Bean
@ConditionalOnMissingBean
ClientSecretCredentialBuilderFactory clientSecretCredentialBuilderFactory(
@Qualifier(DEFAULT_CREDENTIAL_TASK_EXECUTOR_BEAN_NAME) ThreadPoolTaskExecutor threadPoolTaskExecutor,
ObjectProvider<AzureServiceClientBuilderCustomizer<ClientSecretCredentialBuilder>> customizers) {
ClientSecretCredentialBuilderFactory factory = new ClientSecretCredentialBuilderFactory(identityClientProperties);
factory.setExecutorService(threadPoolTaskExecutor.getThreadPoolExecutor());
customizers.orderedStream().forEach(factory::addBuilderCustomizer);
return factory;
}
@Bean
@ConditionalOnMissingBean
ClientCertificateCredentialBuilderFactory clientCertificateCredentialBuilderFactory(
@Qualifier(DEFAULT_CREDENTIAL_TASK_EXECUTOR_BEAN_NAME) ThreadPoolTaskExecutor threadPoolTaskExecutor,
ObjectProvider<AzureServiceClientBuilderCustomizer<ClientCertificateCredentialBuilder>> customizers) {
ClientCertificateCredentialBuilderFactory factory = new ClientCertificateCredentialBuilderFactory(identityClientProperties);
factory.setExecutorService(threadPoolTaskExecutor.getThreadPoolExecutor());
customizers.orderedStream().forEach(factory::addBuilderCustomizer);
return factory;
}
@Bean
@ConditionalOnMissingBean
ManagedIdentityCredentialBuilderFactory managedIdentityCredentialBuilderFactory(
ObjectProvider<AzureServiceClientBuilderCustomizer<ManagedIdentityCredentialBuilder>> customizers) {
ManagedIdentityCredentialBuilderFactory factory = new ManagedIdentityCredentialBuilderFactory(identityClientProperties);
customizers.orderedStream().forEach(factory::addBuilderCustomizer);
return factory;
}
@Bean
@ConditionalOnMissingBean
UsernamePasswordCredentialBuilderFactory usernamePasswordCredentialBuilderFactory(
ObjectProvider<AzureServiceClientBuilderCustomizer<UsernamePasswordCredentialBuilder>> customizers) {
UsernamePasswordCredentialBuilderFactory factory = new UsernamePasswordCredentialBuilderFactory(identityClientProperties);
customizers.orderedStream().forEach(factory::addBuilderCustomizer);
return factory;
}
/**
* The BeanPostProcessor to apply the default token credential to all service client builder factories.
* @return the BPP.
*/
@Bean
static AzureServiceClientBuilderFactoryPostProcessor builderFactoryBeanPostProcessor() {
return new AzureServiceClientBuilderFactoryPostProcessor();
}
@Bean(name = DEFAULT_CREDENTIAL_TASK_EXECUTOR_BEAN_NAME)
@ConditionalOnMissingBean(name = DEFAULT_CREDENTIAL_TASK_EXECUTOR_BEAN_NAME)
ThreadPoolTaskExecutor credentialTaskExecutor() {
return new TaskExecutorBuilder()
.corePoolSize(8)
.allowCoreThreadTimeOut(true)
.threadNamePrefix(DEFAULT_CREDENTIAL_THREAD_NAME_PREFIX)
.build();
}
/**
* Apply the default token credential to service client builder factory.
*/
static class AzureServiceClientBuilderFactoryPostProcessor implements BeanPostProcessor, BeanFactoryAware {
private BeanFactory beanFactory;
@Override
@SuppressWarnings({ "rawtypes", "unchecked" })
public Object postProcessAfterInitialization(Object bean, String beanName) throws BeansException {
if (bean instanceof AbstractAzureCredentialBuilderFactory) {
return bean;
}
if (bean instanceof AbstractAzureServiceClientBuilderFactory
&& beanFactory.containsBean(DEFAULT_TOKEN_CREDENTIAL_BEAN_NAME)) {
AbstractAzureServiceClientBuilderFactory factory = (AbstractAzureServiceClientBuilderFactory) bean;
factory.setDefaultTokenCredential(
(TokenCredential) beanFactory.getBean(DEFAULT_TOKEN_CREDENTIAL_BEAN_NAME));
factory.setTokenCredentialResolver(beanFactory.getBean(AzureTokenCredentialResolver.class));
}
return bean;
}
@Override
public void setBeanFactory(BeanFactory beanFactory) throws BeansException {
this.beanFactory = beanFactory;
}
}
static class IdentityClientProperties extends AbstractAzureHttpConfigurationProperties {
}
IdentityClientProperties getIdentityClientProperties() {
return identityClientProperties;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,879 |
using namespace std;
class HealthCard
{
public:
// Constructor
HealthCard();
HealthCard(string numberParam, Date expiryParam);
// Destructor
~HealthCard();
string getNumber();
void setNumber(const string value);
Date getExpiry();
void setExpiry(const Date value);
Date* getExpiryPtr();
protected:
private:
string number;
Date expiry;
};
#endif
// EOF
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,157 |
\section{Introduction}
\label{s_intro}
Massive stars have gained considerable interest in the last years in various fields of astrophysics. In addition to being well recognized actors in Galactic astronomy (HII regions, bubbles, ISM dynamics and winds, triggered star formation, chemical enrichment) their role in extragalactic physics has considerably grown. Massive stars are known to govern starburst events locally and at high redshift \citep[e.g.][]{steidel96}, being responsible for outflows \citep{veilleux05} and intense episodes of star formation in luminous infrared galaxies \citep{genzel98}. The association of some GRBs with core collapse supernova \citep{galama98,hjorth03} make fast rotating massive stars the likely progenitors of GRBs \citep{ww93}. However, the conditions under which a massive star will actually end up in such an event are still poorly known. Fast rotation is of course a key parameter, but the way the star loses angular momentum, and thus its mass loss history, is equally important. With mass loss decreasing at lower metallicity, GRBs are thought to occur in low Z environments. Observations of GRB hosts tend to confirm this expectation \citep{proch04}. However, the existence of an upper metallicity limit for the formation of GRBs is not established. Massive stars are also thought to be responsible for the re-ionization of the Universe at very high redshift (e.g. Schaerer at al.\ 2002 and references therein). Indeed, the first (population III) stars may have been very massive objects with very high effective temperatures due to their metal free composition \citep{bromm99,nu99}.
In spite of this growing importance of massive stars, their evolution is not fully understood even in the local Universe. It is not clear at present which phases a star of a given initial mass will go through during its life. For example, the initial conditions for a star to enter the red supergiant or luminous blue variable stages are rather blurry. Similarly, the evolutionary connections between various types of Wolf-Rayet stars is poorly constrained. These difficulties result from two facts. The first one is that massive star evolution is governed by mass loss, and our knowledge of mass loss rates through the upper part of the HR diagram is still patchy. The second problem is due to rotation: stars with similar initial masses but different rotational velocities will follow different paths in the HR diagram, and thus will evolve through different states \citep{mm00a}. These two ingredients (mass loss and rotation) are the keys to massive star evolution. An additional complication occurs because both of the above factors depend on metallicity: stars lose less mass at low metallicity \citep{puls00}, and at the same time they rotate faster \citep{martayan07}. Consequently, it is not only crucial to know the mass loss and rotation of all types of massive stars, but it is also important to constrain their metallicity dependence. Recent results indicate a $\dot{M}$\ $\sim Z^{0.83}$ behavior for O stars \citep{mokiem07c}. Theoretical predictions and observational constraints also exist for various types of Wolf-Rayet stars \citep{vink05,goetz08,paul02}. A rather clear picture of the {\it relative} strength of stellar winds at different metallicities is thus emerging. But the question of their {\it absolute} strength is not settled. The reason is clumping.
There is now no doubt that massive--star winds are not homogeneous. Evidence comes from both indirect and direct tracers. The former comprise electron scattering wings of emission lines \citep{hil91}, too--weak observed UV line profiles \citep{paul02b,hil03,massa03,jc03,jc05,fullerton06}, shapes of X-ray lines \citep{oskinova04}, submm excess in continuum emission \citep{blomme02}, linear--polarization continuum variations \citep{stlouis93}. Direct indications are the observations of global moving substructures on emission--line profiles \citep{eversb98,lm99,lm08}. Clumping affects mass--loss determination of massive stars because it changes the density, which in turn modifies the shape of wind lines. In practice, clumping is usually quantified by means of a filling factor $f$. For a given $f$, recombination lines will be fitted, to first order, with a mass loss rate approximately $1/\sqrt{f}$ times lower compared to a homogeneous atmosphere model. Knowledge of the clumping properties of massive--star winds is thus a key to the quantitative understanding of massive star evolution.
Studies of Galactic stars indicate $f$ factors ranging from 0.01 to 0.2 \citep{hk98,hil01,jc05,fullerton06}. Such factors may not be constant throughout the wind of a massive star \citep{puls06}. In addition, if inhomogeneities are due to line driving instabilities \citep{runacres02}, a metallicity dependence can be expected. It is however not constrained at present. A first step in this direction was made by \citet{mar07} (hereafter paper I). In this study, the dynamical properties of clumps directly observed in emission lines of three SMC WR stars were found to be very similar to those of Galactic objects.
In the present study, we analyze the three stars observed in paper I by means of atmosphere models. We constrain the stellar and wind properties, which help us understand their evolutionary status. Being located in the SMC, these stars give us a clearer picture of stellar evolution beyond the main sequence in a low metallicity environment. We find that evolution at very high rotational rate is most likely at work, as suspected for the formation of GRBs. In addition, one can pinpoint the value of the clumping factor $f$ for one star, showing no systematic difference with values found for Galactic stars.
In the following, we briefly review the observational material in Sect.\ \ref{s_spec} before explaining the modelling technique in Sect.\ \ref{s_models}. We then present the results for each star in Sect.\ \ref{s_results} and discuss them in Sect.\ \ref{s_disc}. Sect.\ \ref{s_conc} summarizes our conclusions.
\section{Spectroscopic data}
\label{s_spec}
The three stars analyzed here (SMC-WR1, SMC-WR2, SMC-WR4) were observed with the UV-Visual Echelle Spectrograph (UVES) on VLT/UT2 at ESO. Observations were conducted on August 27-28$^{th}$ 2006. The regions between 3930--6030 was targeted. A slit width of 1.5\arcsec\ was used and observations were conducted under an airmass varying between 1.5 and 2.7, and a seeing fluctuating between 0.8 and 2\arcsec. The data were reduced using the ESO UVES Pipeline. The reduction process
includes bias and inter-order background subtraction, for both the objects
frames and the flat-fields calibration images, optimal extraction of the
objects (that includes removing skylines and cosmics rejection), division by
a flat-field extracted with the same weighed profile as the object,
wavelength calibration and merging of all overlapping orders after a
rebinning to a constant wavelength. No spectrophotometric standard was observed, so our spectra are not flux calibrated. The final spectra have a resolving power of $\sim$ 20000 and a signal to noise ratio between 120 and 160. More details about the observations and data reduction can be found in \cite{mar07}. The spectra we use in the following are the combination of all spectra obtained during the observing run.
We have also used the weighted mean spectra covering the H$_{\alpha}$\ region presented by \citet{foel03a}. They have a low resolution (R$\sim$800) and result from observations with various telescopes and spectrographs \citep[see details in Table 2 of][]{foel03a}. These spectra are not flux calibrated.
In addition to the optical spectra, we used archival IUE data for stars SMC-WR2 and SMC-WR4 (spectra SWP7026 and a weighted mean -- according to exposure time -- of SWP06195 and SWP10727 respectively). They were taken in the low resolution mode (R $\sim$ 250) of the instrument. Two high resolution spectra exist for SMC-WR4, but their S/N is low and they clearly suffer from reduction artifacts (e.g. oscillating continuum). For those two stars we also retrieved FUSE archival data (spectra G036101000 and G03900301000). They have a resolving power of about 20000.
Finally, ESO--SOFI spectra of the sample stars covering the J and H band (also K--band for SMC-WR4) were used. The data on SMC-WR4 were presented by \citet{paul00}. For SMC-WR1 and SMC-WR2, we used unpublished data kindly provided by Paul Crowther. They were obtained in November 2003 (only SMC-WR1) and November 2004 (SMC-WR1 and SMC-WR2) with a 0.6\arcsec\ slit. Their resolution is $\sim$ 500 and they are not flux calibrated. They were reduced following the same procedure as described in \citet{paul00}.
\section{Modelling}
\label{s_models}
The stellar and wind properties of the stars analyzed here have been derived using non-LTE, line blanketed models including winds. The models have been computed with the code CMFGEN. A detailed description of the code is given in \cite{hm98}. Here, we recall only a few points relevant for the understanding of the present study.
\textit{Line-blanketing:} a super-level approximation is used to allow the inclusion of a large number of atomic levels from various elements. In our models, H, He, C, N, O, Ne, P, Si, S, Fe, Ni have been included. In total, the populations of $\sim$1150 levels and $\sim$3500 super-levels were calculated through the resolution of the statistical equilibrium equations. A constant microturbulent velocity (20km s$^{-1}$) was adopted for the calculation of the atmospheric structure. For the formal solution of the transfer equation leading to the emergent spectrum, a microturbulent velocity ranging from 10--50 km s$^{-1}$\ (depending on the star) to 0.1$\times$$v_{\infty}$\ from bottom to top of the atmosphere was chosen. Stellar abundances were chosen to be 1/5$^{th}$ solar \citep{gs98}.
\textit{Clumping:} CMFGEN allows the inclusion of clumping with the following formalism. A volume filling factor $f$ is assumed to vary monotonically with depth, starting from a value of 1 at the photosphere and reaching a maximum value $f_{\infty}$ at the outer boundary of the atmosphere. In practice, $f$ is parametrized as a function of the local wind velocity $v$
\begin{center}
\begin{equation}
f = f_{\infty} + (1-f_{\infty})e^{-\frac{v}{v_{cl}}}
\label{eq_clump}
\end{equation}
\end{center}
\noindent where $v_{cl}$ is a parameter fixing the velocity at which clumping starts to become significant. In practice, we chose $v_{cl}$ = 100 km s$^{-1}$\ so that clumping starts above the sonic point. Lower values have been reported for O stars \citep{jc05} but are difficult to explain theoretically \citep{runacres02}. \cite{puls06} showed that in O stars, clumping varied non monotonically: after an increase ($f$ decreases) up to the region around the sonic point, clumping decreases in the outer wind. Theoretical simulations by \cite{runacres02} tend to predict a similar trend. However, this behavior has not been constrained for Wolf-Rayet stars. Hence, we adopt a conservative approach and keep on working with a monotonic clumping law, keeping in mind that it might not be fully realistic.
\textit{Density structure:} Until recently a major problem with the analysis of W-R stars was that the density structure had to be specified using an assumed velocity law. However work by \citet{goetz05,goetz08} have allowed velocity laws (and mass loss rates) to be derived for some W-R stars using a formulation of radiation driving applicable to optically thick winds. A significant limitation of the work is that the volume filling factor $f$, and its variation with radius, must be assumed since as we have seen clumping properties are still poorly constrained.
Unfortunately, CMFGEN is not yet able to self-consistently compute the velocity law. It still assumes some analytical expression above (approximately) the sonic point: the so-called $\beta$ velocity law is connected to the photospheric structure. Below the sonic point, a photospheric structure is either parametrized by a scale height formalism \citep{hm98} or is taken from alternative atmosphere models, usually TLUSTY \citep{lh03}. In the present analysis, we have improved this formalism. We now require that hydrostatic equilibrium is satisfied below the sonic point. In practice, we proceeded as follows. We first started with a photospheric TLUSTY structure connected to a $\beta$ velocity law. With this structure, we started to converge the models. After a few iterations (usually 10-15), the predicted level populations were used to compute the radiative acceleration. This acceleration was subsequently injected in the hydrostatic equilibrium equation from which an altered density structure was derived at depth, below the sonic point. This ensures a higher degree of consistency between the hydrodynamic and atmospheric structures. Above V(sonic)/2, we adopted the usual beta velocity law formulation (see below). With this revised structure, the models were converged for another 10-15 iterations before another modification of the photospheric structure was performed. We ran 3--4 of these structure iterations before converging the atmosphere models.
In practice, the main inconsistency in the final density structure (lack of radiative acceleration) was found around the sonic point. This reflects the fact that we are specifying the wind velocity law rather than deriving it, that the critical point is close to the sonic point, and that the disagreement depends on the clumping law adopted. \\
With these models, we have constrained the main stellar and wind parameters using the following procedure:
\begin{itemize}
\item[$\bullet$] \textit{Temperature}. As usual for Wolf-Rayet stars, we have derived both an effective temperature representing the radiation flux at a Rosseland optical depth of 2/3, and a temperature T$_{*}$ at $\tau$=20. Both values can be significantly different in the case of a very dense wind for which $\tau$=2/3 is reached in a region where the atmosphere is no longer static. In practice, the temperature of the model was adjusted to correctly reproduce the \textit{relative} strength of \ion{N}{iii} $\lambda\lambda$4635,4641, \ion{N}{iii} $\lambda\lambda$4510,4515, \ion{N}{iv} $\lambda$4058, \ion{N}{v} $\lambda\lambda$4604, 4620 and \ion{N}{v} $\lambda$4944 (when present). Infrared He lines -- \ion{He}{i} $\lambda$1.08 \ifmmode \mu m \else $\mu m$\fi, \ion{He}{ii} $\lambda$1.17 \ifmmode \mu m \else $\mu m$\fi -- were also used and usually gave consistent results (see Sect.\ \ref{s_results}) . In addition for one star (SMC-WR4) we could use the classical \ion{He}{i}/\ion{He}{ii} optical line ratio since the optical \ion{He}{i} lines were strong enough. We estimate the uncertainty on the derived temperatures to be of the order of 3000 K for SMC-WR2 and SMC-WR4, 7000 K for SMC-WR1 (for which \ion{N}{iii} is not present and \ion{N}{iv} is weak).
\item[$\bullet$] \textit{Luminosity and extinction}. For stars SMC-WR2 and SMC-WR4, the luminosity and local extinction was determined from the fit of the SED. The IUE (flux calibrated) spectra have been used in conjunction with optical photometry (our optical spectra are not flux calibrated) to adjust the stellar luminosity and extinction. Table \ref{tabphotom} summarizes the photometric data for each star. FUSE data were not used since 1) the extinction laws in the far-UV are not well constrained and 2) the IUE and FUSE spectra did not show the same flux level in the region around 1200 \AA\ (especially for SMC-WR2), pointing to an uncertainty in the flux calibration (either for FUSE or IUE data). Given point 1, we relied preferentially on IUE spectra. Infrared data were not used since the extinction law in the near-IR is not constrained for the SMC. The Galactic (SMC) extinctions laws of \citet{seaton79} and \citet{howarth83} \citep{bouchet85} were used. The SMC extinction is stronger than the Galactic one in the UV (they are similar in the optical), so that using both IUE and optical data helped constrain the local, SMC extinction and the foreground, Galactic E(B-V). A distance modulus of 19.0 (corresponding to a distance of 63.2 kpc) was assumed. This value was reported by \citet{diben97} and is intermediate between low values (18.80) and high (19.20) values reported in the literature \citep{cole98,caputo99,groen00,harries03,hilditch05,keller08}. For star SMC-WR1, in the absence of (F)UV data, we only used the absolute V magnitude.
From the observed B-V and the theoretical value expected for hot massive stars \citep[(B-V)$_{0}$=-0.28, see][]{mp06} we derive E(B-V)=0.24 which, together with an assumed distance modulus of 19.0, implies M$_{V}$=-4.57. We adjusted the luminosity to reproduce this value.
\item[$\bullet$] \textit{Abundances}. The ratio of He to H was constrained from the relative strength of the optical \ion{He}{ii} lines (especially those at 4686 and 5412 \AA) to the Balmer lines (mainly H$_{\beta}$ and H$_{\delta}$). The carbon abundance was obtained from \ion{C}{iv} $\lambda\lambda$5801, 5812 and, to a lesser extent, from \ion{C}{iv} $\lambda\lambda$1548, 1551. The nitrogen content was derived from the same lines used to constrain the temperature. Note that the latter depends on the \textit{relative} strength of those lines, while the N abundance affects the \textit{absolute} strength of the lines. In addition, \ion{N}{iv} $\lambda$1486 and \ion{N}{iv} $\lambda$1720 were also use as a secondary indicator. \ion{O}{v} $\lambda$5597 and \ion{O}{v} $\lambda$5604 are used to provide upper limits on the oxygen content.
\item[$\bullet$] \textit{Mass loss rate and terminal velocity}. $\dot{M}$\ was derived from the numerous emission and P-Cygni lines present throughout the (F)UV and optical spectrum. We used the blue-ward velocity edge of the absorption part of P-Cygni profiles to derive $v_{\infty}$.
\item[$\bullet$] \textit{Velocity field}. The slope of the velocity field (described by the so-called $\beta$ parameter) was derived from the shape of the \ion{He}{ii} 4686 emission profile. This profile reacts to a combination of change of density due to a different velocity field and to the subsequent change of formation depth of the line implied by the modified atmospheric density structure. In practice, a narrower, more peaked profile results from a larger $\beta$. This is somewhat different from the usual behavior encountered among O stars. The difference is due to the higher wind density affecting the line formation depth in WR stars, while for O stars it is not significantly changed.
\item[$\bullet$] \textit{Clumping}. The clumping factor $f_{\infty}$ was constrained from the strength of the electron scattering wing of \ion{He}{ii} $\lambda$4686. As shown by \cite{hil91}, this part of the spectrum is especially sensitive to the degree of homogeneity in the star's atmosphere: more structured winds lead to weaker electron scattering wings. This method was already used by \cite{paul00} for the study of the star SMC-WR4.
\end{itemize}
Finally, in the FUSE range we have added HI and H$_{2}$ interstellar absorptions on top of our synthetic spectrum. We have not attempted a detailed fit of these interstellar lines. Instead, we have chosen column densities reproducing the main absorption lines.
\section{Results}
\label{s_results}
The results of our analysis in terms of derived parameters are summarized in Table \ref{tab_param}. In the following, we briefly comment on the analysis of each individual star.
\begin{figure*}
\begin{center}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{fit_SED_wr2.eps}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{fit_SED_wr4.eps}
\end{minipage}
\caption{SED fitting for SMC-WR2 (left panel) and SMC-WR4 (right panel). The IUE spectra are reported as well as the fluxes in the UBV bands. E(B-V)=0.03+0.08 (Galactic + SMC) and 0.04+0.10 are required for SMC-WR2 and SMC-WR4 respectively. A distance of 63.2kpc is assumed \citep{diben97}. This SED fitting yields the luminosities: $10^{5.50} L_{\odot}$ for SMC-WR2, $10^{5.90} L_{\odot}$ for SMC-WR4.} \label{fitsed}
\end{center}
\end{figure*}
\begin{table}
\renewcommand{\thefootnote}{\thempfootnote}
\centering
\caption{Photometry of the target stars from \citet{md01} together with the derived extinction.} \label{tabphotom}
\begin{minipage}{0.5\textwidth}
\centering
\begin{tabular}{clrrrr}
\hline
Star & ST & U & B & V & E(B-V) \\
& & & & & Gal.+ SMC \\
\hline
SMC-WR1 & WN3h & 14.22 & 15.13 & 15.17 & 0.24 \\
SMC-WR2 & WN5h & 13.26 & 14.08 & 14.26 & 0.03+0.06 \\
SMC-WR4 & WN6h & 12.48 & 13.25 & 13.37 & 0.04+0.10\\
\hline
\end{tabular}
\end{minipage}
\end{table}
\subsection{SMC-WR4}
\label{s_res_wr4}
The fit of the SED is shown in Fig.\ \ref{fitsed}. The error bars on the optical data reflect the dispersion of measured V magnitudes found in the literature. A luminosity of $10^{5.90} L_{\odot}$ is derived. The best fit of the normalized spectra of SMC-WR4 is shown in Fig.\ \ref{fit_wr4}. The only major difficulty concerns the emission line red-ward of \ion{N}{v} $\lambda$4604, which is not reproduced. But overall, the fit is of excellent quality. An effective temperature of 42000 K gives the best fit of the optical N lines. Slightly lower values ($\sim$ 41000 K) are preferred for the near-IR He lines (especially \ion{He}{i} $\lambda$1.08 \ifmmode \mu m \else $\mu m$\fi), still consistent with the N lines results within the uncertainties. This difference can be explained if the star's spectrum shows variability. Indeed, the near-IR and optical data have been obtained at different epochs. Since the star shows line profile variability (see paper I), simultaneous optical/infrared observations are needed to investigate this discrepancy. The simplified approach to treat clumping might also affect the detailed line profiles. For the FUV interstellar absorption, we found that HI (resp. H$_{2}$) column densities of $10^{22.0} cm^{-2}$ (resp. $10^{19.7}$) gave reasonable results. Individual abundances for He, C, N and O (upper limit) were derived from our analysis. Their determination is illustrated in Figs.\ \ref{ab_wr4}. A medium degree of He enrichment is detected in SMC-WR4, together with a rather strong N overabundance and a carbon depletion. The typical uncertainties on these abundance determinations are of the order 30 to 50$\%$, as seen from Fig.\ \ref{ab_wr4} \footnote{These errors do not include any systematics due to uncertainties in the atomic data}. Finally the mass loss rate determination gives a value of $8 \times\ 10^{-6}$ \ifmmode M_{\odot} \else M$_{\odot}$\fi. Strong optical emission lines would require slightly larger values of the order $10^{-5}$ \ifmmode M_{\odot} \else M$_{\odot}$\fi, while near-IR H and He lines tend to indicate slightly lower values ( $\sim 7 \times\ 10^{-6}$ \ifmmode M_{\odot} \else M$_{\odot}$\fi). \footnote{Note that our best fit model in Fig. \ref{fit_wr4} has a slightly smaller mass loss rate (0.1 dex) than the best fit model of Fig. \ref{ab_wr4} in order to best match both optical/UV and near-infrared lines. In Fig.\ \ref{ab_wr4}, $\dot{M}$\ is scaled to fit \ion{He}{ii} $\lambda$4686}.
The left panel of Fig.\ref{f_effect_wr4} shows various models with different clumping parameters ($f_{\infty}$ ranging from 0.01 to 0.5). For each model, the mass--loss rate is scaled to properly account for the emission of \ion{He}{ii} $\lambda$4686. One clearly sees that the strength of the red electron scattering wing of the line depends on the amount of structure in the wind. From this feature, we see that a model with $f_{\infty} \sim$ 0.1 best represents the observed profile. For a more quantitative statement, we have estimated the goodness of the fit of this red wing by means of a $\chi^2$ analysis of the region between 4702 and 4730 \AA. The results are shown in Fig.\ \ref{f_effect_wr4}, right panel. A value of 0.15$\pm$0.05 is preferred.
SMC-WR4 was studied by \citet{paul00} in the optical and near infrared, using CMFGEN. The parameters we obtain are very close to his. The only notable difference is the slightly larger luminosity we derive, although within the error bars the agreement is acceptable. The difference comes from the different distances to the SMC that were adopted: we use 63.2 kpc while \citet{paul00} assumes a distance of 60.2 kpc. Using his distance, we would derive a luminosity of $10^{5.8} L_{\odot}$, in good agreement with Crowther's result. We also note that Crowther's fit also showed the problem reported for the mass loss rate determination: near-IR lines tend to indicate lower $\dot{M}$\ than optical lines.
\begin{figure*}
\begin{center}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{He_effect_wr4.eps}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{CN_effect_wr4.eps}
\end{minipage}
\caption{He (left) and CN (right) abundance determination for SMC-WR4. The black solid line is the observed spectrum, the interrupted colored lines are models with different He/H (C/H, N/H) ratios.}\label{ab_wr4}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{f_effect_wr4.eps}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{chi2_f_wr4_fit_new.eps}
\end{minipage}
\caption{Determination of clumping parameter $f_{\infty}$. {\it Left}: Effect of the clumping parameter f$_{\infty}$ on the electron scattering wing of HeII 4686 for star SMC-WR4. {\it Right}: Quantification of the goodness of the fit of the red wing of HeII 4686. The solid line is a simple third order polynomial fit. A value of 0.15$\pm$0.05 is preferred for the clumping factor $f_{\infty}$.} \label{f_effect_wr4}
\end{center}
\end{figure*}
\subsection{SMC-WR2}
\label{s_res_wr2}
The fit of the SED of SMC-WR2 is shown in Fig.\ \ref{fitsed}. The error bars on the optical data reflect the dispersion of measured V magnitudes. We derive a luminosity of $10^{5.50} L_{\odot}$. Fig.\ \ref{fit_wr2} shows our best fit of the normalized spectra. The overall quality of the fit is as good as for SMC-WR4. HI and H$_{2}$ interstellar absorption lines have been added in the FUSE range with the following column densities: $10^{21.7} cm^{-2}$ for HI and $10^{19.3} cm^{-2}$ for H$_{2}$. The optical N lines indicate an effective temperature of $\sim$ 42500 K. We note that with this \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ the \ion{He}{i} $\lambda$1.08 \ifmmode \mu m \else $\mu m$\fi\ line is underpredicted. Obtaining a stronger emission requires a reduction of \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ by $\sim$ 5000 K. But in that case the optical spectrum is not reproduced anymore: the \ion{N}{v} lines vanish completely, \ion{N}{iv} $\lambda$4058 \AA\ is strongly reduced and the \ion{N}{iii} lines become much too strong. The same explanation as for SMC-WR4 could be invoked. Clearly, coordinated multiwavelength observations (as soon possible with instruments such as VLT/X--Shooter) are necessary to solve this issue. One can also note that the S/N ratio of the near-IR data is lower than for the optical data, increasing the uncertainty on the temperature determination from NIR lines. Abundance determinations were performed as for SMC-WR4. Only a small He enrichment is required to fit the He and H lines of the spectrum. However, evidence for CNO processing comes from the existence of a C depletion and a N overabundance. We note that both optical and near-IR emission lines are well reproduced by the a single value of the mass loss rate.
Unlike in SMC-WR4, \ion{He}{ii} $\lambda$4686 is a rather weak emission line. Consequently, its red wing is more sensitive to details of the data reduction and normalization. We have attempted to derive the clumping factor $f_{\infty}$ as for SMC-WR4, but it turned out that no conclusions could be drawn due to these observational uncertainties. In particular, the exact position of the continuum is uncertain due to residuals of the flat field correction and to imperfect order merging. Other classical clumping indicators are found in the FUV--UV range. \ion{O}{v} $\lambda$1371 and \ion{N}{iv} $\lambda$1720 have been shown to depend on the degree of inhomogeneities \citep[e.g.][]{jc05}. However, the IUE spectra of SMC-WR2 have a too low spectral resolution and these lines cannot be used. In the FUSE range, \ion{P}{v} $\lambda\lambda$1118, 1128 are known to depend on the amount of clumping. \citet{fullerton06} showed that very low filling factors were necessary to account for the strength of this doublet, while \citet{paul02} and \citet{hil03} obtained satisfactory fits with $f_{\infty} \sim 0.1$. \citet{fullerton06} highlighted the high sensitivity of this doublet to the ionization structure, which can be influenced by shocks and X--rays. This effect is probably reduced in dense WR star winds compared to O stars, but might still be present. Besides, \citet{oskinova07} revealed that depending on which type of clumping was present (micro vs macro--clumping) the line profiles had different shapes. Finally, \citet{paul02} showed that the phosphorus abundance strongly influenced the shape of this unsaturated line. Given these multiple dependencies, we refrained from using \ion{P}{v} $\lambda\lambda$1118, 1128 as a clumping diagnostic in the present study. We simply adopted $f_{\infty}$=0.1 for SMC-WR2.
\subsection{SMC-WR1}
\label{s_res_wr1}
Unfortunately, only optical and near-IR spectra are available for SMC-WR1. The analysis is thus more limited than for the two other stars. In particular, in the absence of UV P-Cygni lines, we adopted a terminal velocity of 1800 km s$^{-1}$\ which turned out to give an excellent fit of the optical emission lines. The uncertainty on \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ (and consequently on L) is larger for this star since only a limited number of lines from successive ionization states is available. In particular, although several \ion{N}{v} lines are used, the only \ion{N}{iv} line (at 4058 \AA) is very weak. It can be used to place a lower limit on \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi. The upper limit comes from the shape of the \ion{N}{v} $\lambda\lambda$4604, 4620 lines which become narrower and weaker at too high effective temperature. In practice, the uncertainty on \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ is $\pm$ 7000 K. This translates into an uncertainty on \ifmmode \log \frac{L}{L_{\odot}} \else $\log \frac{L}{L_{\odot}}$\fi\ of 0.15 dex. In spite of these increased uncertainties, we were able to derive a number of stellar and wind parameters. Fig.\ \ref{fit_wr1} shows our best fit. Overall, the quality is good: optical and near-IR lines can be reproduced with a single set of stellar and wind parameters. The \ion{He}{ii} lines of the Brackett series are slightly too deep and narrow. A change of $\beta$ does not help since it degrades the fit of \ion{He}{ii} $\lambda$4686. Similarly, increasing $v_{\infty}$\ broadens too much the other emission lines. \ion{He}{ii} $\lambda$5412 is slightly underestimated. Improving its fit by increasing $\dot{M}$, \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ or He/H alters the fits of all other lines. Presumably, remaining uncertainties in the density structure are responsible for this discrepancy.
Like SMC-WR2 and SMC-WR4, SMC-WR1 is slightly He enriched and strongly N rich. Fig.\ \ref{He_effect_wr1} illustrates our He/H determination. As we will see later, this quantity is especially important to understand the evolution of SMC-WR1. From the weakness of \ion{C}{iv} and \ion{O}{v} lines in the red optical spectrum, one can put upper limits on the C and O content (see Table \ref{tab_param}), indicating depletion of these two elements. As for SMC-WR2, \ion{He}{ii} $\lambda$4686 is not strong enough to allow a correct clumping factor determination. We adopted $f_{\infty}$=0.1.
\begin{figure}[]
\centering
\includegraphics[width=9cm]{He_effect_wr1.eps}
\caption{Determination of the He to H ratio for star SMC-WR1. The observed spectrum is the black solid line. The mass loss rates of the models with different He/H ratios were adjusted to give a correct fit of the \ion{He}{ii} $\lambda$4686 \AA\ emission line. A value of He/H=0.25 is preferred.}\label{He_effect_wr1}
\end{figure}
\begin{table*}
\renewcommand{\thefootnote}{\thempfootnote}
\centering
\caption{Parameters of our best fit models. Uncertainties are: 3kK (7kK for SMC-WR1) on temperatures, 0.1 dex on luminosities (0.15 for SMC-WR1), 0.2 dex on $\dot{M}$, 100 km s$^{-1}$\ on $v_{\infty}$, 0.05 on He/H, 30 to 50$\%$ on abundances. } \label{tab_param}
\begin{minipage}{\textwidth}
\centering
\begin{tabular}{clrrrrrrrrrrrrr}
\hline
Star & ST & T$_{*}$ & \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi & \ifmmode \log \frac{L}{L_{\odot}} \else $\log \frac{L}{L_{\odot}}$\fi & R$_{*}$ & $\log \dot{M}$ & f$_{\infty}$ & $\beta$ & $v_{\infty}$ & He/H & X & X(C) & X(N) & X(O)\\
& & [kK] & [kK] & & R$_{\odot}$ & [M$_{\odot}$ yr$^{-1}$] & & & [km s$^{-1}$] & \# & (\%) &(\%) & (\%) & (\%)\\
\hline
SMC-WR1 & WN3h & 65.4 & 60.5 & 5.75 & 5.8 & -5.55 & 0.10\footnote{Adopted} & 1.0 & 1800 & 0.25 & 0.50 & $<$0.004 & 0.12 & $<$0.04 \\
SMC-WR2 & WN5h & 44.8 & 42.5 & 5.50 & 9.3 & -5.85 & 0.10\footnotemark[\value{mpfootnote}] & 1.0 & 1000 & 0.15 & 0.62 & 0.005 & 0.35 & -- \\
SMC-WR4 & WN6h & 43.1 & 42.0 & 5.90 & 16.0& -5.00 & 0.15\footnote{Derived from $\chi^{2}$ fit, mass loss rate scaled from model closest to minimum (f$_{\infty}$=0.1).} & 1.0 & 1000 & 0.45 & 0.35 & 0.003 & 0.40 & $<$0.15 \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{Discussion}
\label{s_disc}
In the next sections, we first comment on our results on the clumping factor of SMC-WR4 in the context of a metallicity dependence of clumping properties. We then move to the interpretation of the evolutionary status of the three WNh stars analyzed in the present study.
\subsection{Winds and clumping}
\label{s_wind}
In paper I, a thorough investigation of moving features in strong emission lines revealed that the dynamical properties of clumps in the SMC were very similar to those of Galactic WR stars. Here, we have attempted to constrain the clumping factor at low Z. For star SMC-WR4, we have carried out a detailed study of the clumping factor. We have quantitatively shown that the preferred value for $f_{\infty}$ (see Eq.\ \ref{eq_clump}) was 0.15$\pm$0.05. The $\chi^2$ curve of Fig.\ \ref{f_effect_wr4} unambiguously confirms that lower and larger values are excluded.
Studies of Galactic WR stars have been the first to reveal that clumping was present in massive-star winds \citep[e.g.][]{moffat88,stlouis89}. Quantitative determinations of clumping factors were carried out by various groups using different atmosphere codes. \citet{hk98} analyzed three WN (one WN5, one WN6 and one WN8) stars and one WC star and found that a clumping filling--factor of 0.25 was best suited for WN stars, although 0.06 was still compatible within the uncertainties. The latter value was best for the WC star. Similarly, \citet{schmutz97} found that $f_{\infty}$ = 0.25 led to a good fit of the emission lines of the WN4 star HD50896. \citet{hm99} found that $f_{\infty} = 0.1$ was best suited to fit the UV-optical-IR spectrum of a Galactic WC5 star. \citet{hil01} carried out an extensive study of the spectra of $\eta$ Car and found that a clumping factor of $\sim$0.1 better fitted the electron scattering wings of both H and HeI lines compared to $\sim$0.2-0.3. \citet{herald01} analyzed two Galactic WN8 stars and concluded that $0.05 < f < 0.2$ from the same diagnostics. \citet{paul02,paul06} analyzed WC stars in the LMC and the Galaxy and found that f$_{\infty}$=0.1 was preferred compared to 1.0 and 0.01. However, no other intermediate values were tested. \citet{paco04} found $f_{\infty}$ values between 0.08 and 0.15 for five WNLh stars in the Arches cluster. Hence, a value of 0.1--0.2 seems to be common among Galactic WR stars and related, post main sequence objects.
For SMC-WR4, we have shown that $f_{\infty}$ was definitely between 0.10 and 0.20 (see Fig.\ref{f_effect_wr4}). Keeping in mind that we are dealing with a single measurement for the SMC, it looks as if there is no difference between the low metallicity environment of the SMC and the Galaxy in terms of clumping factor. Or more accurately, if there is any variation, it is not larger than $\sim$ 0.1, which is the uncertainty of current determinations. This is consistent with the results of paper I.
From a theoretical point of view, the hydrodynamical simulations of \citet{owo88}, \citet{runacres02} and \citet{runacres05} show that instabilities grow at a rate proportional to $g_{rad}$, the line radiative acceleration. The radiative acceleration $g_{rad}$ scales with Z$^{1-\alpha}$ \citep[see Eq. 12 and 79 of][]{puls00}, $\alpha$ being the classical CAK parameter. But $g_{rad}$ also depends on $\rho^{-\alpha}$, and since $\rho$ (the density) is directly proportional to $\dot{M}$, which in turn scales roughly as Z$^{\frac{1-\alpha}{\alpha}}$ \citep[Eq.\ 87 of ][]{puls00}, all Z effects cancel out. In this context, no metallicity effects are expected if clumping is driven by radiative linear instabilities. Alternatively, \citet{go95} have shown that in the case of multiple scattering, line overlap leads to a reduction of the linear instability growth rate. Presumably, lowering the metallicity would reduce this overlap and produce an enhanced growth rate (S. Owocki, priv. comm.). In that case, clumping might be stronger at low Z. This is however still widely speculative, and future hydrodynamical simulations, including non linear effects, will certainly shed more light on this issue.
\subsection{Evolutionary status}
\label{s_evol}
We have shown that the three WNh stars analyzed are slightly He rich, strongly N rich, and C (and O for SMC-WR1 and SMC-WR4) poor. This qualitatively indicates that they are massive stars in an early phase of their post--MS evolution. The large quantity of H still present in their atmosphere shows that they are still H--burning objects, or that they have entered the core He--burning phase very recently. This is confirmed by the position of the stars in a diagram showing the H mass fraction as a function of luminosity (Fig.\ \ref{hcno}, left): one sees that the three stars are located on the early parts of evolutionary tracks, still far away from classical H--free WR stars. In this diagram, all stars can in principle be accounted for by evolutionary tracks. This was not the case in the earlier study of SMC-WR4 by \citet{paul00}: he found that SMC-WR4 was underluminous by roughly 50$\%$ compared to the closest track of \citet{meynet94}. Crowther concluded that stellar evolutionary tracks were probably missing an important ingredient. This ingredient was stellar rotation, which is included in the evolutionary tracks of \citet{mm05} used in Fig.\ \ref{hcno} (the initial rotational velocity in these tracks is 300 km s$^{-1}$). From this diagram, one can estimate the initial masses of our program stars by interpolating between existing tracks (at $Z = 0.2 Z_{\odot}$). The results are given in the first column of Table \ref{m_estim}: masses range from about 25 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ (SMC-WR2) to about 40 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ (SMC-WR4, SMC-WR1). The errors take into account the uncertainties on \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ and \ifmmode \log \frac{L}{L_{\odot}} \else $\log \frac{L}{L_{\odot}}$\fi.
The middle panel of Fig.\ \ref{hcno} shows the C versus N mass fraction together with the predictions of \citet{mm05} for various initial masses. The errors on the carbon content are smaller than the symbols. SMC-WR4 and SMC-WR2 are in (or very close to) a phase where X(C) is minimum and X(N) is maximum. SMC-WR1 appears to have a carbon mass fraction consistent with CNO equilibrium, but its N content is a little too low. A similar conclusion is reached if one looks at the right panel of Fig.\ \ref{hcno}: it shows the O versus N mass fraction for stars SMC-WR1 and SMC-WR4. As for carbon, the oxygen content is reduced during the CNO cycle, so that an O underabundance can be expected from evolutionary tracks. The O mass fraction of SMC-WR1 is consistent with the expected minimum, as is the case for carbon. In this figure too the N mass fraction is not as large as expected for the O content. We will see later that the current tracks might not be appropriate for this star. The upper limit on the O content of SMC-WR4 is too high to draw any meaningful conclusion. This limit (not shown) is even higher for star SMC-WR2.
\begin{table}
\begin{center}
\caption{Initial mass estimate (in solar units) from evolutionary diagrams. The second and third column give estimates based respectively on the X(H)--L and HR diagram. The errors take into account uncertainties on \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ and \ifmmode \log \frac{L}{L_{\odot}} \else $\log \frac{L}{L_{\odot}}$\fi. Tracks with $Z = 0.2 Z_{\odot}$ from \citet{mm05} are used.} \label{m_estim}
\begin{tabular}{clrrrrrrrrrr}
\hline
Star & X(H)--L & \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\--L \\
\hline
& & \\
1 & 34.4$^{+10.1}_{-7.8}$ & $>$40 \\
& & \\
2 & 22.8$^{+5.4}_{-5.8}$ & 47.3$^{+7.0}_{-10.6}$ \\
& & \\
4 & 42.2$^{+6.9}_{-7.8}$ & 64.7$^{+16.0}_{-10.1}$ \\
& & \\
\hline
\end{tabular}
\end{center}
\end{table}
Let us now turn to the HR diagrams shown in Fig.\ \ref{hrd}. The position of the three target stars is indicated on top of the \citet{mm05} evolutionary tracks at Z=0.2 Z$_{\odot}$ (appropriate for the SMC, left panel) and Z=0.4 Z$_{\odot}$ (right panel). We have highlighted in bold font the phase of the tracks in which the H mass fraction is larger than 20$\%$. Since SMC-WR1, SMC-WR2 and SMC-WR4 all have X(H) $>$ 0.3, any comparison between the position of the stars and evolutionary models should rely on the bold lines in either panel. Let us consider each star one by one:
\begin{itemize}
\item SMC-WR4: for this star, we find X(H)=0.35$\pm$0.1. From the left panel of Fig.\ \ref{hrd} (i.e. the one with evolutionary tracks at 0.2 \ifmmode Z_{\odot} \else Z$_{\odot}$\fi), one might invoke two scenarios to explain the position of SMC-WR4. In the first one SMC-WR4 is a relatively unevolved star, still close to the main sequence and with an initial mass slightly above 60 \ifmmode M_{\odot} \else M$_{\odot}$\fi. In that case the initial mass can be estimated from the uncertainties on \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ and \ifmmode \log \frac{L}{L_{\odot}} \else $\log \frac{L}{L_{\odot}}$\fi, as given in the second column of Table \ref{m_estim}. The result is somewhat surprising, since it is much larger than the estimate based on the X(H)--L diagram, with no overlap within the errors (42.2$^{+6.9}_{-7.8}$ vs 64.7$^{+16.0}_{-10.1}$). Besides, if SMC-WR4 were indeed at an early post--main sequence phase of its evolution, its mass loss rate would be around $10^{-5.7}$ M$_{\odot}$ yr$^{-1}$\ according to \citet{vink01}. Instead, we find a value one order of magnitude larger (once corrected for clumping). Thus it appears unlikely that SMC-WR4 is just evolving off the main sequence.
Another possibility is that SMC-WR4 is a star with an initial mass of 40--50 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ and evolving blue-ward from a supergiant phase. Indeed, the $Z = 0.2 Z_{\odot}$ 40 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ track ends a little before reaching the position of SMC-WR4. However, the same track at Z=0.4 \ifmmode Z_{\odot} \else Z$_{\odot}$\fi\ passes through the position of SMC-WR4 while still having X(H)$>$0.2 (see right panel of Fig.\ \ref{hrd}). Hence assuming a metal content slightly larger than 0.2 \ifmmode Z_{\odot} \else Z$_{\odot}$\fi\ can explain the characteristics of SMC-WR4. In that case, the 40 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ track would explain the position of SMC-WR4 in both the HR and the X(H)--L diagram (the initial mass derived from the latter diagram is very similar at 0.2 and 0.4 \ifmmode Z_{\odot} \else Z$_{\odot}$\fi\ for SMC-WR4). Note that other effects such as a slightly different initial rotation rate compared to the values used in the evolutionary calculations of \citet{mm03}, or different mass loss rates (uncertainties being large due to clumping) could also modify the tracks at Z=0.2 \ifmmode Z_{\odot} \else Z$_{\odot}$\fi\ and explain the position of SMC-WR4. Given this, we tentatively conclude that SMC-WR4 results from rather normal evolution. \\
\item SMC-WR2: the hydrogen mass fraction is the largest of the three stars, reaching 0.62$\pm$ 0.1. The situation is different from SMC-WR4. There is only one possibility to explain both the effective temperature/luminosity and H content of SMC-WR2: it must be a star just evolving off the main sequence. Indeed, none of the evolutionary tracks coming back from the cool part of the HR diagram and with low enough luminosities can account for the large H mass fraction of SMC-WR2. The 30\ifmmode M_{\odot} \else M$_{\odot}$\fi\ track at Z=0.4\ifmmode Z_{\odot} \else Z$_{\odot}$\fi\ has a too low H content when reaching the position of SMC-WR2 (see the tick on the track at X(H)=0.3). The star has to be in its early evolution. If we accept this, one can estimate the initial mass of the star by interpolation between the red-ward evolving tracks. The results (given in the third column of Fig.\ \ref{m_estim}) indicate M=47.3$^{+7.0}_{-10.6}$ \ifmmode M_{\odot} \else M$_{\odot}$\fi. This is significantly larger than the estimate from the X(H)--L diagram (22.8$^{+5.8}_{-5.4}$ \ifmmode M_{\odot} \else M$_{\odot}$\fi). Such a difference is surprising, since it is based on the same set of tracks. We will discuss possible solutions to this puzzle in Sect.\ \ref{s_nat}. \\
\item SMC-WR1: The situation is even more intriguing for star SMC-WR1. We find convincing evidence that SMC-WR1 is still H--rich, with X(H)=0.50$\pm$0.1. But a look at Fig.\ \ref{hrd} shows that SMC-WR1 is definitely located to the left of the H--burning main sequence, at a temperature of $\sim$65000 K and \ifmmode \log \frac{L}{L_{\odot}} \else $\log \frac{L}{L_{\odot}}$\fi\ of 5.75. The only evolutionary tracks reaching this part of the diagram have initial masses larger than $\sim$ 50 \ifmmode M_{\odot} \else M$_{\odot}$\fi. But this is only at the very end of the star's evolution, when it is H and N free. The properties of SMC-WR1 are thus extremely disturbing, classical stellar evolution being unable to explain them.
\end{itemize}
From this analysis, one can conclude that one star (SMC-WR4) seems to have followed a normal evolution. On the contrary, the two remaining stars (SMC-WR1 and SMC-WR2) have probably been affected by uncommon physical processes. We tentatively identify some of them in Sect.\ \ref{s_nat}.
Before concluding this section, we briefly note that mass estimates are important clues to the nature of the sample stars. One can add another such estimate assuming that the stars have Eddington luminosities. In that case one obtains lower limits to the current masses of the three WNh stars. These values are respectively 9.8, 7.8 and 19.6 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ for SMC-WR1, SMC-WR2 and SMC-WR4 (assuming the only source of opacities is electron scattering). In practice, our modelling indicate that the maximum Eddington factors in their atmosphere are of the order 0.7 and that the total radiative acceleration is 1.2--1.5 times the acceleration due to electron scattering, implying current masses of the order 10-35 \ifmmode M_{\odot} \else M$_{\odot}$\fi. Of course, these values have to be taken as rough estimates since they depend on the exact hydrodynamical structure which still suffers from uncertainties (see Sect.\ \ref{s_models}).
\begin{figure*}
\begin{center}
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=6cm]{XH_L_smc.eps}
\end{minipage}
\hspace{0.05cm}
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=6cm]{CN_wnh.eps}
\end{minipage}
\hspace{0.05cm}
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=6cm]{ON_wnh.eps}
\end{minipage}
\caption{Evolutionary status from chemical composition. {\it Left:} Hydrogen mass fraction as a function of Luminosity. {\it Middle:} Carbon mass fraction as a function of Nitrogen mass fraction for stars SMC-WR2 and SMC-WR4. {\it Right}: O mass fraction as a function of N mass fraction from evolutionary tracks (lines) and as derived for star SMC-WR1. The three stars are still rather H rich, but already show signs of CNO processing. Evolutionary tracks are from \cite{mm05} and have an initial rotational velocity of 300 km s$^{-1}$.}\label{hcno}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{hr_wr_smc_xh0p2_Z0p2.eps}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{hr_wr_smc_xh0p2_Z0p4.eps}
\end{minipage}
\caption{HR diagram with the three sample stars. Evolutionary tracks are from \cite{mm05}, the left panel being for Z=0.2\ifmmode Z_{\odot} \else Z$_{\odot}$\fi, the right one for Z=0.4\ifmmode Z_{\odot} \else Z$_{\odot}$\fi. We have highlighted the parts of the tracks corresponding to the H-rich phase, i.e. X(H)$>$20$\%$, using bold lines. On the Z=0.4\ifmmode Z_{\odot} \else Z$_{\odot}$\fi\ 30 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ track, we have also noted the position where X(H)=0.3.}\label{hrd}
\end{center}
\end{figure*}
\subsection{Evidence for evolution with fast rotation.}
\label{s_nat}
The previous section has shown that classical stellar evolution was able to reproduce the observed properties of SMC-WR4 if its metallicity is a little larger than 0.2\ifmmode Z_{\odot} \else Z$_{\odot}$\fi. However, it was shown that the current set of evolutionary tracks failed to explain SMC-WR1 and SMC-WR2.
Before considering SMC-WR1 and SMC-WR2, let us first reiterate our conclusions regarding SMC-WR4. We showed that both its position in the HR diagram and its surface abundances were compatible with a star of initial mass 40--50 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ in or about to enter the core He burning phase. This is rather similar to the conclusion of \citet{foel03a} who argued that SMC-WR4 most likely resulted from standard evolution of a 70--85 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ star. We stress once more that according to this scenario, SMC-WR4 is coming back from the red part of the HR diagram. Consequently, it is not a core H burning object. This is important in the context of the understanding of the properties of H rich WN stars. Recently, \citet{arches} analyzed 16 WN7--9h stars in the Arches cluster and showed that they were rather young and very massive stars still in the core H burning phase. The WN6h stars in the core of NGC3603 are probably similar objects \citep{cd98}. The initial mass estimates for these objects in the Arches and NGC3603 cluster range between 60 and 130 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ (see \citet{schnurr08} for a recent keplerian mass estimate of NGC3603/A1). Hence, it seems that H--rich WN stars come in different flavors. They can be very massive objects in an early phase of evolution, with ages not larger than 2--3 Myr; but they can also result from less massive objects and be more evolved. In the latter category, one should mention star WR3 in the outer part of the Galaxy. \citet{mar04} showed that it was a H--rich object resulting from the evolution of an initially 40--50 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ object. Since lower metallicity is encountered in the outer regions of the Galaxy, star WR3 might be more similar to the SMC stars analyzed in the present study.
Coming back to SMC-WR1 and SMC-WR2, there are a few possibilities to explain the puzzling results of Sect.\ \ref{s_evol}. The first one is binarity. \citet{well99} showed that mass transfer during binary evolution could dramatically change the evolutionary sequence of both components of the system. In particular, their Fig.\ 1 shows that the primary might evolve almost vertically just after the mass transfer. Consequently, if such a star was analyzed by means of classical single star tracks, its mass would be overestimated. This could explain the inconsistencies encountered for SMC-WR2: its mass derived from the HR diagram is larger than that derived from the X(H)--L diagram. Besides, mass transfer would affect the surface abundances, so that the derived H mass fraction might be different from that of single evolutionary sequences. Hence, mass estimates from the X(H)--L diagram would be affected too. Binarity might thus be an alternative to explain the properties of SMC-WR2. However, \citet{foel03a} thoroughly studied the radial velocities and photometry of all the SMC stars and showed that SMC-WR2 was not varying. They concluded that it was most likely a single star, or maybe a long period binary. But in any case it was not a short period binary prone to experience the kind of evolution described above. The same conclusion was reached for SMC-WR1. Periodic changes of the shape of emission profiles may also reveal the close-binary nature of a massive star.
The line-profile variability of SMC-WR1 and SMC-WR2 closely resembles the behavior of emission lines in the presumably single Galactic W-R stars (Paper I), where the stochastic, seemingly random variability is induced by numerous wind-embedded clumps. Hence, we conclude that practically all the variability patterns seen in SMC-WR1 and SMC-WR2 are caused by stochastic appearance of small-scale overdensities in the winds of the SMC W-R stars.
Binarity thus appears unlikely to be the explanation for the properties of SMC-WR1 and SMC-WR2. Alternatively, there is growing evidence that homogeneous evolution might be at work in at least a fraction of massive stars in the Magellanic Clouds. \citet{jc03} argued that star MPG 355 in NGC 346 could have the same age as the other cluster members if it was evolving homogeneously. \citet{walborn04} explained the position of highly N enriched Magellanic Cloud O2 stars in the HR diagram by large mixing leading to homogeneous evolution. Finally, \citet{mokiem07} explained the correlation of large mass discrepancies \footnote{The ``mass discrepancy'' problem refers to the larger evolutionary masses derived from HR diagrams compared to spectroscopic masses derived from gravities through atmosphere modelling \citep[see][]{her92}.} with He enrichment by the fact that He--rich stars might evolve homogeneously, so that using normal tracks would overestimate their evolutionary mass.
Homogeneous evolution was shown to occur in rapidly rotating stars half a century ago by \citet{sch58}. More recently, \citet{maeder87} and \citet{langer92} ran evolutionary models confirming that fully mixed stars were evolving blue-ward in the HR diagram. The conditions under which such homogeneous chemical evolution occurs depend on several parameters. The basic constraint is that the timescale for rotationally induced mixing should be shorter than the nuclear timescale. In practice, this means that the stars should keep a high rotational rate during their evolution. A key ingredient to keep the high rotational rate is the strength of the stellar wind. Larger mass loss rates will generally lead to larger angular momentum removal, which in turn will slow down the rotation. As radiatively driven winds are weaker at lower metallicity \citep[e.g.][]{puls00}, changes in the surface abundances are expected to be more common at (very) low metallicity. But the downward revision of mass loss rates of massive stars due to clumping \citep[e.g.][]{paul02,jc05} could make homogeneous evolution more common at higher metallicity than previously thought. Besides, mass loss anisotropy implied by fast rotation might affect the details of the subsequent evolution \citep{mm07}. In addition, magnetic fields can modify the way rotation changes during the lifetime of the star, leading to solid body rotation \citep{mm03}. From this, one sees that homogeneous chemical evolution will occur under a combination of effects. A systematic study of these conditions is still lacking, partly because of the uncertainties in the mass loss rates of massive stars, especially Wolf-Rayet stars.
Nonetheless, various works have mentioned the existence of homogeneous evolution. \citet{mm00a} argue that stars initially more massive than 40 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ and with $\frac{\Omega}{\Omega_{crit}} > 0.5$ -- $\Omega$ ($\Omega_{crit}$) being the (critical) rotation rate -- will evolve homogeneously. The 60 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ evolutionary track of \citet{mm05} with an initial $V$ sin$i$\ of 500 km s$^{-1}$, as well as the fast rotating models of \citet{mm07} (which include the effects of a magnetic field) confirm this trend. \citet{yl05} also produce blue-ward evolving stars in their fast rotating models at low metallicity. As mentioned above, a complete set of evolutionary tracks for various metallicities and rotational rates does not exist at present. Only a few tracks are available. In Fig.\ \ref{effect_rot}, we plot the evolutionary tracks of a 60 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ star from \citet{mm05} for both $V$ sin$i$\ = 300 km s$^{-1}$\ (solid line) and 500 km s$^{-1}$\ (dashed line). The latter corresponds to a borderline case where the evolution is almost homogeneous, the star evolving at almost constant \ifmmode T_{\rm eff} \else T$_{\mathrm{eff}}$\fi\ in the first phases. In the X(H)--L diagram, one sees that for fast rotating stars, the track is shifted to lower luminosities. In practice, this means that the mass one would derived from a set of tracks with $V$ sin$i$\ = 500 km s$^{-1}$\ would be larger than for the case where $V$ sin$i$\ = 300 km s$^{-1}$. On the contrary, fast rotating stars are on average more luminous in the HR diagram (due to the effects summarized above). Hence, lower masses are derived for larger $V$ sin$i$\ tracks. This is interesting in the context of the mass problem reported for star SMC-WR2: the large difference between the masses derived from the X(H)--L and HR diagram can, at least qualitatively, be reduced if one uses faster rotating tracks. This strengthens the idea that SMC-WR2 might be a rapidly rotating star, close to homogeneous evolution. It is worth mentioning that using the two methods to estimate masses we have applied in this paper, one might have an indirect way of determining the initial rotational velocity of Wolf-Rayet stars: a set of rotating tracks should be able to give consistent mass estimates in the X(H)--L and HR diagram. If not, the initial rotational velocity is probably not correct.
\begin{figure*}
\begin{center}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{XH_L_hom.eps}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.4\linewidth}
\centering
\includegraphics[width=7cm]{hr_wr_hom.eps}
\end{minipage}
\caption{Illustration of the effect of rotation on the X(H)--L diagram (left panel) and on the HR diagram (right panel). The masses estimated using different rotational velocities will be different in both diagrams. Evolutionary tracks from \citet{mm05}. }\label{effect_rot}
\end{center}
\end{figure*}
The case of SMC-WR1 is more extreme: it is located on the left side of the main sequence, but still contains a significant fraction of hydrogen. Inspection of Fig.\ 1 of \citet{mm07} reveals that homogeneous evolution can reproduce these features. In this figure, SMC-WR1 would lie close to to the anisotropic wind track ($\log T_{\rm eff} = 4.8, \log \frac{L}{L_{\odot}}=5.75$) -- although slightly below due to the lower mass of the star compared to the model --, where the core (and consequently the surface) still contains hydrogen. Qualitatively, SMC-WR1 is thus likely an example of a WR star proceeding from homogeneous evolution of a massive star. Star WR3 in the Galaxy is probably a twin of SMC-WR1. \citet{mar04} showed that it was H--rich and very similar spectroscopically to early WN stars in the SMC. Given our results, it is likely that WR3 also results from evolution with high rotation.
As stated above, no systematic study of the occurrence of homogeneous chemical evolution with metallicity, magnetic field and rotation velocity exists. However, a key ingredient is the high initial rotation rate. One might wonder if initial rotational velocities as large as 500 km s$^{-1}$\ can be reached in the SMC. Recent studies of the projected rotational velocities of B stars in the SMC were carried out by \citet{martayan07} and \citet{hunter08}. Both works show that the $V$ sin$i$\ distribution is shifted towards larger values in the SMC compared to the Galaxy, with averages around 155--175 km s$^{-1}$. According to \citet{hunter08}, about 10$\%$ of the SMC stars have $V$ sin$i$\ larger than 300 km s$^{-1}$. Considering that these values are lower limits on the true surface velocities (the inclinations being unknown), stars rotating at 500 km s$^{-1}$\ might exist in the SMC. We thus argue that SMC-WR1 is most likely the result of homogeneous evolution of a massive star.
We have seen that one could get insight into the initial rotational velocity of our target stars by using both the X(H)--L and HR diagrams. In Fig.\ \ref{vsini_wr12} we constrain the current projected rotational velocities of SMC-WR1 and SMC-WR2 from the shape of narrow absorption/emission lines. One clearly sees that values much larger than 50 km s$^{-1}$\ are excluded, the line profiles becoming much too broad. These values are lower limits on the true rotational velocities. However, it is unlikely that we see both stars pole--on, so that SMC-WR1 and SMC-WR2 probably have equatorial velocities $\lesssim$ 100 km s$^{-1}$. Normal evolutionary tracks with rotation predict equatorial velocities between 0 and 200 km s$^{-1}$\ for 40--60 \ifmmode M_{\odot} \else M$_{\odot}$\fi\ stars in the WR phase \citep{mm05}. On the other hand, the homogeneous tracks of \citet{mm07} predict velocities between 100 and 400 km s$^{-1}$\ in the case of isotropic winds, and around 600 km s$^{-1}$\ for anisotropic winds. The latter predictions (anisotropic winds) are supposed to be more realistic. Since we observe rather modest rotational velocities, and given the strong evidence for homogeneous evolution -- and thus fast initial rotation -- at least for star SMC-WR1, one might argue that braking was more efficient than predicted for these stars.
In the context of GRB formation, fast rotation is the key ingredient (see \citet{wb06} for a review). The other important condition is that the star explodes while it has lost all its hydrogen envelope. Here, we have shown that SMC-WR1 has most likely evolved with a high rotation rate until recently. The fact that it still contains a large H mass fraction might argue against its potential explosion as a GRB. However, according to the model predictions of \citet{mm07}, a star following homogeneous evolution will end its life as a H--free WR star (of the class WC or WO). \citet{hirschi05} argue that GRBs can exist at SMC metallicities and that they should have WO stars as progenitors. Similarly, \citet{yoon06} predict a metallicity threshold of about Z=0.004 for the formation of long--soft GRBs (the exact value depending mainly on the adopted mass loss rates). In their scenario, stars ending their lives as GRBs follow quasi chemically homogeneous evolution. The properties of SMC-WR1 are very reminiscent of this scenario. However, the questions raised in the the previous paragraph prevent us from concluding firmly that SMC-WR1 is on its way to become a GRB progenitor.
\begin{figure}[]
\centering
\includegraphics[width=9cm]{vsini_wr12.eps}
\caption{Estimate of the current projected rotational velocity of stars SMC-WR1 and SMC-WR2. The black solid lines are the observed spectra of SMC-WR2 (top panels) and SMC-WR1 (bottom panel). Colored lines are synthetic spectra convolved with rotational profiles with different $V$ sin$i$. Values larger than 50 km s$^{-1}$\ are excluded. }\label{vsini_wr12}
\end{figure}
\section{Conclusion}
\label{s_conc}
We have conducted a detailed spectroscopic analysis of three WNh stars in the SMC. Using atmosphere models computed with the code CMFGEN, we have determined the stellar and wind parameters. The key results can be summarized as follows:
\begin{itemize}
\item[$\bullet$] the quantitative analysis confirms the spectral classification in the sense that all three stars still contain large amounts of hydrogen in their atmosphere. At the same time, they show clear signs of CNO processing, with C (and O for SMC-WR1) depletion, and N enrichment. SMC-WR1 is a hotter object than the SMC-WR2 and SMC-WR4, as suggested by its earlier spectral type.
\item[$\bullet$] SMC-WR4 can be explained by normal evolutionary tracks including rotation if its global metallicity is slightly above 1/5$^{th}$ solar. In that case, it is most likely a star with an initial mass of $40-50$ \ifmmode M_{\odot} \else M$_{\odot}$\fi\ in or very close to the core He burning phase.
\item[$\bullet$] SMC-WR2 and, more dramatically, SMC-WR1 cannot be accounted for by moderately rotating (i.e. 300 km s$^{-1}$) evolutionary tracks. Their position in both the HR and X(H)--L diagrams can be reproduced only if the stars rotate initially very fast ($>$ 500 km s$^{-1}$). In the case of SMC-WR1, only homogeneous evolution can account for the position of the star to the left of the main sequence and the high H mass fraction in its atmosphere.
\item[$\bullet$] the clumping factor $f_{\infty}$ of star SMC-WR4 is quantitatively constrained to be 0.15$\pm$0.05. Within this uncertainty, no difference is found with Galactic WR stars, confirming the dynamical results of paper I.
\end{itemize}
The present results are based on a limited number of stars. In a subsequent publication, we will present a study of the remaining Wolf-Rayet population of the SMC. This will provide a wider view of the stellar properties of these objects. In particular, we will be able to estimate the fraction of stars potentially following evolution governed by fast rotation. This will be useful in the context of Gamma-Ray Burst formation, thought to be due to very rapidly rotating WR stars.
A more systematic study of the clumping properties of SMC WNh stars is also required to confirm that clumping does not seem to depend on metallicity. This is crucial to understand the physics of inhomogeneity formation in massive--stars winds and, more generally, the mechanisms of mass loss. Our future study will tackle these questions too.
\begin{acknowledgements}
We thank the referee, Paul Crowther, for useful comments which helped to improve the spectroscopic analysis and for sharing near-infrared spectra. We thank Stan Owocki for illuminating discussions about radiative instabilities. FM thanks the CINES for allocation of computing time. AFJM thanks NSERC (Canada) and FQRNT (Quebec) for financial aid. JCB acknowledges support from the French National Research Agency (ANR) through program number ANR-06-BLAN-0105.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,758 |
\section{Introduction}
In cooperative multi-agent reinforcement learning, simultaneously-acting agents must learn to work together to achieve a shared set of goals. A straightforward approach is for each agent to optimise a global objective using single-agent reinforcement learning (RL) methods such as Q-Learning \cite{watkins1992q} or policy gradients \cite{williams1992simple, sutton2000policy}. Unfortunately this suffers various problems in general.
First, from the perspective of any one agent, the environment is non-stationary. This is because as other agents learn, their policies change, creating a partially unobservable influence on the effect of the first agent's actions. This issue has recently been addressed by a variety of deep RL methods, for instance with centralised training of ultimately decentralised policies \cite{lowe2017multi, foerster2018counterfactual}. However, this requires that the centralised critic or critics have access to the actions and observations of all agents during training, which may not always be possible.
A second problem is coordination; agents must learn how to choose actions coherently, even in environments in which many optimal equilibria exist \cite{matignon2012independent}. Whilst particularly challenging when agents are completely independent, such problems can be made more feasible if a form of communication is allowed \cite{vlassis2007concise}. Nevertheless, it is difficult for agents to learn how to communicate relevant information effectively to solve coordination problems, with most approaches relying on further helpful properties such as a differentiable communication channel \cite{sukhbaatar2016learning, foerster2016learning} and/or a model of the world's dynamics \cite{mordatch2018emergence}.
Third, multi-agent methods scale poorly -- the effective state space grows exponentially with the number of agents. Learning a centralised value function therefore suffers the curse of dimensionality, whilst the alternative of decentralised learning often appears inadequate for addressing non-stationarity. Optimising a global objective also becomes challenging at scale, as it becomes difficult to assign credit to each agent \cite{wolpert2002optimal, chang2004all}.
A clue for meeting this myriad of challenges may lie in the way in which human and animal societies are hierarchically structured. In particular, even in broadly cooperative groups, it is frequently the case that different individuals agree to be \emph{assigned} different objectives which they work towards for the benefit of the collective. For example, members of a company typically have different roles and responsibilities. They will likely report to managers who define their objectives, and they may in turn be able to set objectives to more junior members. At the highest level, the CEO is responsible for the company's overall performance.
Inspired by this idea, we propose an approach for the multi-agent domain, which organises many, simultaneously acting agents into a managerial hierarchy. Whilst typically all agents in a cooperative task seek to optimise a shared reward, in Feudal Multi-agent Hierarchies (FMH) we instead only expose the highest-level manager to this `task' reward. The manager must therefore learn to solve the principal-agent problem \cite{jensen1976theory} of communicating subgoals, which define a reward function, to the worker agents under its control. Workers learn to satisfy these subgoals by taking actions in the world and/or by setting their own subgoals for workers immediately below them in the hierarchy.
FMH allows for a diversity of rewards. This can provide individual agents with a rich learning signal, but necessarily implies that interactions between agents will not in general be fully cooperative. However, the intent of our design is to achieve collective behaviours which are apparently cooperative, from the perspective of an outside observer viewing performance on the task objective.
Our idea is a development of a single-agent method for hierarchical RL known as feudal RL \cite{dayan1993feudal,vezhnevets2017feudal}, which involves a `manager' agent defining subgoals for a `worker' agent in order to achieve temporal abstraction. Feudal RL allows for the division of tasks into a series of subtasks, but has largely been investigated with only one worker per manager acting at any one time. By introducing a feudal hierarchy with multiple agents acting simultaneously in FMH, we not only divide tasks over time but also across worker agents. Furthermore, we embrace the full multi-agent setting, in which observations are not in general shared across agents.
We outline a method to implement FMH for concurrently acting, communicating agents trained in a decentralised fashion. Our approach pre-specifies appropriate positions in the hierarchy as well as a mapping from the manager's choice of communication to the workers' reward functions. We show how to facilitate learning of our deep RL agents within FMH through suitable pretraining and repeated communication to encourage temporally extended goal-setting.
We conduct a range of experiments that highlight the advantages of FMH. In particular, we show its ability to address non-stationarity during training, as managerial reward renders the behaviour of workers more predictable. We also demonstrate FMH's ability to scale to many agents and many possible goals, as well as to enable effective coordination amongst workers. It performs substantially better than both decentralised and centralised approaches.
\section{Background}
\subsection{Markov Decision Processes}
Single-agent RL can be formalised in terms of Markov Decision Processes, which consist of sets of states $\mathcal{S}$ and actions $\mathcal{A}$, a reward function $r: \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$ and a transition function $\mathcal{T} : \mathcal{S} \times \mathcal{A} \rightarrow \Delta{(\mathcal{S})}$, where $\Delta{(\mathcal{S})}$ denotes the set of discrete probability distributions over $\mathcal{S}$. Agents act according to a stochastic policy $\pi : \mathcal{S} \times \mathcal{A} \rightarrow [0,1]$. One popular objective is to maximise the discounted expected future reward, defined as $\mathbb{E}_{\pi}[\sum_{t=0}^{\infty} \gamma^t r(s_t, a_t)]$ where the expectation is over the sequence of states and actions which result from policy $\pi$, starting from an initial state distribution $\rho_0: \mathcal{S} \rightarrow [0,1]$. Here $\gamma \in [0,1)$ is a discount factor and $t$ is the time step. This objective can be equivalently expressed as $\mathbb{E}_{s \sim \rho^\pi, a \sim \pi}[r(s,a)]$, where $\rho^\pi$ is the discounted state distribution induced by policy $\pi$ starting from $\rho_0$.
\subsection{Deterministic Policy Gradient Algorithms}
Deterministic policy gradients (DPG) is a frequently used single-agent algorithm for continuous control using model-free RL \cite{silver2014deterministic}. It uses deterministic policies $\mu_\theta : \mathcal{S} \rightarrow \mathcal{A}$, whose parameters $\theta$ are adjusted in an off-policy fashion using an exploratory behavioural policy to perform stochastic gradient ascent on an objective $J(\theta)= \mathbb{E}_{s \sim \rho^\mu, a \sim \mu_\theta}[r(s,a)]$.
We can write the gradient of $J(\theta)$ as:
\begin{equation}
\nabla_\theta J(\theta)= \mathbb{E}_{s \sim \rho^\mu}[ \nabla_\theta \mu_{\theta}(s) \nabla_a Q^\mu (s,a)|_{a=\mu_{\theta}(s)}].
\end{equation}
Deep Deterministic Policy Gradients (DDPG) \cite{lillicrap2015continuous} is a variant with policy $\mu$ and critic $Q^\mu$ being represented via deep neural networks. Like DQN \cite{mnih2015human}, DDPG stores experienced transitions in a replay buffer and replays them during training.
\subsection{Markov Games}
A partially observable Markov game (POMG) \cite{littman1994markov, hu1998multiagent}
for $N$ agents is defined by a set of states $\mathcal{S}$, and sets of actions $\mathcal{A}_1,...,\mathcal{A}_N$ and observations $\mathcal{O}_1,...,\mathcal{O}_N$ for each agent. In general, the stochastic policy of agent $i$ may depend on the set of action-observation histories $H_i \equiv (\mathcal{O}_i \times \mathcal{A}_i)^*$ such that $\pi_i : \mathcal{H}_i \times \mathcal{A}_i \rightarrow [0,1]$. In this work we restrict ourselves to history-independent stochastic policies $\pi_i : \mathcal{O}_i \times \mathcal{A}_i \rightarrow [0,1]$. The next state is generated according to the state transition function $\mathcal{T} : \mathcal{S} \times \mathcal{A}_1 \times ... \times \mathcal{A}_n \rightarrow \Delta{(\mathcal{S})}$. Each agent $i$ obtains deterministic rewards defined as $r_{i} : \mathcal{S} \times \mathcal{A}_1 \times ... \times \mathcal{A}_n \rightarrow \mathbb{R}$ and receives a private observation $\mathbf{o}_i : \mathcal{S} \rightarrow \mathcal{O}_i$. There is an initial state distribution $\rho_0 : \mathcal{S} \rightarrow [0,1]$ and each agent aims to maximise its own discounted sum of future rewards.
\subsection{Centralised and Decentralised Training}
In multi-agent RL, agents can be trained in a centralised or decentralised fashion. In decentralised training, agents have access only to local information: their own action and observation histories during training \cite{tan1993multi}. Agents are often trained using single-agent methods for RL, such as Q-Learning or DDPG.
In centralised training, the action and observation histories of all agents are used, effectively reducing the multi-agent problem to a single-agent problem. Although it may appear restrictive for agents to require access to this full information, this approach has generated recent interest due to the potential for centralised training of ultimately decentralised policies \cite{lowe2017multi,foerster2018counterfactual}. For example, in simulation one can train a Q-function, known as a critic, which exploits the action and observation histories of all agents to aid the training of local policies for each one. These policies can then be deployed in multi-agent systems in the real world, where centralisation may be infeasible.
\subsection{Multi-Agent Deep Determinstic Policy Gradients}
Multi-agent deep deterministic policy gradients (MADDPG) \cite{lowe2017multi} is an algorithm for centralised training and decentralised control of multi-agent systems. It uses deterministic polices, as in DDPG, which condition only on each agent's local observations and actions. MADDPG handles the non-stationarity associated with the simultaneous adaptation of all the agents by introducing a separate centralised critic for each agent. Lowe et al., trained history-independent feedforward networks on a range of mixed cooperative-competitive environments, significantly improving upon agents trained independently with DDPG.
\section{Methods}
We propose FMH, a framework for multi-agent RL which addresses the three major issues outlined in the introduction: non-stationarity, scalability and coordination. In order to increase the generality of our approach, we do so without requiring a number of potentially helpful features: access to the observations of all agents (as in centralised training), a differentiable model of the world's dynamics or a differentiable communication channel between agents.
\subsection{Hierarchies}
We begin by defining the type of hierarchical structure we create for the agents. In its most straightforward setting, this hierarchy corresponds to a rooted directed tree, with the highest level manager as the root node and each worker reporting to only a single manager. Our experiments use this structure, although, for simplicity, using just two-level hierarchies. However, we also note the possibility, in more exotic circumstances, of considering more general directed acyclic graphs in which a single worker reports to multiple managers, and allowing for more than one manager at the highest level of the hierarchy. Acyclicity prevents potentially disastrous feedback cycles for the reward.
Appropriate assignment of agents to positions in a hierarchy should depend on the varied observation and action capabilities of the agents. Agents exposed to key information associated with global task performance are likely to be suited to the role of manager, whereas agents with more narrow information, but which can act to achieve subgoals are more naturally suited to being workers. As we show in our experiments, identifying these differences can often make assigning agents straightforward. Here, we specify positions directly, to focus on demonstrating the resulting effectiveness of the hierarchy. However, extensions in which the hierarchy is itself learned would be interesting.
\subsection{Goal-Setting}
In FMH, managers set goals for workers by defining their rewards. To arrange this, managers communicate a special kind of message to workers that \emph{defines the workers' reward functions} according to a specified mapping. For example, if there are three objects in a room, we might allow the manager to select from three different messages, each defining a worker reward function proportional to the negative distance from a corresponding object. Our experiments explore variants of this scheme, imposing the constraint that the structure of the reward function remain fixed whilst the target is allowed to change. In this context and task, messages therefore correspond to `goals' requiring approach to different objects. The manager must learn to communicate these goals judiciously in order to solve complex tasks. In turn, the workers must learn how to act in the light of the resulting managerial reward, in addition to immediate rewards (or costs) they might also experience.
\begin{figure}[ht]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{FMH_diagram.pdf}}
\caption{An example of a worker-computed Feudal Multiagent Hierarchy with one manager agent and two worker agents. Worker rewards are goal-dependent and computed locally, the manager's reward is provided by the environment.}
\setlength\intextsep{0pt}
\label{FMH_diagram}
\end{center}
\end{figure}
Since a communicated goal is simply another type of action for the manager, our approach is consistent with the formalism of a (partially-observable) Markov game, for which the reward for agent $i$ is defined as $r_{i} : \mathcal{S} \times \mathcal{A}_1 \times ... \times \mathcal{A}_n \rightarrow \mathbb{R}$.
However, the worker's reward need not be treated as part of the environment. For instance, each worker agent $i$ could compute its own reward locally $r_{i} : \mathcal{O}_i \times \mathcal{A}_i \rightarrow \mathbb{R}$, where $\mathcal{O}_i$ includes the observed goal from the manager. We illustrate this `worker-computed' interpretation in Fig.~\ref{FMH_diagram}.
\subsection{Pretraining and Temporally Extended Subgoals}
We next consider the issue of non-stationarity. This frequently arises in multi-agent RL because the policies of other agents may change in unpredictable ways as they learn. By contrast, in FMH we allow manager agents to determine the reward functions of workers, compelling the workers towards more predictable behaviour from the perspective of the manager. However, the same issue applies at the starting point of learning for workers: they will not yet have learned how to satisfy the goals. We would therefore expect a manager to underestimate the value of the subgoals it selects early in training, potentially leading it sub-optimally to discard subgoals which are harder for the worker to learn.
Thus, it would be beneficial for worker agents already to have learned to fulfill managerial subgoals. We address this issue practically in two steps. First, we introduce a bottom-up `pretraining' procedure, in which we initially train the workers before training the manager. Although the manager is not trained during this period, it still acts in the multi-agent environment and sets subgoals for the worker agents. As subgoals are initially of (approximately) equal value, the manager will explore them uniformly. If the set of possible subgoals is sufficiently compact, this will enable workers to gain experience of each potential subgoal.
This pretraining does not completely solve the non-stationarity problem. This is because the untrained manager will, with high probability, vacillate between subgoals, preventing the workers under its command from having any reasonable hope of extracting reward. We therefore want managers not only to try out a variety of subgoals but also to \emph{commit} to them long enough that they have any hope of being at least partially achieved. Thus, the second component of the solution is a communication-repeat heuristic for the manager such that goal-setting is temporally extended. We demonstrate its effectiveness in our experiments.
\subsection{Coordination}
Multi-agent problems may have many equilibria, and good ones can only be achieved through effective coordination of agent behaviour. In FMH, the responsibility of coordination is incumbent upon the manager, which exerts control over its workers through its provision of reward. We show in the simplified setting of a matrix game, how a manager may coordinate the actions of its workers (see Suppl.\ Mat.\ \ref{coordination-game}).
\subsection{FMH-DDPG}
FMH provides a framework for rewarding agents in multi-agent domains, and can work with many different RL algorithms. In our experiments, we chose to apply FMH in combination with the single-agent algorithm DDPG, trained in a fully decentralised fashion. As we do not experiment with combining FMH with any other algorithm, we frequently refer to FMH-DDPG simply as FMH.
\subsection{Parameter Sharing}
We apply our method to scenarios in which a large number of agents have identical properties. For convenience, when training using a decentralised algorithm (FMH-DDPG or vanilla DDPG) we implement parameter sharing among identical agents, in order to train them efficiently. We only add experience from a single agent (among those sharing parameters) into the shared replay buffer, and carry out a single set of updates. We find this gives very similar results to training without parameter sharing, in which each agent is updated using its own experiences stored in its own replay buffer (see Suppl.\ Mat.\ \ref{param-share}).
\section{Experiments and Results}
We used the multi-agent particle environment\footnote{https://github.com/openai/multiagent-particle-envs} as a framework for conducting experiments to test the potential of our method to address the issues of non-stationarity, scalability and coordination, comparing against MADDPG and DDPG. We will release code for both the model and the environments after the reviewing process ends.
The RL agents have both an actor and a critic, each corresponding to a feedforward neural network. We give a detailed summary of all hyperparameters used in Suppl.\ Mat.\ \ref{params}.
\subsection{Cooperative Communication}
\begin{figure*}[!ht]
\begin{center}
\centerline{\includegraphics[width=\textwidth]{Figure_2_final.pdf}}
\caption{Cooperative Communication with 1 listener and 12 landmarks. (a) The speaker (grey circle) sees the colour of the listener (green circle) , which indicates the target landmark (green square). It communicates a message to the listener at every time step. Here there are 12 landmarks and the agent trained using FMH has successfully navigated to the correct landmark. (b) FMH substantially outperforms MADDPG and DDPG. The dotted green line indicates the end of pretraining for FMH. (c) FMH worker reward and the probability of the manager correctly assigning the correct target to the worker. The manager learns to assign the target correctly with probability 1. }
\label{speaker-listener}
\end{center}
\vskip -0.2in
\end{figure*}
We first experiment with an environment implemented in \citet{lowe2017multi} called `Cooperative Communication' (Figure~\ref{speaker-listener}a), in which there are two cooperative agents, one called the `speaker' and the other called the `listener', placed in an environment with many landmarks of differing colours. On each episode, the listener must navigate to a randomly selected landmark; and in the original problem, both listener and speaker obtain reward proportional to the negative distance\footnote{The original implementation used the negative square distance; we found this slightly worse for all algorithms.} of the listener from the correct target. However, whilst the listener observes its relative position from each of the differently coloured landmarks, it does not know to which landmark it must navigate. Instead, the colour of the target landmark can be seen by the speaker, which cannot observe the listener and is unable to move. The speaker can however communicate to the listener at every time step, and so successful performance on the task corresponds to the speaker enabling the listener to reach the correct target. We also note that, although reward is provided during the episode, it is used only for training agents and not directly observed, which means that agents cannot simply learn to follow the gradient of reward.
Although this task seems simple, it is challenging for many RL methods. \citet{lowe2017multi} trained, in a decentralised fashion, a variety of single-agent algorithms, including DDPG, DQN and trust-region policy optimisation \cite{schulman2015trust} on a version of this problem with three landmarks and demonstrated that they all perform poorly on this task. Of these methods, DDPG reached the highest level of performance and so we use DDPG as the strongest commonly used baseline for the decentralised approach. We also compare our results to MADDPG, which combines DDPG with centralised training. MADDPG was found to perform strongly on Cooperative Communication with three landmarks, far exceeding the performance of DDPG.
For our method, FMH, we also utilised DDPG, but reverted to the more generalizable decentralized training that was previously ineffective. Crucially, we assigned the speaker the role of manager and the listener the role of worker. The speaker can therefore communicate messages which correspond to the subgoals of the different coloured landmarks. The listener is not therefore rewarded for going to the correct target but is instead rewarded proportional to the negative distance from the speaker-assigned target. The speaker itself is rewarded according to the original problem definition, the negative distance of the listener from the true target.
We investigated in detail a version of Cooperative Communication with 12 possible landmarks (Figure~\ref{speaker-listener}a). FMH performed significantly better than both MADDPG and DDPG (Figure~\ref{speaker-listener}b) over a training period of 100 epochs (each epoch corresponds to 1000 episodes). For FMH, we pretrained the worker for 10 epochs and enforced extended communication over 8 time steps (each episode is 25 time steps).
In Figure~\ref{speaker-listener}c, the left axis shows the reward received by the FMH worker (listener) over training. This increased during pretraining and again immediately after pretraining concludes. Managerial learning after pretraining resulted in decreased entropy of communication over the duration of an episode (see Suppl.\ Mat.\ \ref{entropy}), allowing the worker to optimise the managerial objective more effectively. This in turn enabled the manager to assign goals correctly, with the rise in the probability of correct assignment occurring shortly thereafter (right axis), then reaching perfection.
Our results show how FMH resolves the issue of non-stationarity. Initially, workers are able to achieve reward by learning to optimise managerial objectives, even whilst the manager itself is not competent. This learning elicits robust behaviour from the worker, conditioned on managerial communication, which makes workers more predictable from the persepective of the manager. This then enables the manager to achieve competency -- learning to assign goals so as to solve the overall task.
\begin{table*}[t]
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{cc|rrrr|rrr}
\hline
\multicolumn{2}{c|}{Number of} & \multicolumn{4}{c|}{Final Reward} & \multicolumn{3}{c}{Epochs until Convergence}\\
Listeners & Landmarks & FMH & MADDPG & DDPG &CoM&FMH & MADDPG & DDPG\\
\hline
$1$ &$3$ &$\mathbf{-6.63 \pm 0.05}$&$\mathbf{-6.58\pm0.03}$&$-14.26\pm0.07$&$-17.28$&$56$&$24$&$55$ \\
$1$ & $6$ &$-6.91\pm0.07$&$\mathbf{-6.69\pm0.06}$&$-18.10\pm0.07$&$-18.95$&$57$&$66$&$42$ \\
$1$ &$12$ &$\mathbf{-7.79\pm0.06}$&$-15.96\pm0.09$&$-19.32\pm0.11$&$-19.56$ &$-$&$-$&$36$ \\
$3 $ &$6$ &$\mathbf{-7.10\pm0.04}$&$-11.13\pm0.03$&$-18.90\pm0.05$&$-18.95$&$77$&$-$&$50$ \\
$5$ & $6$ &$\mathbf{-7.17\pm0.03}$&$-18.47\pm0.04$&$-19.73\pm0.06$ &$-18.95$&$79$ &$75$ &$53$ \\
$10$ &$6$&$\mathbf{-8.96\pm0.03} $&$-19.80\pm 0.06$ &$-21.19\pm0.03$&$-18.95$&$-$&59&32 \\
\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\caption{Performance of FMH, MADDPG and DDPG for versions of Cooperative Communication with different numbers of listeners and landmarks. Final reward is determined by training for 100 epochs and evaluating the mean reward per episode in the final epoch. We indicate no convergence with a $-$ symbol. For further details see Sup. Mat.~\ref{table-scaling}.}
\label{tab:agent-scaling}
\end{table*}
Our implementation of FMH used both pretraining and extended goal-setting with a communication repeat heuristic. In Suppl.\ Mat.\ \ref{pretrain}, we show that pretraining the worker improved performance on this task, although even without pretraining FMH still performed better than MADDPG and DDPG. The introduction of extended communication is however more critical. In Figure~\ref{comm-extend} we show the performance of FMH with goal-setting over various number of time steps (and fixed pretraining period of 10 epochs). When there were no communication repeats (Comm 1), performance was similar to MADDPG, but introducing even a single repeat greatly improved performance. By contrast, introducing communication repeats to MADDPG did not improve performance (see Suppl.\ Mat.\ ~\ref{comm-extend-maddpg}).
\begin{figure}[ht]
\begin{minipage}[c]{0.32\textwidth
\centerline{\includegraphics[width=\columnwidth]{extended_comm_FMH_3.pdf}}
\end{minipage}
\begin{minipage}[c]{0.1533\textwidth}
\caption{\\Communication sent by the manager is repeated for extended goal-setting over various numbers of time steps.}
\label{comm-extend
\end{minipage}
\end{figure}
\subsubsection{Scaling to many agents}
We next scaled Cooperative Communication to include many listener agents. At the beginning of an episode each listener is randomly assigned a target out of all the possible landmarks and the colour of each listener's target is observed by the speaker. The speaker communicates a single message to each listener at every time step. To allow for easy comparison with versions of Cooperative Communication with only one listener, we normalised the reward by the number of listeners. As discussed in the methods, we shared parameters across listeners for FMH and DDPG.
In Table~\ref{tab:agent-scaling} we show the performance of FMH, MADDPG and DDPG for variants of Cooperative Communication with different numbers of listener agents and landmarks. Consider the version with 6 landmarks, which we found that MADDPG could solve with a single listener within 100 epochs (unlike for 12 landmarks). On increasing the number of listeners up to a maximum of 10, we found that FMH scales much better than MADDPG; FMH was able to learn an effective policy with 5 listener agents whereas MADDPG could not. Further, FMH even scaled to 10 listeners, although it did not converge over the 100 epochs.
To aid interpretation of the reward values in Table~\ref{tab:agent-scaling} we also compare performance to a policy which simply moves to the centroid of the landmarks. This `Centre of Mass' (CoM) agent was trained using MADDPG until convergence on a synthetic task in which reaching the centroid is maximally rewarded, and then evaluated on the true version of the task. We find that both MADDPG and DDPG do not perform better than the CoM agent when there are 10 listeners and 6 landmarks.
In Figure~\ref{10-listeners} we show the final state achieved on an example episode of this version of the task, for agents trained using MADDPG and FMH. After training for 100 epochs, MADDPG listeners do not find the correct targets by the end of the episode whereas FMH listeners manage to do so.
\begin{figure}[ht]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{FMH_MADDPG_10_listeners_final_time_step.pdf}}
\caption{Scaling Cooperative Communication to 10 listeners with 6 landmarks - final time step on example episode.}
\setlength\intextsep{0pt}
\label{10-listeners}
\end{center}
\end{figure}
\begin{figure*}[!ht]
\begin{center}
\centerline{\includegraphics[width=\textwidth]{Figure_Cooperative_Coordination.pdf}}
\caption{Cooperative Coordination (a) Three listeners (light grey agents) must move to cover the green landmarks whilst ignoring the blue landmarks. However, only the speaker (dark grey agent) can see the landmarks' colours; it communicates with the listeners at every time step. In this example, FMH agents have successfully coordinated to cover the three correct targets. (b) FMH performs significantly better than MADDPG and DDPG. The dotted green line indicates the end of pretraining for FMH. (c) Agents trained using FMH cover on average more targets, by the end of the episode, than MADDPG and DDPG (d) Agents trained using FMH avoid collisions more effectively than MADDPG and DDPG over the duration of an episode.}
\label{coop-coord}
\end{center}
\vskip -0.2in
\end{figure*}
\subsection{Cooperative Coordination}
We then designed a task to test coordination called `Cooperative Coordination' (Figure~\ref{coop-coord}a). In this task, there are 6 landmarks. At the beginning of each episode 3 landmarks are randomly selected to be green targets and 3 blue decoys. A team of 3 agents must navigate to cover the green targets whilst ignoring the blue decoys, but they are unable to see the colours of the landmarks. A fourth agent, the speaker, can see the colours of the landmarks and can send messages to the other agents (but cannot move). All agents can see each others' positions and velocities, are large in size and face penalties if they collide with each other. The task shares aspects with `Cooperative Navigation' from \cite{lowe2017multi} but is considerably more complex due to the greater number of potential targets and the hidden information.
We apply FMH to this problem, assigning the speaker agent the role of manager. One consideration is whether the manager, the worker, or both should receive the negative penalties due to worker collisions. Here we focus on the case in which the manager only concerns itself with the `task' reward function. Penalties associated with collisions are therefore experienced only by the workers themselves, which seek to avoid these whilst still optimising the managerial objective.
In Figure~\ref{coop-coord}b we compare the performance of FMH to MADDPG and DDPG. As with Cooperative Communication, FMH does considerably better than both after training for 150 epochs. This is further demonstrated when we evaluate the trained policies over a period of 10 epochs: Figure~\ref{coop-coord}c shows the mean number of targets covered by the final time step of the episode, for which FMH more than doubles MADDPG. Figure~\ref{coop-coord}d shows the mean number of collisions (which are negatively rewarded) during an episode. FMH collides very rarely whereas MADDPG and DDPG collide over 4 times more frequently.
We also implement a version of Cooperative Coordination in which the manager is responsible not only for coordinating its workers but must also navigate to targets itself. We find that it can learn to do both roles effectively (see Suppl.\ Mat.\ \ref{manager-moves}).
\subsubsection{Exploiting Diversity}
One role of a manager is to use the diversity it has available in its workers to solve tasks more effectively. We tested this in a version of Cooperative Coordination in which one of the listener agents was lighter than the other two and so could reach farther targets more quickly.
We trained FMH (without parameter sharing due to the diversity) on this task and then evaluated the trained policies on a `Two-Near, One-Far' (TNOF) version of the task in which one target is far away and the remaining two are close (see Suppl.\ Mat.\ \ref{diversity}). We did this to investigate whether FMH, which was trained on the general problem, had learned the specific strategy of assigning the farthest target to the fastest agent. We found this this to be true 100 percent of the time (evaluating over 10 epochs); we illustrate this behaviour in Figure~\ref{fig-diversity}.
\begin{figure}[ht]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{Figure_Coop_Coord_Diversity.pdf}}
\caption{FMH solves the TNOF task (example episode). Left: Agents are initialised at the bottom of the environment, two targets are close by, and one far away. Right: By the end of the episode, the faster (red) agent covers the farther target on the right, despite starting on the left. }
\setlength\intextsep{0pt}
\label{fig-diversity}
\end{center}
\end{figure}
\section{Discussion}
We have shown how cooperative multi-agent problems can be solved efficiently by defining a hierarchy of agents. Our hierarchy was reward-based, with manager agents setting goals for workers, and with the highest level manager optimising the overall task objective. Our method, called FMH, was trained in a decentralised fashion and showed considerably better scaling and coordination than both centralised and decentralised methods that used a shared reward function.
Our work was partly inspired by the feudal RL architecture (FRL) \cite{dayan1993feudal}, a single-agent method for hierarchical RL \cite{barto2003recent} which was also developed in the context of deep RL by \citet{vezhnevets2017feudal}. In particular, FMH addresses the `too many chiefs' inefficiency inherent to FRL, namely that each manager only has a single worker under its control. A much wider range of management possibilities and benefits arises when multiple agents operate at the same time to achieve one or more tasks \cite{busoniu2008comprehensive}. We focused on the cooperative setting \cite{panait2005cooperative}; however, unlike the fully-cooperative setting, in which all agents optimise a shared reward function \cite{boutilier1996planning} our approach introduces a diversity of rewards, which can help with credit-assignment \cite{wolpert2002optimal, chang2004all} but also introduces elements of competition. This competition need not always be deleterious; for example, in some cases, an effective way of optimising the task objective might be to induce competition amongst workers, as in generative adversarial networks \cite{goodfellow2014generative}.
We followed FRL (though not all its successors; \cite{vezhnevets2017feudal} in isolating the workers from much of the true environmental reward, making them focus on their own narrower tasks. Such reward hiding was not complete -- we considered individualised or internalised costs from collisions that workers experienced directly, such that their reward was not purely managerially dependent. A more complete range of possibilities for creating and decomposing rewards between managers and workers when the objectives of the two are not perfectly aligned, could usefully be studied under the aegis of principal-agent problems \cite{jensen1976theory, laffont2009theory}.
Goal-setting in FMH was achieved by specifying a relationship between the chosen managerial communication and the resulting reward function. The communication was also incorporated into the observational state of the worker; however, the alternative possibility of a goal-embedding would be worth exploring \cite{vezhnevets2017feudal}. We also specified goals directly in the observational space of the workers, whereas Vezhnevets et al. specified goals in a learned hidden representation. This would likely be of particular value for problems in which defining a set of subgoals is challenging, such as those which require learning directly from pixels. More generally, work on the way that agents can learn to construct and agree upon a language for goal-setting would be most important.
To train our multi-agent systems we leveraged recent advances in the field of deep RL; in particular the algorithm DDPG \cite{lillicrap2015continuous}, which can learn in a decentralised fashion. Straightforward application of this algorithm has been shown to achieve some success in the multi-agent domain \cite{gupta2017cooperative} but also shown it to be insufficient in handling more complex multi-agent problems \cite{lowe2017multi}. By introducing a managerial hierarchy, we showed that FMH-DDPG has the potential to greatly facilitate multi-agent learning whilst still retaining the advantage of decentralised training. Our proposed approach could also be combined with centralised methods, and this would be worth further exploration.
Other work in multi-agent RL has also benefitted from ideas from hierarchical RL. The MAXQ algorithm \cite{dietterich2000hierarchical} has been used to train homogenous agents \cite{makar2001hierarchical}, allowing them to coordinate by communicating subtasks rather than primitive actions, an idea recently re-explored in the context of deep RL \cite{tang2018hierarchical}. A meta-controller which structures communication between agent pairs in order to achieve coordination has also been proposed \cite{kumar2017federated} (and could naturally be hybridized with FMH), as well as the use of master-slave architecture which merges the actions of a centralised master agent with those of decentralised slave agents \cite{kong2017revisiting}. Taken together, these methods represent interesting alternatives for invoking hierarchy which are unlike our primarily reward-based approach.
There are a number of additional directions for future work. First, the hierarchies used were simple in structure and specified in advance, based on our knowledge of the various information and action capabilities of the agents. It would be interesting to develop mechanisms for the formation of complex hierarchies. Second, in settings in which workers can acquire relevant task information, it would be worthwhile investigating how a manager might incentivise them to provide this. Third, it would be interesting to consider how a manager might learn to allocate resources, such as money, computation or communication bandwidth to enable efficient group behaviour. Finally, we did not explore how managers should train or supervise the workers beneath them, such as through reward shaping \cite{ng1999policy}. Such an approach might benefit from recurrent networks, which could enable managers to use the history of worker performance to better guide their learning.
\section{Acknowledgements}
We would like to thank Heishiro Kanagawa, Jorge A. Menendez and Danijar Hafner for helpful comments on a draft version of the manuscript. Sanjeevan Ahilan received funding from the Gatsby Computational Neuroscience Unit and the Medical Research Council. Peter Dayan received funding from the Max Planck Society.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,812 |
\section{Introduction}
\label{introduction}
The main results of this work are lower bounds for energy functionals of mappings from real, complex and quaternionic projective spaces to Riemannian manifolds and characterizations of mappings which minimize energy in these results. For real projective space, we will prove:
\begin{theorem}
\label{rpn_thm}
Let $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ be the $n$-dimensional real projective space with its canonical (constant curvature) metric $g_{0}$, with $n \geq 2$. Let $F: (\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0}) \rightarrow (M^{m},g)$ be a Lipschitz mapping to a Riemannian manifold $(M,g)$. Let $L^{\star}$ be the infimum of the lengths of paths in the free homotopy class of $F_{*}(\gamma)$, where $\gamma$ represents the non-trivial class in $\pi_{1}(\mathbb R} \newcommand{\T}{\mathbb T P^{n})$, and let $E_{p}(F)$ be the $p$-energy of $F$, as in Definition \ref{p_energy_def} below. For all $p \geq 1$,
\begin{equation}
\label{rpn_thm_eqn}
\displaystyle E_{p}(F) \geq \frac{\sigma(n)}{2}\left(\frac{2\sqrt{n}}{\pi}L^{\star}\right)^{p}, \medskip
\end{equation}
where $\sigma(n)$ is the volume of the unit $n$-sphere. \\
If $p > 1$ and equality holds for $p$, then $F$ is a homothety onto a totally geodesic submanifold of $(M,g)$ and equality holds for all $p \geq 1$. If $F$ is a smooth immersion, equality for $p=1$ also implies these conditions.
\end{theorem}
This implies in particular that the identity mapping of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ minimizes $p$-energy in its homotopy class for all $p \geq 1$. \\
The case $p=2$ of Theorem \ref{rpn_thm} was proven by Croke in \cite{Cr1}. The $2$-energy, referred to simply as the energy of a mapping, is the classical energy functional of mappings of Riemannian manifolds and a fundamental invariant in the theory of harmonic maps. It is a generalization of the energy of a path in a Riemannian manifold and the Dirichlet integral of a real-valued function. The fact that the identity mapping of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ minimizes energy in its homotopy class, which was first established by Croke in this result, is different from the round sphere $(S^{n},g_{0})$, $n \geq 3$: in that case, conformal dilations give energy-decreasing deformations of the identity mapping. In fact, they give 1-parameter families of mappings for which the infimum of the energy is $0$. More generally, work of White \cite{Wh1} implies that in any closed Riemannian manifold $(M,g)$ with $\pi_{1}(M) = 0$, $\pi_{2}(M) = 0$, the identity mapping is homotopic to mappings with arbitrarily small energy. Croke also noted in \cite{Cr1} that the results of Smith \cite{Sm1} imply there are metrics arbitrarily close to the canonical metric on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$, obtained by conformal deformations, for which the identity mapping is not even a stable critical point of the energy functional. \\
Croke observed in \cite{Cr1} that his argument could be adapted to prove that the identity mapping of complex projective space $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ with its canonical metric $g_{0}$ minimizes $2$-energy in its homotopy class, but that this was already known because $(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, g_{0})$ is a K\"ahler manifold -- in fact, it is a result of Lichnerowicz \cite{Li1} that any holomorphic mapping of compact K\"ahler manifolds minimizes energy in its homotopy class. \\
A natural extension of Theorem \ref{rpn_thm} to complex projective space would give lower bounds for the energy of mappings $F: (\mathbb C} \newcommand{\N}{\mathbb N P^{N}, g_{0}) \rightarrow (M^{m},g)$ to Riemannian manifolds in terms of the infimum $A^{\star}$ of the areas of mappings $f:S^{2} \rightarrow M$ which represent the homotopy or homology class of $F_{*}(\mathbb C} \newcommand{\N}{\mathbb N P^{1})$. However unlike the strong characterization of equality in Theorem \ref{rpn_thm}, basic properties of the energy functional and K\"ahler manifolds imply that in any such optimal result, the equality case must be broad enough to include any holomorphic mapping from $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ to a compact K\"ahler manifold $(X,h)$ -- we will discuss this at the beginning of Section \ref{complexes} below. Also, although conformal deformations of the canonical metric give Riemannian metrics on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ for which the identity mapping is not a stable critical point of the energy functional, as with $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$, on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ the result of Lichnerowicz cited above also gives an infinite-dimensional family of metrics, obtained by K\"ahler deformations of the canonical metric, for which the identity is energy-minimizing in its homotopy class. \\
In Theorem \ref{cpn_thm} we will state and prove lower bounds for the $p$-energy of Lipschitz mappings $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow (M,g)$, $p \geq 2$, where $(M,g)$ is a Riemannian manifold and $\widetilde{g}$ is any K\"ahler metric on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, which are analogous to Theorem \ref{rpn_thm} for $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$. This result generalizes an inequality for mappings of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ with its canonical metric in \cite[Theorem 3]{Cr1}. The full characterization of equality in Theorem \ref{cpn_thm} is somewhat technical and involves several partial results under weaker assumptions, but for the complex projective plane, this result implies that holomorphic mappings are essentially the only energy-minimizing maps of this type. We record this in the following:
\begin{theorem}
\label{kahler_thm}
Let $\widetilde{g}$ be a K\"ahler metric on $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ and $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{2}, \widetilde{g}) \rightarrow (M^{m}, g)$ a Lipschitz mapping to a Riemannian manifold $(M,g)$. Let $A^{\star}$ be the infimum of the areas of mappings $f:S^{2} \rightarrow (M,g)$ in the free homotopy class of $F_{*}(\mathbb C} \newcommand{\N}{\mathbb N P^{1})$, and let $E_{p}(F)$ be the $p$-energy of $F$. Then for all $p \geq 2$,
\begin{equation}
\label{kahler_thm_eqn}
\displaystyle E_{p}(F) \geq \frac{\pi^{2}}{2} \left( \frac{4}{\pi} A^{\star} \right)^{\frac{p}{2}}. \medskip
\end{equation}
Equality for $p = 2$ implies that $F$ is smooth and, for any neighborhood $\mathcal{V}$ of $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ on which $F$ is an immersion, $F^{*}g|_{\mathcal{V}}$ is a K\"ahler metric, $F(\mathcal{V})$ is minimal in $(M,g)$ and the second fundamental form of $F(\mathcal{V})$ in $(M,g)$ can be diagonalized by a unitary basis. If $p > 2$ and equality holds for $p$, then $F$ has constant energy density and equality holds for all $p \geq 2$. If $F$ is an immersion and equality holds for $p > 2$, then $F$ is a homothety onto its image.
\end{theorem}
It would be interesting to determine whether Theorem \ref{kahler_thm} is true for mappings of $(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})$, $N \geq 3$. The conclusion of Theorem \ref{kahler_thm} is similar to a result of Burns, Burstall, de Bartolomeis and Rawnsley \cite[Theorem 3]{BBdBR1} that if $\phi:(M^{4},g) \rightarrow (\mathcal{Z},h)$ is stable harmonic map from a closed, real-analytic Riemannian $4$-manifold $(M^{4},g)$ to a Hermitian symmetric space $(\mathcal{Z},h)$ and there is a point of $M$ at which the rank of $d\phi$ is at least $3$, then there is a unique K\"ahler structure on $M$ with respect to which $\phi$ is holomorphic. \\
When the mapping of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ in Theorem \ref{cpn_thm} is to a simply connected, compact K\"ahler manifold, we will prove a stronger characterization of equality -- we record this equality case along with the general inequality in Theorem \ref{cpn_thm} in the following:
\begin{theorem}
\label{cpn_holom_thm}
Let $\widetilde{g}$ be a K\"ahler metric on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow (M,g)$ a Lipschitz mapping to a Riemannian manifold and $A^{\star}$ the infimum of the areas of mappings $f:S^{2} \rightarrow M$ in the free homotopy class of $F_{*}(\mathbb C} \newcommand{\N}{\mathbb N P^{1})$. Then for $p \geq 2$,
\begin{equation}
\label{general_cpn_estimate}
\displaystyle E_{p}(F) \geq \frac{\pi^{N}}{N!} \left( \frac{2N}{\pi} A^{\star} \right)^{\frac{p}{2}}. \medskip
\end{equation}
Suppose in addition that $(M,g)$ is a compact, simply connected K\"ahler manifold and the class of $F_{*}(\mathbb C} \newcommand{\N}{\mathbb N P^{1})$ in $H_{2}(X;\mathbb Z)$ can be represented by a rational curve. Then if equality holds for $p=2$, $F$ is holomorphic or antiholomorphic. If $p > 2$ and equality holds for $p$, then $F$ is a homothety onto its image and equality holds for all $p \geq 2$.
\end{theorem}
Ohnita \cite{Oh1} has proven that holomorphic and antiholomorphic mappings are the only stable harmonic maps between $(\mathbb C} \newcommand{\N}{\mathbb N P^{N_{1}},g_{0})$ and $(\mathbb C} \newcommand{\N}{\mathbb N P^{N_{2}},g_{0})$ with the canonical metric $g_{0}$. The equality conditions in Theorem \ref{cpn_holom_thm} and the more general results in Theorem \ref{cpn_thm} are similar to those in several results of Ohnita \cite{Oh1} and Burns, Burstall, de Bartolomeis and Rawnsley \cite{BBdBR1}, which we will discuss in Section \ref{complexes}. \\
White's results \cite{Wh1} cited above imply that for any Riemannian metric $g$ on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ the identity mapping of $(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, g)$ is homotopic to maps with arbitrarily small $p$-energy for all $ 1 \leq p < 2$. They likewise imply that the identity mapping of quaternionic projective space $\mathbb{H} P^{N}$ is homotopic to maps with arbirarily small $p$-energy for all $1 \leq p < 4$ and the identity mapping of the Cayley projective plane $\mathcal{C}a P^{2}$ is homotopic to maps with arbitrarily small $p$-energy for all $1 \leq p < 8$, for any Riemannian metrics on these spaces. In this sense, Theorems \ref{kahler_thm} and \ref{cpn_holom_thm} are optimal, and the strongest results one can hope to establish in $\mathbb{H} P^{N}$ are lower bounds for the $p$-energy of mappings for $p \geq 4$. Some aspects of the results for $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ and $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ above make such lower bounds for $\mathbb{H} P^{N}$ a natural extension of Theorems \ref{rpn_thm}, \ref{kahler_thm} and \ref{cpn_holom_thm} -- we will discuss this below -- but the optimal result for $\mathbb{H} P^{N}$ cannot be as strong as our results for $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ and $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$: the work of Wei \cite{We1} implies that for $N \geq 2$, the identity mapping of $(\mathbb{H} P^{N},g_{0})$ is not a stable critical point of the $4$-energy, and in particular does not minimize $4$-energy in its homotopy class. We will show that for $p \geq 4$, the $p$-energies of mappings of $(\mathbb{H} P^{N},g_{0})$ nonetheless do satisfy lower bounds like our results for mappings of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ and $(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})$:
\begin{theorem}
\label{hpn_thm}
Let $(\mathbb{H}P^{N},g_{0})$ be the quaternionic projective space with its canonical metric, $N \geq 2$. Let $F:(\mathbb{H}P^{N},g_{0}) \rightarrow (M^{m},g)$ be a non-constant Lipschitz map to a closed Riemannian manifold and $B^{\star}$ the minimum mass of an integral 4-current $T$ in $M$ which represents the class of $F_{*}(\mathbb{H}P^{1})$ in $H_{4}(M;\mathbb Z)$. Then for all $p \geq 4$,
\begin{equation}
\label{hpn_thm_eqn}
\displaystyle E_{p}(F) > \frac{\pi^{2N}}{(2N+1)!} \left( K_{N}B^{\star} \right)^{\frac{p}{4}}, \medskip
\end{equation}
where $K_{N}$ is a positive constant which depends only on $N$ and is given by (\ref{hpn_const_formula}) below.
\end{theorem}
The minimum $B^{\star}$ in Theorem \ref{hpn_thm} is over all integral currents homologous to $F_{*}(\mathbb{H}P^{1})$, which may be larger than the set of mappings $f:S^{4} \rightarrow M$ homotopic to $F_{*}(\mathbb{H}P^{1})$ as in Theorems \ref{rpn_thm}, \ref{kahler_thm} and \ref{cpn_holom_thm}. However a theorem of White \cite{Wh2} shows that if $M$ is simply connected, $B^{\star}$ is equal to the infimum of the areas of mappings $f:S^{4} \rightarrow M$ in the free homotopy class of $F_{*}(\mathbb{H}P^{1})$. We will discuss this after the proof of Theorem \ref{hpn_thm}, in Section \ref{quaternions}. \\
The proof of Theorem \ref{hpn_thm} does show that $Id:(\mathbb{H}P^{N},g_{0}) \rightarrow (\mathbb{H}P^{N},g_{0})$ minimizes energy in its homotopy class among maps which satisfy an additional hypothesis -- we will also discuss after the proof of Theorem \ref{hpn_thm}. The strongest conjecture which is consistent with Wei's result \cite[Theorem 5.1]{We1} in all dimensions is that the identity mapping of $(\mathbb{H} P^{N},g_{0})$ minimizes $p$-energy in its homotopy class for $p \geq 6$. More precisely, Wei's results imply that the identity mapping of $(\mathbb{H} P^{N},g_{0})$ is an unstable critical point of the $p$-energy for $1 \leq p < 2 + 4(\frac{N}{N+1})$ and a stable critical point for $p \geq 2 + 4(\frac{N}{N+1})$. It would be interesting to find optimal lower bounds for the $p$-energy of mappings of $(\mathbb{H} P^{N},g_{0})$ and $(\mathcal{C}a P^{2},g_{0})$ and determine the $p$-energies for which the identity mappings are energy-minimizing in their homotopy classes -- for the Cayley plane with its canonical metric, Wei's results imply that the identity mapping is an unstable critical point of the $p$-energy for $1 \leq p < 10$ and a stable critical point for $p \geq 10$. At the end of Section \ref{quaternions}, we will sketch one possible approach to this problem for $\mathbb{H} P^{N}$. \\
For mappings homotopic to the identity of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, the lower bound in Theorem \ref{cpn_holom_thm} follows from a lower bound for the area of surfaces homologous to $\mathbb C} \newcommand{\N}{\mathbb N P^{1}$ in $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, which follows from the calibrated structure of the K\"ahler $2$-form of $(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})$. The characterization of energy-minimizing maps as holomorphic or antiholomorphic in Theorem \ref{cpn_holom_thm} is related to the fact that a map which realizes the lower bound in (\ref{general_cpn_estimate}) must map all linearly embedded $\mathbb C} \newcommand{\N}{\mathbb N P^{1} \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{N}$ to cycles which are calibrated by the K\"ahler form of the image of $F$. We will explain at the end of Section \ref{reals} how the canonical $1$-form on the unit tangent bundle of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ gives a calibration-like structure, and how the energy-minimizing property of $Id:(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0}) \rightarrow (\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ can be derived using this structure. \\
Quaternionic projective space also carries a calibration, by a parallel $4$-form described in \cite{Be2,Kr1,Kr2}, and this plays a part in the proof of Theorem \ref{hpn_thm}. Despite these similarities, however, the fact that the identity mapping of $(\mathbb{H} P^{N},g_{0})$ does not minimize $4$-energy in its homotopy class, contrary to our results for the $1$-energy of mappings of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ and the $2$-energy of mappings of $(\mathbb C} \newcommand{\N}{\mathbb N P^{N},g_{0})$, mirrors a surprising difference between the systolic geometry of the projective planes $\mathbb R} \newcommand{\T}{\mathbb T P^{2}$, $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ and $\mathbb{H} P^{2}$. These systolic results have several connections to the results in this paper. Our results also have some connections to the Blaschke conjecture and its generalizations to $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, $\mathbb{H} P^{N}$ and $\mathcal{C}a P^{2}$. We will finish this introduction by briefly discussing the relationships between these results and giving an outline of the rest of the paper. \\
Pao Ming Pu's inequality, the first published result in systolic geometry, gives a lower bound for the area of a Riemannian metric on the real projective plane in terms of the minimum length of its non-contractible curves:
\begin{theorem}[Pu's Inequality, \cite{Pu}]
\label{pu_thm}
Let $g$ be a Riemannian metric on $\mathbb R} \newcommand{\T}{\mathbb T P^{2}$. Let $A(\mathbb R} \newcommand{\T}{\mathbb T P^{2},g)$ be its area and $L^{\star}(g)$ its systole, that is, the minimum length of a non-contractible closed curve in $(\mathbb R} \newcommand{\T}{\mathbb T P^{2},g)$. Then:
\begin{equation}
\label{pu_eqn}
\displaystyle A(\mathbb R} \newcommand{\T}{\mathbb T P^{2},g) \geq \left( \frac{2}{\pi} \right) L^{\star}(g)^{2}. \medskip
\end{equation}
Equality holds if and only if $(\mathbb R} \newcommand{\T}{\mathbb T P^{2},g)$ has constant curvature.
\end{theorem}
Pu's theorem follows from Croke's proof of the $p=2$ case of Theorem \ref{rpn_thm} in \cite{Cr1} -- we will discuss this in Section \ref{energies}. Croke's reasoning in the last section of \cite{Cr1} can also be used to show that the canonical metric on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ is infinitesimally optimal for an inequality of the form $Vol(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g) \geq G_{n}L^{\star}(g)^{n}$ for all $n \geq 3$. The results of Gromov \cite{Gr3} imply that such an inequality holds with a positive constant $G_{n}$ for all $n \geq 2$, but for $n \geq 3$ the optimal constant $G_{n}$ is currently not known. \\
Gromov has also proven an inequality for complex projective space which is analogous to Pu's inequality, in terms of an invariant known as the stable $2$-systole. For a Riemannian metric $g$ on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, this can be defined as follows: let $\mu_{k}(g)$ be the minimum area of a $2$-dimensional current representing $k \in H_{2}(\mathbb C} \newcommand{\N}{\mathbb N P^{N};\mathbb Z) \cong \mathbb Z$. Then the stable $2$-systole $A^{\star}(g)$ of $g$ is:
\begin{equation}
\label{stable_2_systole}
\displaystyle A^{\star}(g) = \lim\limits_{k \to \infty} \left( \frac{1}{k} \right) \displaystyle \mu_{k}(g). \bigskip
\end{equation}
\begin{theorem}[Gromov's Stable Systolic Inequality for $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, \cite{Gr2}, see also \cite{Gr1,BKSW}]
\label{gromov_thm}
Let $g$ be a Riemannian metric on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, let $Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{N},g)$ be its volume and $A^{\star}(g)$ its stable $2$-systole as above. Then:
\begin{equation}
\label{gromov_eqn}
\displaystyle Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{N},g) \geq \frac{A^{\star}(g)^{N}}{N!}.
\end{equation}
\end{theorem}
As in Theorem \ref{pu_thm}, equality holds for the canonical metric $g_{0}$ on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ in Theorem \ref{gromov_thm}. Unlike the rigidity of the equality case in Pu's inequalty (\ref{pu_eqn}), however, equality also holds for all K\"ahler metrics on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. As with the broader characterization of equality in Theorem \ref{cpn_holom_thm} for complex projective space, compared to Theorem \ref{rpn_thm} for real projective space, this is related to the fact that for any K\"ahler metric $\widetilde{g}$ on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, complex curves are calibrated and thus area minimizing in their homology classes. \\
A result analogous to Pu's and Gromov's inequalities (\ref{pu_eqn}) and (\ref{gromov_eqn}) holds for the quaternionic projective plane. Like the result for $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ in Theorem \ref{gromov_thm}, this inequality is in terms of the stable $4$-systole, which is defined by a limit for $H_{4}(\mathbb{H}P^{2};\mathbb Z) \cong \mathbb Z$ as in (\ref{stable_2_systole}). However unlike the results for $\mathbb R} \newcommand{\T}{\mathbb T P^{2}$ and $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ in Pu's and Gromov's inequalities, Bangert, Katz, Shnider and Weinberger have shown that the canonical metric on $\mathbb{H} P^{2}$ is not optimal for this inequality:
\begin{theorem}[\cite{BKSW}]
Let $g$ be a Riemannian metric on $\mathbb{H}P^{2}$ and $B^{\star}(g)$ the stable $4$-systole of $g$ (as defined in \cite{Gr2,BKSW}). There is a positive constant $D_{2}$, independent of $g$, such that:
\begin{equation}
\label{hpn_systolic_eqn}
\displaystyle Vol(\mathbb{H}P^{2},g) \geq D_{2} B^{\star}(g)^{2}. \medskip
\end{equation}
The optimal constant in (\ref{hpn_systolic_eqn}) satisfies $\frac{1}{6} \geq D_{2} \geq \frac{1}{14}$, which excludes the value $\frac{3}{10}$ of the canonical metric.
\end{theorem}
The proof of the characterization of equality in Theorem \ref{rpn_thm} for $p=1$ is different from the proof for $p > 1$ and uses the characterization of the canonical metric as the only Riemannian metric on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ for which the first conjugate locus of each point $x_{0}$ consists of a single point (in fact $x_{0}$ itself). Blaschke conjectured that this was the case and it was proven by the combined work of Berger, Green, Kazdan and Yang, cf. \cite{Gn, Be1}. The Blaschke conjecture therefore implies that, among mappings which are immersions, the equality case in Theorem \ref{rpn_thm} for $p = 1$ is the same as for $p > 1$. There are conjectures similar to the Blaschke conjecture for $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, $\mathbb{H} P^{N}$ and $\mathcal{C}a P^{2}$ which are open. However, the statement which follows from the Blaschke conjecture for $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ in Theorem \ref{rpn_thm} is false for $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$: there are holomorphic mappings of $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ which are not isometries of the canonical metric. These mappings minimize $2$-energy in their homotopy class but by Theorem \ref{kahler_thm}, they do not minimize $p$-energy for $p > 2$. Therefore, the equality case for $p=2$ in our results for $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ is strictly larger than for $p > 2$. \\
An outline of this paper is as follows: \\
In Section \ref{energies} we will define the $p$-energy of a mapping of a Riemannian manifold and establish some of its basic properties. In Section \ref{reals} we will prove Theorem \ref{rpn_thm}. In Section \ref{complexes} we will prove Theorem \ref{cpn_thm}, of which Theorem \ref{kahler_thm} is a special case, and Theorem \ref{cpn_holom_thm}. In Section \ref{quaternions}, we will prove Theorem \ref{hpn_thm}. The proof of Theorem \ref{hpn_thm} uses the twistor fibration $\Psi:\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1} \rightarrow \mathbb{H} P^{N}$, and we will discuss some background related to the twistor fibration at the beginning of Section \ref{quaternions}. \\
Throughout, we will write $\sigma(m)$ for the volume of the unit sphere in $\mathbb R} \newcommand{\T}{\mathbb T^{m+1}$.
\subsection*{Acknolwedgements:} I am very happy to thank Christopher Croke, Joseph H.G. Fu, Mikhail Katz, Frank Morgan and Michael Usher for their input and feedback about this work.
\section{Energy Functionals of Mappings}
\label{energies}
In this section, we will define and study the $p$-energy of a mapping. We will prove two elementary results, Lemmas \ref{elementary_lemma} and \ref{energy_formula_lemma}, which we will use in the proof of our main theorems below. \\
The energy of a Lipschitz mapping $F:(M^{m},g) \rightarrow (N^{n},h)$ of Riemannian manifolds is:
\begin{equation}
\label{energy_eqn}
\displaystyle E_{2}(F) = \int\limits_{M} |dF_{x}|^{2} dVol_{g}, \bigskip
\end{equation}
where $|dF_{x}|$ is the norm of $dF:T_{x}M \rightarrow T_{F(x)}N$ at a point $x$ where $F$ is differentiable. We note that many authors define the energy to be one half the expression in (\ref{energy_eqn}). \\
There are many equivalent ways to define the energy of a mapping -- these are discussed by Eells and Sampson in \cite{ES1}. In this work, they initiated the study of the critical points of the energy as a functional on mappings from $(M,g)$ to $(N,h)$. These critical mappings are known as harmonic maps and have many important connections to minimal submanifold theory, K\"ahler geometry and several other topics in differential geometry and analysis. \\
The energy in (\ref{energy_eqn}) fits naturally into a $1$-parameter family of functionals:
\begin{definition}
\label{p_energy_def}
Let $F:(M^{m},g) \rightarrow (N^{n},h)$ be a Lipschitz mapping of closed Riemannian manifolds. For $p \geq 1$, the $p$-energy of $F$ is:
\begin{equation}
\label{p-energy_eqn}
\displaystyle \int\limits_{M} |dF_{x}|^{p} dVol_{g}.
\end{equation}
\end{definition}
The pointwise quantity $|dF_x|^{p}$, where defined, is called the $p$-energy density. We will denote this $e_{p}(F)_{x}$. It will be helpful to keep in mind that wherever this can be defined, i.e. at all $x \in M$ at which $F$ is differentiable, $F^{*}h$ is a positive semidefinite, symmetric bilinear form on $T_{x}M$ which can be diagonalized relative to $g$. Letting $e_{1}, e_{2}, \cdots, e_{m}$ be an orthonormal basis for $T_{x}M$ (relative to $g$) of eigenvectors for $F^{*}h$, we then have:
\begin{equation}
\label{energy_frame_eqn}
\displaystyle |dF_{x}|^{2} = \sum\limits_{i=1}^{m} |dF(e_{i})|^{2}. \bigskip
\end{equation}
This gives the following elementary lower bound for the $p$ energy of $F:(M^{m},g) \rightarrow (N^{n},h)$ for $p \geq dim(M)$:
\begin{lemma}
\label{elementary_lemma}
Let $F:(M^{m},g) \rightarrow (N^{n},h)$ be a Lipschitz mapping, and let $Vol_{g}(M,F^{*}h)$ be:
\begin{equation}
\label{pullback_vol_eqn}
\displaystyle \int\limits_{M}|\det(dF_{x})| dVol_{g}, \medskip
\end{equation}
where $det(dF_{x})$ is $0$ if $rk(dF_{x}) < m$ and is the determinant of $dF:T_{x}M \rightarrow dF(T_{x}) \subseteq T_{F(x)}N$ if $rk(dF_{x}) = m < n$. \\
Then for $p \geq m$,
\begin{equation}
\label{g_dim_energy_eqn}
\displaystyle E_{p}(F) \geq m^{\frac{p}{2}} \frac{Vol_{g}(M,F^{*}h)^{\frac{p}{m}}}{Vol(M,g)^{\frac{p-m}{m}}}. \medskip
\end{equation}
For $p=m$, equality holds if and only if $dF_{x}$ is a homothety at almost all $x \in M$ at which $F$ is differentiable. For $p > m$, equality holds if and only if $dF_{x}$ is a homothety, by a constant factor $C_{F}$, at almost all $x \in M$ at which $F$ is differentiable.
\end{lemma}
\begin{proof}
For $x$ in $M$ at which $F$ is differentiable, let $e_{1}, e_{2}, \cdots e_{m}$ be an orthonormal basis of eigenvectors for $F^{*}h$ as in (\ref{energy_frame_eqn}). By (\ref{energy_frame_eqn}) and the arithmetic-geometric mean inequality,
\begin{equation*}
\displaystyle e_{m}(F)_{x} = \left( \sum\limits_{i=1}^{m} |dF(e_{i})|^{2} \right)^{\frac{p}{2}} \geq \left( m \sqrt[m]{\prod\limits_{i=1}^{m}|dF(e_{i})|^{2}} \right)^{\frac{p}{2}} \bigskip
\end{equation*}
\begin{equation}
\label{energy_lemma_pf_eqn_1}
\displaystyle = m^{\frac{p}{2}}|\det(dF_{x})|^{\frac{p}{m}}. \bigskip
\end{equation}
For $p=m$, this immediately implies (\ref{g_dim_energy_eqn}). Equality holds if and only if it holds a.e. in (\ref{energy_lemma_pf_eqn_1}), and equality in (\ref{energy_lemma_pf_eqn_1}) holds at $x$ precisely if $dF_{x}$ is a homothety. \\
For $p > m$, by (\ref{energy_lemma_pf_eqn_1}) and H\"older's inequality,
\begin{equation}
\label{energy_lemma_pf_eqn_2}
\displaystyle E_{p}(F) \geq m^{\frac{p}{2}} \int\limits_{M} |\det(dF_{x})|^{\frac{p}{m}} dVol_{g} \geq m^{\frac{p}{2}} \frac{Vol_{g}(M,F^{*}h)^{\frac{p}{m}}}{Vol(M,g)^{\frac{p-m}{m}}}. \bigskip
\end{equation}
Equality holds if and only if it holds a.e. in (\ref{energy_lemma_pf_eqn_1}), as in the case $p = m$, and in H\"older's inequality in (\ref{energy_lemma_pf_eqn_2}). We have equality in H\"older's inequality precisely if $|\det(dF_{x})|$ is a.e. equal to a constant $C_{F}$. \end{proof}
Note that in Lemma \ref{elementary_lemma}, equality for any $p > m$ implies equality for all $p \geq m$. \\
If $F$ is a smooth mapping, the equality condition for $p=m$ in Lemma \ref{elementary_lemma} says that $F$ is a semiconformal mapping, that is $F^{*}h = \varphi(x)g$ for a nonnegative function $\varphi(x)$ on $M$, and the equality condition for $p > m$ says that $F$ is a homothety, i.e. $F^{*}h$ is a rescaling of $g$. This generalizes the well-known fact that for mappings of surfaces, the energy is pointwise bounded below by the area of the image, with equality precisely where the mapping is conformal. The uniformization theorem implies that every Riemannian metric $g$ on $\mathbb R} \newcommand{\T}{\mathbb T P^{2}$ is conformally equivalent to a constant curvature metric $g_{0}$, which is unique up to scale. Letting $g = \varphi(x) g_{0}$, Lemma \ref{elementary_lemma} implies that $Id:(\mathbb R} \newcommand{\T}{\mathbb T P^{2},g_{0}) \rightarrow (\mathbb R} \newcommand{\T}{\mathbb T P^{2},g)$ is energy-minimizing in its homotopy class and has energy equal to $2A(\mathbb R} \newcommand{\T}{\mathbb T P^{2},g)$ (with the energy defined with our normalization in (\ref{energy_eqn})). Pu's Theorem \ref{pu_thm} is then a special case of Croke's lower bound for the $2$-energy of mappings $F:(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0}) \rightarrow (M,g)$. \\
The following formula for the energy density is used in Croke's results in \cite{Cr1} and will also be used throughout the proofs of our results below:
\begin{lemma}[See \cite{Cr1}]
Let $F:(M^{m},g) \rightarrow (N^{n},h)$ be a Lipschitz mapping of Riemannian manifolds and $x \in M$ with $F$ differentiable at $x$.
\begin{equation}
\label{norm_squared_formula}
\displaystyle |dF_x|^{2} = \frac{m}{\sigma(m-1)} \int\limits_{U_{x}M} |dF(\vec{u})|^{2} d\vec{u}.
\end{equation}
\end{lemma}
The identity in (\ref{norm_squared_formula}) is the basis for the following formula for the $2$-energy of a mapping $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow (M,g)$, where $\widetilde{g}$ is a K\"ahler metric on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. This result is an elementary example of the type of arguments and calculations we will employ below and is also an important lemma in several of them:
\begin{lemma}
\label{energy_formula_lemma}
Let $g_{0}$ be the canonical metric on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, normalized so that its sectional curvature $K$ satisfies $1 \leq K \leq 4$, and let $\widetilde{g}$ be a K\"ahler metric cohomologous to $g_{0}$. Let $\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ be the family of linearly embedded $1$-dimensional complex projective subspaces $\mathbb C} \newcommand{\N}{\mathbb N P^{1} \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{N}$, let $U(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})$ be the unit tangent bundle of $(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, \widetilde{g})$ and let $\mathsf{T}:U(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ be the mapping which sends a unit tangent vector $\vec{u}$ to $(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})$ to the unique element of $\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ to which it is tangent. Let $d\widetilde{\mathcal{M}}$ be the measure on $\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ which is pushed forward from the Riemannian volume on $U(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})$ via $\mathsf{T}$. \\
Let $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow (M,g)$ be a Lipschitz mapping to a Riemannian manifold $(M,g)$, and for $\mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$, let $F|_{\mathcal{P}}$ be the mapping restricted to $\mathcal{P}$. Note that $F|_{\mathcal{P}}:(\mathcal{P},\widetilde{g}) \rightarrow (M,g)$ is also Lipschitz. Then:
\begin{equation}
\displaystyle E_{2}(F) = \frac{2\pi N}{\sigma(2N-1)} \int\limits_{\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})} E_{2}(F|_{\mathcal{P}}) d\widetilde{\mathcal{M}}.
\end{equation}
\end{lemma}
\begin{proof}
By (\ref{norm_squared_formula}), we have:
\begin{equation*}
\displaystyle E_{2}(F) = \int\limits_{\mathbb C} \newcommand{\N}{\mathbb N P^{N}} |dF_{x}|^{2} dVol_{\widetilde{g}} = \frac{2N}{\sigma(2N-1)} \int\limits_{U(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})} |dF(\vec{u})|^{2} d\vec{u} \bigskip
\end{equation*}
\begin{equation*}
\displaystyle = \frac{2N}{\sigma(2N-1)} \int\limits_{\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})} \int\limits_{U(\mathcal{P},\widetilde{g})} |dF(\vec{u})|^{2} \ d\vec{u} \ d\widetilde{\mathcal{M}}, \bigskip
\end{equation*}
where $U(\mathcal{P},\widetilde{g})$ is the unit tangent bundle of a complex projective line $\mathcal{P}$ in the metric $\widetilde{g}|_{\mathcal{P}}$. Using (\ref{norm_squared_formula}) again,
\begin{equation*}
\displaystyle \frac{2N}{\sigma(2N-1)} \int\limits_{\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})} \int\limits_{U(\mathcal{P},\widetilde{g})} |dF(\vec{u})|^{2} \ d\vec{u} \ d\widetilde{\mathcal{M}} = \frac{2\pi N}{\sigma(2N-1)} \int\limits_{\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})} \int\limits_{\mathcal{P}} |dF|_{\mathcal{P}_{x}}|^{2} \ dx \ d\widetilde{\mathcal{M}} \bigskip
\end{equation*}
\begin{equation*}
\displaystyle = \frac{2\pi N}{\sigma(2N-1)} \int\limits_{\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})} E_{2}(F|_{\mathcal{P}}) d\widetilde{\mathcal{M}}.
\end{equation*}
\end{proof}
One can derive similar formulas for the energy of mappings of $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$, $\mathbb{H} P^{N}$ and the Cayley plane $\mathcal{C}a P^{2}$ as integrals over the spaces of linearly embedded $\mathbb R} \newcommand{\T}{\mathbb T P^{1}$, $\mathbb{H} P^{1}$ and $\mathcal{C}a P^{1} \cong S^{8}$, and over the space of geodesics in the the sphere $(S^{n},g_{0})$. The formula of this type for the energy of mappings $F:(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0}) \rightarrow (M,g)$ plays a key part in Croke's proof of the $p=2$ case of Theorem \ref{rpn_thm} in \cite{Cr1}. Even in the cases $\mathbb{H}P^{N}$, $\mathcal{C}a P^{2}$ and $S^{n},n \geq 3$, where one knows that the identity mapping is homotopic to maps with arbitrarily small energy, one can use such a formula to show that a family of mappings whose energies decay to $0$ must also have energies decaying to $0$ when restricted to almost all linear subpsaces $\mathbb{H}P^{k} \subseteq \mathbb{H}P^{N}$, $\mathcal{C}a P^{1} \subseteq \mathcal{C}a P^{2}$, and almost all great subspheres $S^{k} \subseteq S^{n}$. \\
We end this section with a few comments about the regularity of harmonic and energy-minimizing maps: \\
Continuous, weakly harmonic maps are smooth \cite[Ch.10]{Aub1}. In particular, any continuous map which minimizes energy in its homotopy class is smooth. Assuming only Lipschitz regularity, mappings which realize equality for $p=2$ in Theorems \ref{rpn_thm}, \ref{kahler_thm}, \ref{cpn_holom_thm} and \ref{cpn_thm} are therefore $C^{\infty}$. However, unless one has established that equality holds for $p=2$, one cannot assume smoothness because for $p \neq 2$ there are $p$-energy minimizing maps which are $C^{1,\alpha}$ for $\alpha < 1$ but are not $C^{2}$ \cite[Section 3]{EL2}. In dimensions $3$ and greater, $p$-energy minimizing maps also need not be continuous \cite[Section 3]{EL2}. White has shown \cite{Wh3} that $p$-energy minimizing sequences of maps of compact Riemannian manifolds converge, in an appropriate topology, to mappings which belong to a Sobolev space of mappings and have well-defined homotopy classes when restricted to lower-dimensional skeleta of their domain. However a $p$-energy minimizing sequence of maps in one homotopy class can converge, in a weak sense, to a map in another homotopy class -- for example, on $(S^{n},g_{0})$, a family of conformal dilations with energy decaying to $0$, in the homotopy class of the identity, converges weakly to a constant map. In light of these results, Lipschitz regularity is a stronger assumption than is natural for $p$-energy minimizing maps in general. However, the Lipschitz condition works well in our setting because it is inherited by the restriction of maps $F:(M,g) \rightarrow (N,h)$ to any submanifold of $M$. In our case, this implies that a Lipshitz mapping $F:(\mathbb{K}P^{N},g_{0}) \rightarrow (M,g)$ will be Lipshitz when restricted to all $\mathbb{K}P^{1} \subseteq \mathbb{K}P^{N}$, where $\mathbb{K}$ is $\mathbb R} \newcommand{\T}{\mathbb T$, $\mathbb C} \newcommand{\N}{\mathbb N$ or $\mathbb{H}$. For $\mathbb{H} P^{N}$, we will also consider the restriction of $F$ to totally geodesic submanifolds isometric to $(\mathbb C} \newcommand{\N}{\mathbb N P^{2},g_{0})$. This will allow us to draw conclusions about the mapping of $\mathbb{K}P^{N}$ from its behavior along lower-dimensional subspaces.
\section{Real Projective Space}
\label{reals}
In this section, we will prove Theorem \ref{rpn_thm}. We will write $g_{0}$ for the canonical metric on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$, normalized to have constant sectional curvature $1$. We define the space of oriented geodesics in $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ to be the quotient of the unit tangent bundle $U(\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0})$ by the geodesic flow. We denote this $\mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ and we equip $\mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ with the measure $d\gamma$ pushed forward from the measure on $U(\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0})$, so that:
\begin{equation}
\displaystyle Vol(\mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})) = \frac{Vol(U(\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0}))}{\pi}. \bigskip
\end{equation}
Because $Vol(U(\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0})) = \frac{\sigma(n)\sigma(n-1)}{2}$, we have:
\begin{equation}
\label{Vol_G_equation}
\displaystyle Vol(\mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})) = \frac{\sigma(n)\sigma(n-1)}{2\pi}. \bigskip
\end{equation}
\begin{proof}[Proof of Theorem \ref{rpn_thm}]
By the formula (\ref{norm_squared_formula}) for the energy density and the Cauchy-Schwarz inequality,
\begin{equation*}
\displaystyle E_{p}(F) = \int\limits_{\mathbb R} \newcommand{\T}{\mathbb T P^{n}} |dF_{x}|^{p} dVol_{g_{0}} = \int\limits_{\mathbb R} \newcommand{\T}{\mathbb T P^{n}} \left( \frac{n}{\sigma(n-1)} \int\limits_{U_{x}\mathbb R} \newcommand{\T}{\mathbb T P^{n}} |dF(\vec{u})|^{2} d\vec{u} \right)^{(\frac{p}{2})} dVol_{g_{0}} \bigskip
\end{equation*}
\begin{equation}
\label{rpn_pf_eqn_1}
\displaystyle \geq \frac{n^{\frac{p}{2}}}{\sigma(n-1)^{p}} \int\limits_{\mathbb R} \newcommand{\T}{\mathbb T P^{n}} \left( \int\limits_{U_{x}\mathbb R} \newcommand{\T}{\mathbb T P^{n}} |dF(\vec{u})| d\vec{u} \right)^{p} dVol_{g_{0}}. \bigskip
\end{equation}
For $p = 1$, this says:
\begin{equation}
\label{rpn_pf_eqn_2}
\displaystyle E_{1}(F) \geq \frac{\sqrt{n}}{\sigma(n-1)} \int\limits_{U(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})} |dF(\vec{u})| d\vec{u}. \bigskip
\end{equation}
For $p > 1$, by (\ref{rpn_pf_eqn_1}) and H\"older's inequality,
\begin{equation}
\label{rpn_pf_eqn_3}
\displaystyle E_{p}(F) \geq \frac{2^{p-1} n^{\frac{p}{2}}}{\sigma(n-1)^{p} \sigma(n)^{p-1}} \left( \int\limits_{U(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})} |dF(\vec{u})| d\vec{u} \right)^{p}. \bigskip
\end{equation}
For each $\gamma \in \mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$, $F \circ \gamma$ is an oriented Lipschitz $1$-cycle in $(M,g)$. Letting $|F \circ \gamma|$ be its mass, and writing $\gamma: [0,\pi] \rightarrow \mathbb R} \newcommand{\T}{\mathbb T P^{n}$ for a unit-speed parametrization of $\gamma$ and $F \circ \gamma: [0,\pi] \rightarrow M$ for the associated parametrization of $F \circ \gamma$, we then have:
\begin{equation*}
\displaystyle |F \circ \gamma| = \int\limits_{0}^{\pi} |(F \circ \gamma)'(t)| dt, \bigskip
\end{equation*}
where we have used that the Lipschitz mapping $F \circ \gamma: [0,\pi] \rightarrow M$ is differentiable almost everywhere. For all $t$ such that $F$ is differentiable at $\gamma(t)$, $(F \circ \gamma)'(t) = dF(\gamma'(t))$. By Fubini's theorem, the right-hand sides of (\ref{rpn_pf_eqn_2}) and (\ref{rpn_pf_eqn_3}) can therefore be rewritten as an integral over $\mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0})$, which implies:
\begin{equation}
\displaystyle E_{p}(F) \geq \frac{2^{p-1} n^{\frac{p}{2}}}{\sigma(n-1)^{p} \sigma(n)^{p-1}} \left( \int\limits_{\mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})} |F \circ \gamma| d\gamma \right)^{p}. \bigskip
\end{equation}
Because each geodesic $\gamma$ represents a generator of $\pi_{1}(\mathbb R} \newcommand{\T}{\mathbb T P^{n})$, $|F \circ \gamma| \geq L^{\star}$, and therefore,
\begin{equation*}
\displaystyle E_{p}(F) \geq \frac{2^{p-1} n^{\frac{p}{2}}}{\sigma(n-1)^{p} \sigma(n)^{p-1}} \left( Vol(\mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})) L^{\star} \right)^{p}, \bigskip
\end{equation*}
which is (\ref{rpn_thm_eqn}). \\
Suppose equality holds for $p=1$. \\
This implies that equality holds in the Cauchy-Schwarz inequality in (\ref{rpn_pf_eqn_1}) for a.e. $x \in \mathbb R} \newcommand{\T}{\mathbb T P^{n}$. For all $x$ at which $F$ is differentiable and for which this equality holds, $|dF_{x}(\vec{u})|$ depends only on $x$. This implies that $F^{*}g$ is a.e. equal to $\varphi(x) g_{0}$, where $\varphi(x)$ is a nonnegative function on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$. Because all $\gamma \in \mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ map to rectifiable currents $F \circ \gamma$ with well-defined lengths in $(M,g)$, equality also implies that for almost all $\gamma$, $|F \circ \gamma| = L^{\star}$. Because $|F \circ \gamma|$ is lower semicontinuous on $\mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$, we in fact have $|F \circ \gamma| = L^{\star}$ for all $\gamma$. The image via $F$ of each geodesic $\gamma$ in $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ is therefore a closed geodesic in $(M,g)$, of minimal length $L^{\star}$ in its free homotopy class, although a priori $F \circ \gamma$ may not be parametrized by arc length. \\
If $F$ is a smooth immersion, then because each geodesic $\gamma$ in $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ maps to a closed geodesic in $M$, the image of $F$ is a totally geodesic submanifold, and because $F \circ \gamma$ is of minimal length in its free homotopy class in $(M,g)$, $F^{*}g$ is a Blaschke metric on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$, cf. Remark \ref{blaschke_remark} below. By the Berger-Green-Kazdan-Yang proof of the Blaschke conjecture \cite{Gn, Be1}, $F^{*}g$ is therefore isometric to a round metric. This does not yet imply that $F$ is an isometry or a homothety. However, letting $\psi: (\mathbb R} \newcommand{\T}{\mathbb T P^{n}, F^{*}g) \rightarrow (\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0})$ be an isometry (or homothety), we then have $\psi^{*}g_{0} = \varphi(x) g_{0}$, where $\varphi(x)$ is the semi-conformal factor as above and is in fact a conformal factor, i.e. is everywhere-defined and positive, because $F$ is an immersion. By the classification of conformal diffeomorphisms of the round sphere $(S^{n}, g_{0})$, the mapping $\psi$ is therefore an isometry of $g_{0}$, up to rescaling, and $F$ is an isometry or homothety onto its image. \\
Now suppose $p > 1$ and equality holds for $p$. \\
Supposing only that $F$ is Lipschitz, this implies all of the conditions which follow for Lipshitz mappings which realize equality for $p=1$ and also implies equality in H\"older's inequality in (\ref{rpn_pf_eqn_3}). Equality in H\"older's inequality implies that the semiconformal factor $\varphi(x)$ is a.e. equal to a constant $C_{F}$. This implies that equality holds for all $p \geq 1$. Because equality holds for $p = 2$, $F$ is smooth, so $\varphi(x)$ is an everywhere-defined constant function and $F$ is a homothety onto its image. Because all $\gamma \in \mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ maps to a geodesic of $(M,g)$, the image of $F$ is a totally geodesic submanifold. \end{proof}
\begin{remark}
\label{blaschke_remark}
To see that the conditions for equality when $p=1$ and $F$ is an immersion imply that $F^{*}g$ is a Blaschke metric, i.e. that the first conjugate locus of each point $x_{0}$ in $(\mathbb R} \newcommand{\T}{\mathbb T P^{N},F^{*}g)$ is a single point, in fact $x_{0}$, note that each unit-speed geodesic $c:[0,\infty) \rightarrow (\mathbb R} \newcommand{\T}{\mathbb T P^{N},F^{*}g)$ has a conjugate point at $c(L^{\star}) = c(0)$, where all geodesics based at $c(0)$ intersect, and that this must be the first conjugate point to $c(0)$ along $c$ because $c([0,L^{\star}])$ is length-minimizing in its homotopy class.
\end{remark}
Also, curiously, this part of Theorem \ref{rpn_thm} is false without the stipulation that $n \geq 2$: any diffeomorphism of $\mathbb R} \newcommand{\T}{\mathbb T P^{1} = \mathbb R} \newcommand{\T}{\mathbb T / \pi\mathbb Z$ has $1$-energy equal to $\pi$. In fact, any homeomorphism of $\mathbb R} \newcommand{\T}{\mathbb T P^{1} \cong S^{1}$ is differentiable a.e. and has $1$-energy equal to the identity mapping. \\
In the next section, we will prove a lower bound similar to Theorem \ref{rpn_thm} for mappings of any K\"ahler manifold biholomorphic to $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. In light of this result, it might be natural to ask whether the result of Theorem \ref{rpn_thm} applies to a larger family of metrics on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$. The proof of Theorem \ref{rpn_thm} would give the same lower bounds for mappings of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n}, \widehat{g})$, where $\widehat{g}$ is any metric on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ all of whose geodesics are closed and non-contractible, but the results of Lin and Schmidt in \cite{LS1} show that for $n \neq 3$, the only metrics on $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ all of whose geodesics are closed are those of constant curvature. A proof of the Berger conjecture in dimension $n=3$, as in \cite{GG1} for $n=2$ and \cite{RW1} for $n \geq 4$, would imply that this is the case for $\mathbb R} \newcommand{\T}{\mathbb T P^{3}$ as well. \\
We note that for $n \geq 2$, non-isometric projective linear transformations of $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ have the length-preserving property for $\gamma \in \mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0})$ which is implied by equality for $p=1$ in Theorem \ref{rpn_thm} but do not minimize $1$-energy in their homotopy class. In complex projective space, however, non-isometric projective complex linear transformations are holomorphic and minimize $2$-energy in the homotopy class of the identity. \\
We end this section by re-interpreting the fact that the identity mapping of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ minimizes $1$-energy in its homotopy class in terms of a calibration-like property of the canonical $1$-form on the unit tangent bundle of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$: \\
For any oriented integral $1$-chain $\sum_{i} a_{i}\tau_{i}$ in $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$, where $a_{i} \in \mathbb Z$ and $\tau_{i}: \Delta^{1} \rightarrow \mathbb R} \newcommand{\T}{\mathbb T P^{n}$ are oriented $1$-simplices, one can define a $1$-current in the unit tangent bundle $U(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ by associating each point $\tau_{i}(t)$ at which $\tau_{i}$ is differentiable to the unit vector in $T_{\tau_{i}(t)}\mathbb R} \newcommand{\T}{\mathbb T P^{N}$ tangent to $\tau_{i}$ in the oriented direction, with multiplicity $a_{i}$. Letting $\alpha$ be the canonical $1$-form on the unit tangent bundle of $(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$, the mass of this current is its pairing with $\alpha$. The current of this type associated to any non-contractible closed curve in $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ has mass greater than or equal to $\pi$, and any Lipshitz mapping $F:(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0}) \rightarrow (\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ homotopic to the identity sends each $\gamma \in \mathcal{G}(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0})$ to such a closed curve. This implies the lower bound for the $1$-energy of mappings homotopic to $Id:(\mathbb R} \newcommand{\T}{\mathbb T P^{n},g_{0}) \rightarrow (\mathbb R} \newcommand{\T}{\mathbb T P^{n}, g_{0})$ in Theorem \ref{rpn_thm}. \\
Unlike a calibration, however, $\alpha$ is not closed: $d\alpha$ is the Liouville symplectic form on the tangent bundle.
\section{Complex Projective Space}
\label{complexes}
In this section, we will prove lower bounds for the energy of mappings from K\"ahler manifolds biholomorphic to $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ to Riemannian manifolds, similar to the estimate for mappings of real projective space in Theorem \ref{rpn_thm}. The equality case in these results is much broader than in Theorem \ref{rpn_thm}, but for mappings from $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ it implies Theorem \ref{kahler_thm}. For mappings from $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ to simply connected, compact K\"ahler manifolds, we will prove a stronger characterization of equality in Theorem \ref{cpn_holom_thm}. \\
As in Lemma \ref{energy_formula_lemma}, we will let $\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ be the space of linearly embedded $\mathbb C} \newcommand{\N}{\mathbb N P^{1} \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{N}$. We will use the fact that all K\"ahler metrics on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ are cohomologous, up to rescaling, to the canonical metric $g_{0}$, and that for any K\"ahler metric $\widetilde{g}$ cohomologous to $g_{0}$, $Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, \widetilde{g}) = \frac{\pi^{N}}{N!}$ and for all $\mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$, $A(\mathcal{P},\widetilde{g}) = \pi$. As in Lemma \ref{energy_formula_lemma}, we will let $d\widetilde{\mathcal{M}}$ be the measure on $\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ pushed forward from the measure on the unit tangent bundle $U(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, \widetilde{g})$ of $(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, \widetilde{g})$ via the mapping $\mathsf{T}:U(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, \widetilde{g}) \rightarrow \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ which sends non-zero tangent vectors to $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ to the unique element of $\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ to which they are tangent. The total volume of $\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ in the measure $d\widetilde{\mathcal{M}}$ is:
\begin{equation}
\displaystyle Vol(\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N}),d\widetilde{\mathcal{M}}) = \frac{Vol(U(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, \widetilde{g}))}{Vol(U(\mathbb C} \newcommand{\N}{\mathbb N P^{1}, \widetilde{g}))} = \frac{\sigma(2N-1)\pi^{N-2}}{2N!}. \bigskip
\end{equation}
It is a fundamental fact of K\"ahler geometry that closed complex submanifolds of a compact K\"ahler manifold $(X,h)$ are area-minimizing in their homology classes -- this follows from the calibrated structure given by the K\"ahler form \cite{HL1}. Moreover, for any compact complex curve $\Sigma$ and any holomorphic mapping $F$ of $\Sigma$ to a compact K\"ahler manifold $(X,h)$, it follows from Lemma \ref{elementary_lemma} that:
\begin{equation}
\displaystyle E_{2}(F) = 2Area(\Sigma,F^{*}h), \bigskip
\end{equation}
where $Area(\Sigma,F^{*}h)$ is equal to the area of $F(\Sigma)$ and is the minimum area of any cycle representing the class of $F(\Sigma)$ in $H_{2}(X;\mathbb Z)$. For any holomorphic mapping $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow (X,h)$, letting $A^{\star}$ this minimum area for $F(\mathcal{P})$, $\mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$, Lemma \ref{energy_formula_lemma} then implies:
\begin{equation*}
\displaystyle E_{2}(F) = \frac{2\pi N}{\sigma(2N-1)} \int\limits_{\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})} E_{2}(F|_{\mathcal{P}}) d\widetilde{\mathcal{M}}
\end{equation*}
\begin{equation}
\displaystyle = \frac{2\pi N}{\sigma(2N-1)} \times Vol(\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N}),d\widetilde{\mathcal{M}}) \times 2A^{\star} = \frac{2 \pi^{N-1}}{(N-1)!} A^{\star}. \bigskip
\end{equation}
We will see that this coincides with a sharp lower bound for the energy of any Lipschitz mapping from a K\"ahler manifold $(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})$ to a Riemannian manifold $(M,g)$:
\begin{theorem}
\label{cpn_thm}
Let $\widetilde{g}$ be a K\"ahler metric on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, which we suppose without loss of generality is cohomologous to the canonical metric $g_{0}$ with sectional curvature $K$ satisfying $1 \leq K \leq 4$. Let $F: (\mathbb C} \newcommand{\N}{\mathbb N P^{N}, \widetilde{g}) \rightarrow (M^{m},g)$ be a Lipschitz mapping to a Riemannian manifold $(M,g)$ and $A^{\star}$ the infimum of the areas of mappings $f:S^{2} \rightarrow (M,g)$ in the free homotopy class of $F_{*}(\mathbb C} \newcommand{\N}{\mathbb N P^{1})$. \\
For all $p \geq 2$,
\begin{equation}
\label{cpn_thm_eqn}
\displaystyle E_{p}(F) \geq \frac{\pi^{N}}{N!} \left( \frac{2N}{\pi} A^{\star} \right)^{\frac{p}{2}}. \medskip
\end{equation}
If equality holds for $p = 2$, then $F^{*}g$ is a positive semidefinite Hermitian bilinear form on $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. In particular, on any neighborhood $\mathcal{V}$ on which $F$ is an immersion, $F^{*}g$ is a Hermitian metric. Moreover, for any neighborhood $\mathcal{V}$ on which $F$ is an immersion, $F(\mathcal{V})$ is a minimal submanifold of $(M,g)$, the second fundamental form of $F(\mathcal{V})$ in $(M,g)$ can be diagonalized by a unitary basis of $F^{*}g$ and, letting $\omega^{*}$ denote the K\"ahler form of $F^{*}g$, $d\omega^{*}$ vanishes on all subspaces of $T_{x}\mathcal{V}$ of complex dimension $2$, for all $x \in \mathcal{V}$. \\
If $p > 2$ and equality holds for $p$, then equality holds for all $p \geq 2$ and $F$ has constant energy density. \\
If $p > 2$, $\mathbb C} \newcommand{\N}{\mathbb N P^{2} \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{N}$ is a linearly embedded subspace and $F|_{\mathbb C} \newcommand{\N}{\mathbb N P^{2}}$ is an immersion and realizes equality for $p$, then $F|_{\mathbb C} \newcommand{\N}{\mathbb N P^{2}}$ is a homothety onto its image. \end{theorem}
\begin{proof}
Throughout the proof, we will let $I$ denote the complex structure of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. By (\ref{norm_squared_formula}),
\begin{equation}
\label{cpn_pf_eqn_1}
\displaystyle E_{p}(F) = \int\limits_{\mathbb C} \newcommand{\N}{\mathbb N P^{N}} |dF_{x}|^{p} dVol_{\widetilde{g}} = \left( \frac{2N}{\sigma(2N-1)} \right)^{\frac{p}{2}} \int\limits_{\mathbb C} \newcommand{\N}{\mathbb N P^{N}} \left( \int\limits_{U_{x}\mathbb C} \newcommand{\N}{\mathbb N P^{N}} |dF(\vec{u})|^{2} d\vec{u} \right)^{\frac{p}{2}} dVol_{\widetilde{g}}. \medskip
\end{equation}
For $x \in \mathbb C} \newcommand{\N}{\mathbb N P^{N}$, let $G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)$ be the space of complex lines in $T_{x}\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. $G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)$ is then a copy of $\mathbb C} \newcommand{\N}{\mathbb N P^{N-1}$, and the fibration $\mathsf{T}:U(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, \widetilde{g}) \rightarrow \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ factors through a quotient mapping which is given fibrewise by $U_{x}(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)$. For $\Lambda \in G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)$, let $U(\Lambda,\widetilde{g})$ be the unit circle in $\Lambda$ in the metric $\widetilde{g}$. For each such $\Lambda$ at each point $x$ where $F$ is differentiable, $F^{*}g|_{\Lambda}$ is a positive semidefinite, symmetric $2$-form and can be diagonalized relative to the metric $\widetilde{g}|_{\Lambda}$. Letting $\vec{u}_{1}, \vec{u}_{2}$ be a set of eigenvectors for $F^{*}g|_{\Lambda}$ which are orthonormal in $\widetilde{g}$, by the arithmetic-geometric mean inequality,
\begin{equation*}
\displaystyle \int\limits_{U(\Lambda,\widetilde{g})} |dF(\vec{u})|^{2} d\vec{u} = \int\limits_{0}^{2\pi} \left( \cos^{2}(\theta) |dF(\vec{u}_{1})|^{2} + \sin^{2}(\theta) |dF(\vec{u}_{2})|^{2} \right) d\theta \medskip
\end{equation*}
\begin{equation}
\label{cpn_pf_eqn_2}
\displaystyle = \pi \left( |dF(\vec{u}_{1})|^{2} + |dF(\vec{u}_{2})|^{2} \right) \geq 2\pi |dF(\vec{u}_{1})||dF(\vec{u}_{2})| = 2\pi |\det(dF|_{\Lambda})|. \bigskip
\end{equation}
For $p = 2$, (\ref{cpn_pf_eqn_1}), (\ref{cpn_pf_eqn_2}) and Fubini's theorem imply:
\begin{equation*}
\displaystyle E_{2}(F) = \frac{2N}{\sigma(2N-1)} \int\limits_{\mathbb C} \newcommand{\N}{\mathbb N P^{N}} \int\limits_{G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)} \int\limits_{U(\Lambda,\widetilde{g})} |dF(\vec{u})|^{2} \ d\vec{u} \ d\Lambda \ dVol_{\widetilde{g}} \bigskip
\end{equation*}
\begin{equation}
\label{cpn_pf_eqn_3}
\displaystyle \geq \frac{4N\pi}{\sigma(2N-1)} \int\limits_{\mathbb C} \newcommand{\N}{\mathbb N P^{N}} \int\limits_{G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)} |\det(dF|_{\Lambda})| d\Lambda dVol_{\widetilde{g}} = \frac{4N\pi}{\sigma(2N-1)} \int\limits_{\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})} |F(\mathcal{P})| d\widetilde{\mathcal{M}}, \bigskip
\end{equation}
where $|F(\mathcal{P})|$ is the area of the image via $F$ of a complex projective line $\mathcal{P}$ in $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ and where the measure on $G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)$ is its canonical measure as a quotient of the unit sphere $U_{x}(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g})$. As in the proof of Theorem \ref{rpn_thm}, we have used the fact that $F|_{\mathcal{P}}$ is Lipschitz, so that $F(\mathcal{P})$ represents a rectifiable $2$-current with a well-defined mass $|F(\mathcal{P})|$, for all $\mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$. \\
For $p > 2$, (\ref{cpn_pf_eqn_1}), (\ref{cpn_pf_eqn_2}), Fubini's theorem and H\"older's inequality imply:
\begin{equation}
\label{cpn_pf_eqn_4}
\displaystyle E_{p}(F) \geq \frac{(4N\pi)^{\frac{p}{2}}}{\sigma(2N-1)^{\frac{p}{2}}Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{N})^{\frac{p-2}{2}}} \left( \int\limits_{\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})} |F(\mathcal{P})| d\widetilde{\mathcal{M}} \right)^{\frac{p}{2}}. \medskip
\end{equation}
Since $|F(\mathcal{P})| \geq A^{\star}$, (\ref{cpn_pf_eqn_3}) and (\ref{cpn_pf_eqn_4}) imply:
\begin{equation}
\label{cpn_pf_eqn_5}
\displaystyle E_{p}(F) \geq \frac{(4N\pi)^{\frac{p}{2}}}{\sigma(2N-1)^{\frac{p}{2}}Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{N})^{\frac{p-2}{2}}} \left( A^{\star} Vol(\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, g_{0})) \right)^{\frac{p}{2}}, \medskip
\end{equation}
which is (\ref{cpn_thm_eqn}). \\
Suppose equality holds for $p=2$. \\
$F$ is therefore smooth. Equality holds in (\ref{cpn_pf_eqn_2}) for all $\Lambda \in G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)$, at all $x \in \mathbb C} \newcommand{\N}{\mathbb N P^{N}$. Equality in (\ref{cpn_pf_eqn_2}) implies that $|dF(\vec{u})|$ is $S^{1}$-invariant on $\Lambda$, where $S^{1}$ represents the unit complex numbers acting by multiplication on $T_{x}\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. $F^{*}g$ is therefore a positive semidefinite Hermitian bilinear form on $T_{x}\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. In particular, $F^{*}g$ is a Hermitian metric on any neighborhood of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ on which $F$ is an immersion. We also have that $|F(\mathcal{P})| = A^{\star}$ for all $\mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ and, by Lemma \ref{energy_formula_lemma}, that $F|_{\mathcal{P}}$ minimizes energy in its homotopy class of mappings $F:\mathbb C} \newcommand{\N}{\mathbb N P^{1} \rightarrow (M,g)$ for all $\mathcal{P} \in \mathcal{L}$. \\
Now let $\mathcal{V}$ be a neighborhood of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ such that $F|_{\mathcal{V}}$ is an immersion. We will show that $F(\mathcal{V})$ is a minimal submanifold of $(M,g)$, whose second fundamental form can be diagonalized by a unitary basis of $F^{*}g$. \\
For $x_{0} \in \mathcal{V}$, let $\vec{n}$ be a unit normal vector in $(M,g)$ to $F(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ at $F(x_{0})$. Let $S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}$ be the shape operator of $F(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ in the normal direction $\vec{n}$. Let $\vec{u}_{0}$ be a principal vector for $S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}$, with $|S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}(\vec{u}_{0})|$ maximal, and let $\mathcal{P}_{0} = \mathsf{T}(\vec{u}_{0})$. Let $S_{\vec{n}}^{\mathcal{P}_{0}}$ be the shape operator of $\mathcal{P}_{0}$ in the normal direction $\vec{n}$. Then $\vec{u}_{0}$ is also a principal vector for $S_{\vec{n}}^{\mathcal{P}_{0}}$ \\
$F(\mathcal{P}_{0} \cap \mathcal{V})$ is minimal in $(M,g)$ because $F(\mathcal{P}_{0})$ minimizes area in its homotopy class of mappings $\mathbb C} \newcommand{\N}{\mathbb N P^{1} \rightarrow (M,g)$. By the minimality of $F(\mathcal{P}_{0} \cap \mathcal{V})$ and the fact that the metric $F^{*}g$ on $\mathcal{V}$ is Hermitian, $I(\vec{u}_{0})$ is also a principal vector for $S_{\vec{n}}^{\mathcal{P}_{0}}$ and $g\left(S_{\vec{n}}^{\mathcal{P}_{0}}(I(\vec{u}_{0})),I(\vec{u}_{0})\right) = -g \left(S_{\vec{n}}^{\mathcal{P}_{0}}(\vec{u}_{0}),\vec{u}_{0}\right)$. In particular, $|S_{\vec{n}}^{\mathcal{P}_{0}}(I(\vec{u}_{0}))| = |S_{\vec{n}}^{\mathcal{P}_{0}}(\vec{u}_{0})|$. Because $|S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}(I(\vec{u}_{0}))| \geq |S_{\vec{n}}^{\mathcal{P}_{0}}(I(\vec{u}_{0}))|$ and $|S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}(\vec{u}_{0})|$ is maximal, this implies that $I(\vec{u}_{0})$ is also a principal vector for $S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}$, and that the principal curvature of $S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}$ along $I(\vec{u}_{0})$ is the negative of its principal curvature along $\vec{u}_{0}$. \\
Now let $\vec{u}_{1}$ be a principal vector for $S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}$ which maximizes $|S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}(\vec{u}_{1})|$ in the subspace of $T_{x_{0}}\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ which is orthogonal via $F^{*}g$ to $span(\vec{u}_{0},I(\vec{u}_{0}))$ and let $\mathcal{P}_{1}$ be the linear $\mathbb C} \newcommand{\N}{\mathbb N P^{1}$ to which $\vec{u}_{1}$ is tangent. As above, $\vec{u}_{1}$ is a principal vector for $S_{\vec{n}}^{\mathcal{P}_{1}}$, which implies that $I(\vec{u}_{1})$ is a principal vector for $S_{\vec{n}}^{\mathcal{P}_{1}}$ whose principal curvature is the negative of the principal curvature of $\vec{u}_{1}$, and therefore that $\vec{u},I(\vec{u})$ are principal vectors of $S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}$ whose principal curvatures are equal in magnitude and opposite in sign. Continuing in this way diagonalizes $S_{\vec{n}}^{\mathbb C} \newcommand{\N}{\mathbb N P^{N}}$ with a unitary basis for $T_{x_{0}}\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. \\
Now let $\nabla^{*}$ and $\omega^{*}$ be the connection and K\"ahler form of $F^{*}g$ on $\mathcal{V}$, that is, for $\vec{v},\vec{w} \in T_{x_{0}}\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, $\omega^{*}(\vec{v},\vec{w}) = F^{*}g(I(\vec{v}),\vec{w})$. Let $\vec{v}$ be a unit tangent vector to $(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, F^{*}g)$ at a point $x_{0}$ of $\mathcal{V}$ and let $\vec{w}, I(\vec{w})$ be a pair of unit vectors which span a complex $1$-dimensional subspace orthogonal to $\vec{v}$ in $T_{x_{0}}\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, where the norm and orthogonality of $\vec{v}$, $\vec{w}$, $I(\vec{w})$ are again with respect to $F^{*}g$. Let $\mathsf{T}(\vec{w}) = \mathsf{T}(I(\vec{w})) = \mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$. Choose a system of normal coordinates for $\mathcal{P}$ about $x_{0}$, in the metric $F^{*}g$, based on the frame $\vec{w}, I(\vec{w})$. Extend $\vec{v}$ to an orthonormal frame for the normal space to $\mathcal{P}$ at $x_{0}$, then extend this frame to an orthonormal frame field for the normal bundle of $\mathcal{P}$ in $M$, on the normal coordinate neighborhood of $\mathcal{P}$ defined above. Use this local orthonormal frame field to define a system of Fermi coordinates about $x_{0}$ in $(\mathcal{V},F^{*}g)$. Let $V$ be the coordinate vector field which coincides with $\vec{v}$ and $W_{1}, W_{2}$ the coordinate vector fields which coincide with $\vec{w},I(\vec{w})$ at $x_{0}$. We then have:
\begin{equation*}
\displaystyle d\omega^{*}(\vec{v}, \vec{w}, I(\vec{w})) = V(\omega^{*}(W_{1},W_{2})) - W_{1}(\omega^{*}(V,W_{2})) + W_{2}(\omega^{*}(V,W_{1})) \bigskip
\end{equation*}
\begin{equation}
\label{cpn_pf_eqn_6}
\displaystyle = V(F^{*}g(I(W_{1}),W_{2})) - W_{1}(F^{*}g(I(V),W_{2})) + W_{2}(F^{*}g(I(V),W_{1})). \bigskip
\end{equation}
Because $W_{1},W_{2}$ are tangent and $V$ is normal to to the complex submanifold $\mathcal{P}$ of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ and $F^{*}g$ is Hermitian, the terms $W_{1}(\omega^{*}(V,W_{2}))$ and $W_{2}(\omega^{*}(V,W_{1}))$ vanish, and we have:
\begin{equation}
\label{cpn_pf_eqn_7}
\displaystyle d\omega^{*}(\vec{v}, \vec{w}, I(\vec{w})) = V(\omega^{*}(W_{1},W_{2})) = V(F^{*}g(I(W_{1}),W_{2})). \bigskip
\end{equation}
Although $I(W_{1})$ may not be equal to $W_{2}$ at $x \neq x_{0}$, by (\ref{cpn_pf_eqn_7}),
\begin{equation*}
\displaystyle d\omega^{*}(\vec{v}, \vec{w}, I(\vec{w})) = F^{*}g(\nabla^{*}_{V}I(W_{1}),I(W_{1})) + F^{*}g(W_{2},\nabla^{*}_{V}W_{2}) \bigskip
\end{equation*}
\begin{equation}
\label{cpn_pf_eqn_8}
\displaystyle = \frac{1}{2} V\left( F^{*}g(I(W_{1}),I(W_{1})) + F^{*}g(W_{2},W_{2}) \right). \bigskip
\end{equation}
Using the fact that $F^{*}g$ is Hermitian, so that $F^{*}g(I(W_{1}),I(W_{1})) \equiv F^{*}g(W_{1},W_{1})$, and that $\nabla^{*}_{V}W_{1} - \nabla^{*}_{W_{1}}V = [V,W_{1}] = 0$, $\nabla^{*}_{V}W_{2} - \nabla^{*}_{W_{2}}V = [V,W_{2}] = 0$ because $V,W_{1},W_{2}$ are coordinate vector fields from the same Fermi coordinate system defined above, we then have:
\begin{equation*}
\displaystyle d\omega^{*}(\vec{v}, \vec{w}, I(\vec{w})) = F^{*}g(\nabla^{*}_{V}W_{1},W_{1}) + F^{*}g(\nabla^{*}_{V}W_{2},W_{2}) \bigskip
\end{equation*}
\begin{equation}
\displaystyle = F^{*}g(\nabla^{*}_{W_{1}}V,W_{1}) + F^{*}g(\nabla^{*}_{W_{2}}V,W_{2}). \bigskip
\end{equation}
This is the negative of the mean curvature of $\mathcal{P}$ at $x_{0}$ in the normal direction determined by $\vec{v}$ in the metric $F^{*}g$. Because $F(\mathcal{P} \cap \mathcal{V})$ is minimal, $d\omega^{*}$ vanishes on $\vec{v}, \vec{w}, I(\vec{w})$, and therefore on any subspace of $T_{x_{0}}\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ of complex dimension $2$. \\
Now suppose $p > 2$ and equality holds for $p$. \\
Assuming only that $F$ is Lipschitz, equality then holds in (\ref{cpn_pf_eqn_2}) for almost all $\Lambda \in G_{1}^{\mathbb C} \newcommand{\N}{\mathbb N}(x)$, at almost all $x \in \mathbb C} \newcommand{\N}{\mathbb N P^{N}$. This implies that equality holds for $p=2$, and therefore that $F$ is smooth. Equality must also hold in H\"older's inequality in (\ref{cpn_pf_eqn_4}). This implies that $\int_{U_{x}\mathbb C} \newcommand{\N}{\mathbb N P^{N}} |dF(\vec{u})|^{2} d\vec{u}$ is a constant function of $F$, which implies that $F$ has constant energy density. \\
Suppose $\mathbb C} \newcommand{\N}{\mathbb N P^{2} \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{N}$ is a linear subspace, $p > 2$, $F|_{\mathbb C} \newcommand{\N}{\mathbb N P^{2}}$ is an immersion and realizes equality in (\ref{cpn_thm_eqn}) for $p$, and therefore for all $p \in [2,\infty)$. The equality for $p = 2$ implies that $F^{*}g$ is a K\"ahler metric on $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$. Letting $\omega^{*}$ be the K\"ahler form of $F^{*}g$ as above, for all $\mathbb C} \newcommand{\N}{\mathbb N P^{1} \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{2}$, $\int_{\mathbb C} \newcommand{\N}{\mathbb N P^{1}}\omega^{*} = A^{\star}$. Letting $\widetilde{\omega}$ be the K\"ahler form of $\widetilde{g}$ on $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$, this implies that $\omega^{*}$ is cohomologous to $(\frac{A^{\star}}{\pi})\widetilde{\omega}$, and therefore that $Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{2},F^{*}g) = \frac{A^{\star^{2}}}{2}$. For $p >4$, the lower bound in Theorem \ref{cpn_thm} therefore coincides with the lower bound in Lemma \ref{elementary_lemma}. Because equality in (\ref{cpn_thm_eqn}) holds for all $p \geq 2$, including $p > 4$, $F$ realizes equality in Lemma \ref{elementary_lemma} for $p > 4$ and is therefore a homothety onto its image. \end{proof}
\begin{remark}
The proof Theorem \ref{cpn_thm} includes a proof of the following fact: if $h$ is a Hermitian metric on a complex surface $X$ such that all complex curves in $X$ are minimal, then $h$ is a K\"ahler metric. In fact, one need only assume that all complex lines $\Pi \subset T_{p}X$ are tangent to a complex curve which is minimal in $(X,h)$. The author does not know of a reference for this fact.
\end{remark}
We note that Ohnita \cite{Oh1} has proven that if $\phi:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},g_{0}) \rightarrow (M,g)$ is a stable harmonic map from $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ with its canonical metric $g_{0}$ to any Riemannian manifold $(M,g)$, then $\phi$ is pluriharmonic -- for continuous maps, this is equivalent to the statement that $\phi|_{\Sigma}$ is harmonic for all complex curves $\Sigma \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{N}$. \\
For $\mathbb C} \newcommand{\N}{\mathbb N P^{1}$, the lower bound in Theorem \ref{cpn_thm} coincides with the bound implied by Lemma \ref{elementary_lemma}, so equality for $p > 2$ for $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{1},\widetilde{g}) \rightarrow (M,g)$ in Theorem \ref{cpn_thm} implies that $F$ is a homothety onto its image. For mappings of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$, $N \geq 2$, let $Vol_{\widetilde{g}}(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, F^{*}g)$ be the invariant of the mapping $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow (M,g)$ as defined in Lemma \ref{elementary_lemma}. We can then ask whether $Vol_{\widetilde{g}}(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, F^{*}g)$ satisfies the following inequality along the lines of Pu's and Gromov's inequalities in Theorems \ref{pu_thm} and \ref{gromov_thm}:
\begin{equation}
\label{pu_gromov_type_inequality}
\displaystyle Vol_{\widetilde{g}}(\mathbb C} \newcommand{\N}{\mathbb N P^{N}, F^{*}g) \geq \frac{A^{\star^{N}}}{N!}. \bigskip
\end{equation}
For mappings for which (\ref{pu_gromov_type_inequality}) holds, the lower bound for $E_{p}(F)$ in Lemma \ref{elementary_lemma} is bounded below by the lower bound in Theorem \ref{cpn_thm}. Mappings which realize equality in Theorem \ref{cpn_thm} for $p > 2$ and satisfy (\ref{pu_gromov_type_inequality}) must therefore be homotheties. However, it is known that one cannot replace the limit (\ref{stable_2_systole}) in Theorem \ref{gromov_thm} by the minimum area of a current representing a generator of $H_{2}(\mathbb C} \newcommand{\N}{\mathbb N P^{N};\mathbb Z)$, so (\ref{pu_gromov_type_inequality}) need not hold. In fact, there are Riemannian metrics on $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ of arbitrarily small volume for which the minimum area of a cycle generating $H_{2}(\mathbb C} \newcommand{\N}{\mathbb N P^{N};\mathbb Z)$ is $1$. This is an example of a phenomenon known as systolic freedom and is discussed in \cite{CK1}. \\
Also, unlike in the proof of Theorem \ref{rpn_thm} for real projective space, where one knows that all length-minimizing paths in a free homotopy class are smoothly immersed closed geodesics, the infima of the area and energy in a homotopy class of mappings $f:S^{2} \rightarrow (M,g)$, even if they are realized, may not be realized by a smooth immersion. This is discussed by Sacks and Uhlenbeck in \cite{SU}, where they prove that if $(M,g)$ is a compact Riemannian manifold with $\pi_{2}(M) \neq 0$, one can find a generating set for $\pi_{2}(M)$ which consists of conformal, branched minimal immersions of $S^{2}$ which minimize area and energy in their homotopy classes. \\
When one has more information about the regularity of mappings which minimize area and energy in the homotopy class of $F(\mathbb C} \newcommand{\N}{\mathbb N P^{1})$ in Theorem \ref{cpn_thm}, one may be able to draw stronger conclusions about the regularity of energy-minimizing maps $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow (M,g)$. This is the basis for our proof of Theorem \ref{cpn_holom_thm}. The conclusion of Theorem \ref{cpn_holom_thm} for mappings which realize equality for $p=2$, that they must be holomorphic or antiholomorphic, will then allow us to conclude that mappings which realize equality for $p > 2$ in Theorem \ref{cpn_holom_thm} must be homotheties, by an argument similar to the argument sketched above for mappings which satisfy (\ref{pu_gromov_type_inequality}).
\begin{proof}[Proof of Theorem \ref{cpn_holom_thm}]
Let $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},\widetilde{g}) \rightarrow (X^{d},h)$ be a Lipschitz mapping to a compact, simply connected K\"ahler manifold, and suppose equality holds in Theorem \ref{cpn_thm} for $p = 2$. This implies $F$ is smooth. By the Hurewicz theorem, the natural mapping $\pi_{2}(X) \rightarrow H_{2}(X;\mathbb Z)$ is an isomorphism, so the family of mappings $f:S^{2} \rightarrow X$ which is homotopic to $F|_{\mathbb C} \newcommand{\N}{\mathbb N P^{1}}$ coincides with the family which is homologous. Because $F_{*}([\mathbb C} \newcommand{\N}{\mathbb N P^{1}]) \in H_{2}(X;\mathbb Z)$ can be represented by a rational curve, the infimum $A^{\star}$ of the areas in this homotopy class is in fact a minimum and is realized exclusively by cycles which are calibrated by the K\"ahler form of $(X,h)$. By Lemma \ref{energy_formula_lemma}, for all $\mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$, $F|_{\mathcal{P}}$ must be area-minimizing and energy-minimizing in its homotopy class and must therefore be a holomorphic or antiholomorphic map to a complex curve in $(X,h)$. Letting $\omega_{h}$ be the K\"ahler form of $(X,h)$, $\int_{\mathcal{P}} F^{*}\omega_{h}$ is constant on $\mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$. $F|_{\mathcal{P}}$ is either holomorphic for all $\mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$ or antiholomorphic for all $\mathcal{P} \in \mathcal{L}(\mathbb C} \newcommand{\N}{\mathbb N P^{N})$, depending on whether this constant is positive or negative. \\
Reversing the complex structure of $(X,h)$ if necessary, we can suppose $F|_{\mathcal{P}}$ is holomorphic. Choose affine coordinates $(z^{1}, z^{2}, \cdots, z^{N})$ on a neighborhood $\mathcal{U}$ of a point $x_{0}$ in $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$ and holomorphic coordinates on a neighborhood of $F(x_{0})$ in $X$. For each $i = 1, 2, \dots, N$ and each fixed $z_{0}^{1}, z_{0}^{2}, \dots, z_{0}^{i-1}, z_{0}^{i+1}, \dots, z_{0}^{N}$, the affine line $(z_{0}^{1}, \dots, z^{i}, \dots, z_{0}^{N})$ in these coordinates on $\mathcal{U}$ corresponds to a complex $2$-plane through the origin in $\mathbb C} \newcommand{\N}{\mathbb N^{N+1}$, via the association $(z^{1}, z^{2}, \cdots, z^{N}) \rightarrow [1:z^{1}:z^{2}:\cdots :z^{N}]$ of affine and homogeneous coordinates on $\mathcal{U}$, and thus to a complex projective line $\mathcal{P}$ in $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. Because $F|_{\mathcal{P}}$ is holomorphic, each of the functions $F_{1}, F_{2}, \dots, F_{d}$ representing $F$ in these coordinates on $X$ is holomorphic when restricted to any such affine line in $\mathcal{U}$. Osgood's Lemma \cite{GR_0} then implies that $F$ is holomorphic on $\mathcal{U}$, and therefore that $F$ is a holomorphic mapping of $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. \\
If $p > 2$ and equality holds for $p$, then by Theorem \ref{cpn_thm}, equality holds for all $p \in [2,\infty)$. Equality for $p=2$ implies that, reversing orientation if necessary, $F$ is holomorphic. Because $\int_{\mathbb C} \newcommand{\N}{\mathbb N P^{1}}F^{*}\omega_{h} = A^{\star}$ for linear $\mathbb C} \newcommand{\N}{\mathbb N P^{1} \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{N}$, $F^{*}\omega_{h}$ is cohomologous to $(\frac{A^{\star}}{\pi})\widetilde{\omega}$, where $\widetilde{\omega}$ again represents the K\"ahler form of $\widetilde{g}$. This implies that, in the notation of Lemma \ref{elementary_lemma}, $Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{N},F^{*}h) = \frac{A^{\star^{N}}}{N!}$, and that for $p>2N$, the lower bound in Theorem \ref{cpn_thm} coincides with the lower bound in Lemma \ref{elementary_lemma}. This implies that $F$ realizes equality in Lemma \ref{elementary_lemma} for $p > 2N$, and therefore that $F$ is a homothety onto its image. \end{proof}
We note that Burns, Burstall, de Bartolomeis and Rawnsley \cite{BBdBR1} have proven that any stable harmonic map $\phi:(\mathbb C} \newcommand{\N}{\mathbb N P^{N},g_{0}) \rightarrow (\mathcal{Z},h)$ to a compact, simple Hermitian symmetric space $(\mathcal{Z},h)$ is holomorphic or antiholomorphic. This generalizes the result of Ohnita \cite{Oh1} for mappings $F:(\mathbb C} \newcommand{\N}{\mathbb N P^{N_{1}},g_{0}) \rightarrow (\mathbb C} \newcommand{\N}{\mathbb N P^{N_{2}},g_{0})$ mentioned in the introduction.
\section{Quaternionic Projective Space}
\label{quaternions}
In this section, we will prove Theorem \ref{hpn_thm}. In the proof of Theorem \ref{hpn_thm}, we will make use of the twistor fibration $\Psi: \mathbb C} \newcommand{\N}{\mathbb N P^{2N+1} \rightarrow \mathbb{H}P^{N}$. This mapping gives a parametrization by $\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}$ of the complex structures on tangent spaces to $\mathbb{H}P^{N}$ which satisfy a local compatibility condition with the the canonical metric $g_{0}$. We will describe this below, emphasizing the features which we will use, and refer to \cite{Sa1} and \cite[Ch. 14]{Be2} for more background. The results in Theorem \ref{hpn_thm} for $(\mathbb{H} P^{N},g_{0})$, $N \geq 2$ are different from the corresponding results for $(\mathbb{H} P^{1},g_{0})$, which is isometric to a rescaling of $(S^{4},g_{0})$. These results for $(\mathbb{H} P^{1},g_{0})$ therefore follow from Lemma \ref{elementary_lemma}, and throughout the discussion below we will restrict to the case $N \geq 2$. \\
We fix a basis $1, I, J, K$ for $\mathbb{H}$ and a Euclidean inner product on $\mathbb{H}^{N+1}$, which we view as a right $\mathbb{H}$-module. Throughout, we will identify $\mathbb C} \newcommand{\N}{\mathbb N$ with the subfield $\mathbb R} \newcommand{\T}{\mathbb T + \mathbb R} \newcommand{\T}{\mathbb T I$ of $\mathbb{H}$. In this way we will view $\mathbb{H}^{N+1}$ and all of its $\mathbb{H}$-submodules as complex vector spaces. Letting $U(1)$ denote the unit complex numbers and $Sp(1)$ the group of unit quaternions, we identify $\mathbb{H} P^{N}$ with the quotient $\mathbb{H}^{N+1} / \mathbb{H}^* = S^{4N + 3} / Sp(1)$, and we identify $\mathbb C} \newcommand{\N}{\mathbb N P^{2N + 1}$ with $\mathbb{H}^{N+1} / \mathbb C} \newcommand{\N}{\mathbb N^* = S^{4N + 3} / U(1)$. Given a $1$-dimensional $\mathbb{H}$-submodule of $\mathbb{H}^{N+1}$, we will write $l$ to denote both the associated point in $\mathbb{H} P^{N}$ and the subspace of $\mathbb{H}^{N+1}$. We will write $Z_{l}$ for the fibre of the twistor fibration $\Psi : \mathbb C} \newcommand{\N}{\mathbb N P^{2N + 1} \rightarrow \mathbb{H} P^{N}$, which can be defined as follows: \\
The choice of a point $p$ in the unit sphere $S^{3}$ in $l \subseteq \mathbb{H}^{N+1}$ determines an $\mathbb R} \newcommand{\T}{\mathbb T$-linear isomorphism between the subspace $l^{\perp}$ orthogonal to $l$ in $\mathbb{H}^{N+1}$ and the tangent space $T_{l} \mathbb{H} P^{N}$ to $\mathbb{H} P^{N}$ at $l$, via the differential $d\Phi_{p}$ of the Hopf fibration $\Phi : S^{4N + 3} \rightarrow \mathbb{H} P^{N}$. This isomorphism can be used to transfer the $\mathbb C} \newcommand{\N}{\mathbb N$-vector space and $\mathbb{H}$-module structures of $l^{\perp} \subseteq \mathbb{H}^{N+1}$ to $T_{l} \mathbb{H} P^{N}$, but the structures induced on $T_{l} \mathbb{H} P^{N}$ in this way depend on the choice of the point $p$. In particular, because the action of $Sp(1)$ on $l^{\perp}$ is not $U(1)$-equivariant, the complex structure on $T_{l} \mathbb{H} P^{N}$ induced in this way depends on $p$. The subgroup of $Sp(1)$ which acts $U(1)$-equivariantly is $U(1)$ itself. The complex structures induced on $T_{l} \mathbb{H} P^{N}$ in this way are therefore parametrized by $Sp(1)/U(1)$, which is canonically identified with the projectivization of $l \cong \mathbb C} \newcommand{\N}{\mathbb N^{2}$ as a complex vector space. The collection of complex structures induced on $T_{l}\mathbb{H}P^{N}$ as above, paramatrized by $Sp(1)/U(1)$, is the fibre $Z_{l}$ at $l$ of the twistor fibration $\Psi:\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1} \rightarrow \mathbb{H} P^{N}$. Viewing $\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}$ and $\mathbb{H}P^{N}$ as quotients of $S^{4N+3}$ by $U(1)$, resp. $Sp(1)$, the twistor fibration corresponds to the mapping $S^{4N+3}/U(1) \rightarrow S^{4N+3}/Sp(1)$. The canonical metrics on $\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}$ and $\mathbb{H} P^{N}$ are the base metrics of Riemannian submersions from $(S^{4N+3},g_{0})$ and the twistor fibration is itself a Riemannian submersion $(\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1},g_{0}) \rightarrow (\mathbb{H} P^{N},g_{0})$, with totally geodesic fibres $\mathbb C} \newcommand{\N}{\mathbb N P^{1} \subseteq \mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}$ isometric to round $2$-spheres with curvature $4$ and area $\pi$. \\
Later, it will important that although the induced $\mathbb C} \newcommand{\N}{\mathbb N$ vector space and $\mathbb{H}$-module structures on $T_{l} \mathbb{H} P^{N}$ depend on $p$, the $1$-dimensional $\mathbb{H}$-submodule of $T_{l} \mathbb{H} P^{N}$ determined by a tangent vector $v \in T_{l} \mathbb{H} P^{N}$ is well-defined independent of $p$. Likewise, for a complex structure $\tau \in Z_{l}$ and a complex $1$-dimensional subspace $\lambda$ of the complex vector space $(T_{l}\mathbb{H}P^{N},\tau)$, the $1$-dimensional $\mathbb{H}$-submodule of $T_{l} \mathbb{H} P^{N}$ containing $\lambda$ is independent of $p$. We will denote the $1$-dimensional quaternionic subspace determined by $v \in T_{l} \mathcal{H} P^{N}$ by $v\mathbb{H}$ and the subspace determined by a complex line $\lambda$ in $(T_{l}\mathbb{H}P^{N},\tau)$ by $\lambda\mathbb{H}$. \\
For $\tau \in Z_{l}$, we define the following family of bases of the complex vector space $(T_{l}\mathbb{H}P^{N},\tau)$:
\begin{definition}
\label{goodframes}
Given a point $l$ in $\mathbb{H} P^{N}$ and $\tau \in Z_{l}$, let $\mathcal{F}_{l}(\tau)$ be the set of ordered orthonormal frames $e_{1}, e_{2}, e_{3}, \cdots, e_{4N-1}, e_{4N}$ for $T_{l}\mathbb{H}P^{N}$ such that:
\bigskip
\begin{enumerate}
\item $e_{2i} = \tau(e_{2i-1})$ for all $i = 1, 2, \cdots, 2N$.
\bigskip
\item $e_{4j-3}, e_{4j-2}, e_{4j-1}, e_{4j}$ span a quaternionic line in $T_{l}\mathbb{H}P^{N}$ for each $j = 1, 2, \cdots, N$.
\bigskip
\end{enumerate}
\end{definition}
We will write $\mathcal{F}(N)$ for the volume of the space of frames $\mathcal{F}_{l}(\tau)$ in Definition \ref{goodframes}, when viewed as a subset of the Steifel manifold of frames for $\mathbb R} \newcommand{\T}{\mathbb T^{4N}$. We then have:
\begin{equation}
\displaystyle \mathcal{F}(1) = \sigma(3)\sigma(1) = 4\pi^{3}, \medskip
\end{equation}
and for $N \geq 2$,
\begin{equation}
\displaystyle \mathcal{F}(N) = Vol(\mathbb{H}P^{N-1})\mathcal{F}(N-1)\mathcal{F}(1). \bigskip
\end{equation}
By convention, we define $\mathcal{F}(0) = 1$. Using that the volume of $\mathbb{H}P^{N}$ in the metric $g_{0}$ is equal to $\frac{\pi^{2N}}{(2N+1)!}$, we then have:
\begin{equation}
\label{Vol_F_formula}
\displaystyle \mathcal{F}(N) = \frac{\pi^{N(N+2)}4^{N}}{\prod_{j=1}^{N-1}(2j+1)!}. \bigskip
\end{equation}
It will also be helpful to note that:
\begin{equation}
\label{Vol_F_ratios}
\displaystyle \frac{\mathcal{F}(1)\mathcal{F}(N-1)}{\mathcal{F}(N)} = \frac{(2N-1)!}{\pi^{2N-2}}, \medskip
\end{equation}
\begin{equation}
\label{Vol_F_ratios_2}
\displaystyle \frac{\mathcal{F}(N-2)}{\mathcal{F}(N)} = \frac{(2N-2)!(2N-3)!}{16 \pi^{4N}}. \bigskip
\end{equation}
Given a mapping $F:(\mathbb{H}P^{N},g_{0}) \rightarrow (M^{m},g)$, the following formula for the $4$-energy of $F$ at a point $l \in \mathbb{H}P^{N}$ is an elementary consequence of (\ref{energy_frame_eqn}):
\begin{equation*}
\displaystyle e_{4}(F)_{l} = \frac{1}{\mathcal{F}(N)} \int\limits_{\mathcal{F}_{l}(\tau)} \left( \sum\limits_{i=1}^{2N} |dF(e_{2i-1})|^{2} + |dF(e_{2i})|^{2}\right)^{2} de \bigskip
\end{equation*}
\begin{equation}
\label{H_formula}
\displaystyle = \frac{1}{\pi\mathcal{F}(N)} \int\limits_{Z_{l}}\int\limits_{\mathcal{F}_{l}(\tau)} \left( \sum\limits_{i=1}^{2N} |dF(e_{2i-1})|^{2} + |dF(e_{2i})|^{2}\right)^{2} de \ d\tau. \bigskip
\end{equation}
\begin{definition}
\label{hyperlagrangian}
Given $l \in \mathbb{H}P^{N}$ and $\tau \in Z_{l}$ as above, let $\mathcal{C}_{2}(\tau)$ be the family of subspaces $V$ of $T_{l} \mathbb{H} P^{N}$ which are complex subspaces of dimension $2$ with respect to the complex structure $\tau$, with the following property: for any complex $1$-dimensional subspace $\lambda$ of $V$, the orthogonal complement of $\lambda$ in the quaternionic line $\lambda\mathbb{H}$ is also orthogonal to $V$.
\end{definition}
Each such subspace $V \in \mathcal{C}_{2}(\tau)$ again determines a well-defined subspace of $T_{l}\mathbb{H}P^{N}$ of real dimension $8$ which is an $\mathbb{H}$-submodule for any $\mathbb{H}$-module structure induced on $T_{l}\mathbb{H}P^{N}$ as above. We will denote this $V\mathbb{H}$. $\mathcal{C}_{2}(\tau)$ is the base of a fibration $\mathcal{F}_{l}(\tau) \rightarrow \mathcal{C}_{2}(\tau)$, which sends a frame $e_{1}, e_{2}, e_{3}, \cdots, e_{4N-1}, e_{4N}$ to $span(e_{1}, e_{2}, e_{5}, e_{6})$. We will equip $\mathcal{C}_{2}(\tau)$ with the measure pushed forward from $\mathcal{F}_{l}(\tau)$ via this mapping -- we then have:
\begin{equation}
\label{Vol_C_eqn}
\displaystyle Vol(\mathcal{C}_{2}(\tau)) = \frac{\mathcal{F}(N)}{4\pi^{2}\mathcal{F}(1)\mathcal{F}(N-2)} = \frac{(2N-2)!(2N-3)!}{256\pi^{4(N+1)+1}},
\end{equation}
where we have used (\ref{Vol_F_ratios}) and the fact that, for each frame $e_{1}, e_{2}, e_{5}, e_{6}$ for $V$, the volume of the space of frames $e_{3}, e_{4}, e_{7}, e_{8}$ for $V^{\perp} \subseteq V\mathbb{H}$ which give a frame $e_{1}, e_{2}, \cdots, e_{7}, e_{8}$ for $V\mathbb{H}$ as in Definition \ref{goodframes} is equal to $4\pi^{2}$. \\
We will let $Gr_{1}^{\mathbb{H}}(l)$ denote the family of quaternionic $1$-dimensional subspaces of $T_{l}\mathbb{H}P^{N}$, that is, the family of $1$-dimensional $\mathbb{H}$-modules determined by $v \in T_{l}\mathbb{H} P^{N}$ -- this space is a copy of $\mathbb{H}P^{N-1}$. The fundamental pointwise result which is the basis for Theorem \ref{hpn_thm} is:
\begin{lemma}
\label{twistor_lemma}
Let $F:(\mathbb{H}P^{N},g_{0}) \rightarrow (M^{m},g)$ be a Lipschitz mapping and $l \in \mathbb{H}P^{N}$ a point at which $F$ is differentiable. Then:
\begin{equation}
\label{twistor_lemma_eqn}
\displaystyle e_{4}(F)_{l} \geq \scriptstyle \frac{16N^{2}(2N-2)!}{\pi^{2N-2}} \displaystyle \left( \int\limits_{Gr_{1}^{\mathbb{H}}(l)} |\det(dF|_{\Lambda})| d\Lambda + \scriptstyle \frac{(2N-2)!}{(2N-1)\pi^{2N-2}} \displaystyle \int\limits_{Z_{l}}\int\limits_{\mathcal{C}_{2}(\tau)} |\det(dF|_{V})| dV d\tau \right).
\end{equation}
Equality holds if and only $dF_{l}$ is a homothety.
\end{lemma}
\begin{proof}
We begin with the identity for $e_{4}(F)_{l}$ from (\ref{H_formula}). By Newton's inequality for symmetric functions (cf. \cite{ES1} p.113) and the arithmetic-geometric mean inequality,
\begin{equation*}
= \displaystyle e_{4}(F)_{l} \geq \frac{16N}{(2N-1)\pi\mathcal{F}(N)} \int\limits_{Z_{l}}\int\limits_{\mathcal{F}_{l}(\tau)} \sum\limits_{i=1}^{2N-1} \sum\limits_{j=i+1}^{2N} |dF(e_{2i-1})||dF(e_{2i})|dF(e_{2j-1})||dF(e_{2j})| de d\tau \bigskip
\end{equation*}
\begin{equation*}
\displaystyle = \frac{16N}{(2N-1)\pi\mathcal{F}(N)} \text{\Huge{[}} \int\limits_{Z_{l}}\int\limits_{\mathcal{F}_{l}(\tau)} \sum\limits_{i=1}^{N} |dF(e_{4i-3})||dF(e_{4i-2})||dF(e_{4i-1})||dF(e_{4i})| de d\tau
\end{equation*}
\begin{equation}
\label{twistor_lemma_pf_eqn_1}
\displaystyle + \int\limits_{Z_{l}}\int\limits_{\mathcal{F}_{l}(\tau)} \text{\Large{(}} \sum\limits_{j=1}^{N-1} \sum\limits_{k=j+1}^{N} |dF(e_{4j-3})||dF(e_{4j-2})||dF(e_{4k-3})||dF(e_{4k-2})|
\end{equation}
\begin{align*}
\displaystyle + \sum\limits_{j=1}^{N-1} \sum\limits_{k=j+1}^{N} |dF(e_{4j-3})||dF(e_{4j-2})||dF(e_{4k-1})||dF(e_{4k})| \\ + \sum\limits_{j=1}^{N-1} \sum\limits_{k=j+1}^{N} |dF(e_{4j-1})||dF(e_{4j})||dF(e_{4k-3})||dF(e_{4k-2})| \\ + \sum\limits_{j=1}^{N-1} \sum\limits_{k=j+1}^{N} |dF(e_{4j-1})||dF(e_{4j})||dF(e_{4k-1})||dF(e_{4k})| \text{\Large{)}} de d\tau \text{\Huge{]}}. \bigskip
\end{align*}
For each $\Lambda \in Gr_{1}^{\mathbb{H}}(l)$ and each orthonormal frame $e_{4i-3}, e_{4i-2}, e_{4i-1}, e_{4i}$ for $\Lambda$, we have:
\begin{equation}
\label{twistor_lemma_pf_eqn_2}
\displaystyle |dF(e_{4i-3})||dF(e_{4i-2})||dF(e_{4i-1})||dF(e_{4i})| \geq |\det(dF|_{\Lambda})|. \bigskip
\end{equation}
Therefore, for each $\tau \in Z_{l}$,
\begin{equation}
\label{twistor_lemma_pf_eqn_3}
\displaystyle \int\limits_{\mathcal{F}_{l}(\tau)} \sum\limits_{i=1}^{N} |dF(e_{4i-3})||dF(e_{4i-2})||dF(e_{4i-1})||dF(e_{4i})| de \bigskip \geq \scriptstyle N \mathcal{F}(1)\mathcal{F}(N-1) \displaystyle \int\limits_{Gr_{1}^{\mathbb{H}}(l)} |\det(dF|_{\Lambda})| d\Lambda, \bigskip
\end{equation}
Note that the constant $N \mathcal{F}(1)\mathcal{F}(N)$ in (\ref{twistor_lemma_pf_eqn_3}) is the product of the number of indices $i = 1, 2, \cdots, N$ in the summation, the volume $\mathcal{F}(1)$ of the space of frames for $\Lambda$ and the volume $\mathcal{F}(N-1)$ of the space of frames for $\Lambda^{\perp}$. \\
Similarly, for each $\tau \in Z_{l}$ and each $V \in \mathcal{C}_{2}(\tau)$, the space of frames $e_{4j-3}, e_{4j-2}, e_{4k-3}, e_{4k-2}$ for $V$ has volume $\sigma(3)\sigma(1) = 4\pi^{3}$. The volume of the space of frames $e_{4j-1}, e_{4j}, e_{4k-1}, e_{4k}$ for $V^{\perp} \subseteq V\mathbb{H}$ has volume $\sigma(1)^{2} = 4\pi^{2}$, the space of frames for $V\mathbb{H}^{\perp}$ has volume $\mathcal{F}(N-2)$ and there are $\binom{N}{2}$ terms in the summation $\sum_{j=1}^{N-1} \sum_{k=j+1}^{N}$ at which the frame may occur. The same is true for the other possible indexings of each frame, i.e. $e_{4j-3}, e_{4j-2}, e_{4k-1}, e_{4k}$; $e_{4j-1}, e_{4j}, e_{4k-3}, e_{4k-2}$ and $e_{4j-1}, e_{4j}, e_{4k-1}, e_{4k}$. For each $\tau \in Z_{l}$, the second integral term in (\ref{twistor_lemma_pf_eqn_1}) is therefore bounded below by:
\begin{equation}
\label{twistor_lemma_pf_eqn_4}
\displaystyle 32N(N-1)\pi^{5}\mathcal{F}(N-2) \int\limits_{\mathcal{C}_{2}(\tau)} |\det(dF|_{V})| dV. \bigskip
\end{equation}
Combining (\ref{twistor_lemma_pf_eqn_1}) with the lower bounds in (\ref{twistor_lemma_pf_eqn_3}) and (\ref{twistor_lemma_pf_eqn_4}), noting that the integration over $Z_{l}$ in (\ref{twistor_lemma_pf_eqn_1}) adds a factor $\pi$ to the constant in (\ref{twistor_lemma_pf_eqn_3}) and using the identities (\ref{Vol_F_ratios}) gives (\ref{twistor_lemma_eqn}). \\
Equality requires equality in Newton's inequality and the arithmetic-geometric mean inequality in (\ref{twistor_lemma_pf_eqn_1}). This implies that for all $\tau \in Z_{l}$, $\lbrace e_{1}, e_{2}, \dots, e_{4N} \rbrace \in \mathcal{F}_{l}\tau$, for all $i,j = 1, 2, \cdots 2N$, $|dF(e_{2i-1})|^{2} + |dF(e_{2i})|^{2} = |dF(e_{2j-1})|^{2} + |dF(e_{2j})|^{2}$, and that $|dF(e_{2i-1})| = |dF(e_{2i})|$. This implies $|dF(\vec{u})|$ is the same for all unit vectors $\vec{u}$ tangent to $\mathbb{H}P^{N}$ at $l$, so that $dF$ is a homothety on $T_{l}\mathbb{H}P^{N}$. \end{proof}
For each $\tau \in Z_{l}$ and each $V \in \mathcal{C}_{2}(\tau)$, $V$ is tangent to a totally geodesic submanifold of $\mathbb{H}P^{N}$ which is isometric to $\mathbb C} \newcommand{\N}{\mathbb N P^{2}$ with its canonical metric, cf. \cite[Ch. 5]{Be1}. In the following, we describe these totally geodesic subspaces and collect several of their properties which we will use in the proof of Theorem \ref{hpn_thm}:
\begin{lemma}
\label{tot_geo_lemma}
Let $\tau \in Z_{l}, V \in \mathcal{C}_{2}(\tau)$ as above.
\bigskip
\begin{enumerate}
\item Let $\lambda_{\tau}$ be the complex line in $l \cong \mathbb C} \newcommand{\N}{\mathbb N^{2} \subseteq \mathbb{H}^{N+1}$ such that $\lambda_{\tau} \cap S^{3} \subseteq l$ corresponds to $\tau$ by the twistor fibration. Let $p_{1}, p_{2} \in \lambda_{\tau} \cap S^{3}$, and let $V_{1}, V_{2} \subseteq l^{\perp}$ such that $d\Phi_{p_{i}}(V_{i}) = V$. Then $V_{1} = V_{2}$. We will denote this subspace of $l^{\perp}$ by $V_{\tau}$.
\bigskip
\item For each subspace $\lambda' \cong \mathbb C} \newcommand{\N}{\mathbb N$ in $V_{\tau} \oplus \lambda_{\tau} \cong \mathbb C} \newcommand{\N}{\mathbb N^{3} \subseteq \mathbb{H}^{N+1}$, the orthogonal complement $\lambda'^{\perp}$ to $\lambda'$ in $ \lambda'\mathbb{H}$ is orthogonal to $V_{\tau} \oplus \lambda_{\tau}$ in $\mathbb{H}^{N+1}$.
\bigskip
\item Let $\widetilde{X}_{\tau}$ be the image of $V_{\tau} \oplus \lambda_{\tau}$ in $\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}$ via the projectivization of $\mathbb C} \newcommand{\N}{\mathbb N^{2N+2} = \mathbb{H}^{N+1}$. Then $\widetilde{X}_{\tau}$ maps injectively and isometrically to $\mathbb{H}P^{N}$ via the twistor fibration. Its image is a totally geodesic submanifold of $\mathbb{H} P^{N}$ which we denote $X_{\tau}$.
\bigskip
\item $V$ is tangent to $X_{\tau}$ at $l$.
\bigskip
\item For each complex $1$-dimensional subspace $\lambda'$ of $V_{\tau} \oplus \lambda_{\tau}$, let $l' = \lambda'\mathbb{H} \subseteq \mathbb{H}^{N+1}$ and let $\tau'$ be the complex structure on $T_{l'}\mathbb{H} P^{N}$ associated to $\lambda'$ by the twistor fibration, by the identification of the twistor fibre with the projectivization of $l'$ as above. Then $T_{l'}X_{\tau}$ is $\tau'$-invariant. In particular, $\widetilde{X}_{\tau}$ is a section of the twistor fibration over $X_{\tau}$, and the complex structure on $T\mathbb{H} P^{N}|_{X_{\tau}}$ defined in this way preserves $T X_{\tau}$.
\bigskip
\end{enumerate}
\end{lemma}
We will write $\mathcal{C}(\mathbb{H}P^{N},g_{0})$ for the space of all such totally geodesic $X_{\tau}$ in $\mathbb{H}P^{N}$. We will equip $\mathcal{C}(\mathbb{H}P^{N},g_{0})$ with the measure pushed forward from the total space of the bundle over $\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}$ whose fibre over $\tau$ is $\mathcal{C}_{2}(\tau)$, via the natural fibration $V \mapsto X_{\tau}$ of this space over $\mathcal{C}(\mathbb{H}P^{N},g_{0})$. We then have:
\begin{equation}
\label{Vol_tot_geo_space}
\displaystyle Vol \left(\mathcal{C}(\mathbb{H}P^{N},g_{0})\right) = \frac{Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1},g_{0})Vol(\mathcal{C}_{2}(\tau))}{Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{2})} = \frac{(2N-3)!}{256N(4N^{2}-1)\pi^{2N+6}}. \bigskip
\end{equation}
We will let $\mathcal{H}(\mathbb{H}P^{N},g_{0})$ be the space of linearly embedded $\mathbb{H}P^{1} \subseteq \mathbb{H}P^{N}$, with the measure pushed forward from the unit tangent bundle $U(\mathbb{H}P^{N},g_{0})$. Then:
\begin{equation}
\displaystyle Vol\left(\mathcal{H}(\mathbb{H}P^{N},g_{0})\right) = \frac{Vol\left(U(\mathbb{H}P^{N},g_{0})\right)}{Vol\left(U(\mathbb{H}P^{1},g_{0})\right)} = \frac{6\sigma(4N-1)\pi^{2N-1}}{\sigma(3)(2N+1)!}. \bigskip
\end{equation}
The constant $K_{N}$ in Theorem \ref{hpn_thm} can be defined in terms of $\mathcal{H}(\mathbb{H}P^{N},g_{0})$ and $\mathcal{C}(\mathbb{H}P^{N},g_{0})$:
\begin{equation*}
\displaystyle K_{N} = \frac{16N^{2}(2N+1)!(2N-2)!}{\pi^{4N-2}} \left( Vol\left(\mathcal{H}(\mathbb{H}P^{N},g_{0})\right) + \frac{2(N-1)(2N-3)!}{(2N-1)\pi^{2N-2}} Vol \left(\mathcal{C}(\mathbb{H}P^{N},g_{0})\right) \right) \bigskip
\end{equation*}
\begin{equation}
\label{hpn_const_formula}
\displaystyle = \frac{32N^{2}(2N+1)!(2N-2)!}{\pi^{4N-2}} \left( \frac{3\sigma(4N-1)\pi^{2N-1}}{\sigma(3)(2N+1)!} + \frac{(N-1)(2N-3)!^{2}}{256N(2N-1)(4N^{2}-1)\pi^{4(N+1)}} \right). \bigskip
\end{equation}
Given a triple of complex structures $I,J,K$ on a neighborhood of $\mathbb{H}P^{N}$ which satisfy the quaternion relations $I^{2} = J^{2} = K^{2} = IJK = -Id$, we can form their associated K\"ahler forms $\omega_{I}$, $\omega_{J}$, $\omega_{K}$. The 4-form $\omega_{I}^{2} + \omega_{J}^{2} + \omega_{K}^{2}$ is independent of the choice of $I,J,K$. This form is therefore a local representation of a canonical, globally-defined closed 4-form on $\mathbb{H}P^{N}$, known as the fundamental 4-form or Kraines 4-form (cf. \cite{Be1,Kr1,Kr2}), whose powers generate the cohomology of $\mathbb{H}P^{N}$ and calibrate the linear subspaces $\mathbb{H}P^{k} \subseteq \mathbb{H}P^{N}$. \\
More precisely, we define $\Omega$ to be the form which coincides with $\frac{1}{\pi^{2}}\left( \omega_{I}^{2} + \omega_{J}^{2} + \omega_{K}^{2} \right)$ for any choice of $I,J,K$ as above. $\Omega$ is a closed, parallel form satisfying $\langle \Omega, \mathbb{H}P^{1} \rangle = 1$. In $H^{4}(\mathbb{H}P^{N};\mathbb R} \newcommand{\T}{\mathbb T)$, the cohomology class of $\Omega$ is the image of a generator of $H^{4}(\mathbb{H}P^{N};\mathbb Z)$ via the natural homomorphism $H^{4}(\mathbb{H}P^{N};\mathbb Z) \rightarrow H^{4}(\mathbb{H}P^{N};\mathbb R} \newcommand{\T}{\mathbb T)$. For any orthonormal frame $e,I(e),J(e),K(e)$ for a quaternionic line in $T_{l}\mathbb{H}P^{N}$, $\Omega(e,I(e),J(e),K(e)) = \frac{6}{\pi^{2}}$. More generally, $(\frac{\pi^{2}}{6})\Omega$ has comass $\equiv 1$ and gives a calibration of $(\mathbb{H}P^{N},g_{0})$, whose calibrated submanifolds are precisely the linearly embedded $\mathbb{H}P^{1}$ in $\mathbb{H}P^{N}$. The powers of $\Omega$, approriately rescaled, likewise give calibrations whose calibrated submanifolds are linear subspaces $\mathbb{H}P^{k} \subseteq \mathbb{H} P^{N}$. \\
If $X \in \mathcal{C}(\mathbb{H}P^{N},g_{0})$, then by choosing $I$ to coincide with the complex structure $\tau$ on $T_{l}X$ along $X$ as in Lemma \ref{tot_geo_lemma} (5), we have that for any orthonormal frame $e_{1}, I(e_{1}), e_{2}, I(e_{2})$ for $T_{l}X$, $\Omega(e_{1}, I(e_{1}), e_{2}, I(e_{2})) = \frac{2}{\pi^{2}}$. Since $Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{2}) = 3Vol(\mathbb{H}P^{1})$, this implies that $X$ also represents a generator of $H_{4}(\mathbb{H}P^{N};\mathbb Z)$ and is ``one third calibrated" by $\Omega$.
\begin{proof}[Proof of Theorem \ref{hpn_thm}]
Let $F:(\mathbb{H}P^{N},g_{0}) \rightarrow (M^{m},g)$ be a mapping as above and $p \geq 4$. By Lemma \ref{twistor_lemma},
\begin{equation*}
\displaystyle E_{p}(F) = \int\limits_{\mathbb{H}P^{N}} |dF|^{p} dVol_{g_{0}} \bigskip
\end{equation*}
\begin{equation}
\label{hpn_thm_pf_eqn_1}
\displaystyle \geq \scriptstyle \left( \frac{16N^{2}(2N-2)!}{\pi^{2N-2}} \right)^{\frac{p}{4}} \displaystyle \int\limits_{\mathbb{H}P^{N}} \left( \int\limits_{Gr_{1}^{\mathbb{H}}(x)} |\det(dF|_{\Lambda})| d\Lambda + \scriptstyle \frac{(2N-2)!}{(2N-1)\pi^{2N-2}} \displaystyle \int\limits_{Z_{x}}\int\limits_{\mathcal{C}_{2}(\tau)} |\det(dF|_{V})| dV \right)^{\frac{p}{4}} dVol_{g_{0}}. \bigskip
\end{equation}
For $p = 4$ this immediately implies:
\begin{equation}
\label{hpn_thm_pf_eqn_2}
\displaystyle E_{4}(F) \geq \scriptstyle \left( \frac{16N^{2}(2N-2)!}{\pi^{2N-2}} \right) \displaystyle \left( \int\limits_{\mathcal{H}(\mathbb{H}P^{N},g_{0})} |F(\mathcal{Q})| d\mathcal{Q} + \scriptstyle \frac{(2N-2)!}{(2N-1)\pi^{2N-2}} \displaystyle \int\limits_{\mathcal{C}(\mathbb{H}P^{N},g_{0})} |F(X)| dX \right). \bigskip
\end{equation}
For $p > 4$, (\ref{hpn_thm_pf_eqn_1}) and H\"older's inequality imply:
\begin{equation}
\label{hpn_thm_pf_eqn_3}
\displaystyle E_{p}(F) \geq \scriptstyle \left( \frac{((2N+1)!(2N-2)!16N^{2})^{p}}{(2N+1)!^{4}\pi^{(4N-2)p-8N}} \right)^{\frac{1}{4}} \displaystyle \left( \int\limits_{\mathcal{H}(\mathbb{H}P^{N},g_{0})} |F(\mathcal{Q})| d\mathcal{Q} + \scriptstyle \frac{(2N-2)!}{(2N-1)\pi^{2N-2}} \displaystyle \int\limits_{\mathcal{C}(\mathbb{H}P^{N},g_{0})} |F(X)| dX \right)^{\frac{p}{4}}. \bigskip
\end{equation}
Because all $\mathcal{Q} \in \mathcal{H}(\mathbb{H}P^{N},g_{0})$ and all $X \in \mathcal{C}(\mathbb{H}P^{N},g_{0})$ represent generators of $H_{4}(\mathbb{H} P^{N};\mathbb Z)$ we have $|F(\mathcal{Q})|, |F(X)| \geq B^{\star}$, which implies (\ref{hpn_thm_eqn}), as a non-strict inequality. \\
For (\ref{hpn_thm_eqn}) to be an equality, equality would have to hold a.e. in (\ref{hpn_thm_pf_eqn_1}), and therefore in Lemma \ref{twistor_lemma}. This would imply that $F^{*}g$ is equal a.e. to $\varphi(x)g_{0}$, where $\varphi(x)$ is an a.e.-defined non-negative function on $\mathbb{H}P^{N}$. Equality would also require that $F$ take almost all $\mathcal{Q},X \subseteq \mathbb{H}P^{N}$ to area-minimizing currents in their homology class in $H_{4}(M;\mathbb Z)$. Together, these conditions would imply that for almost all $\mathcal{Q},X$,
\begin{equation}
\label{hpn_thm_pf_eqn_4}
\displaystyle \int\limits_{\mathcal{Q}} \varphi(x)^{2} dx = B^{\star}, \int\limits_{X} \varphi(x)^{2} dx = B^{\star}. \bigskip
\end{equation}
Letting $\zeta: U(\mathbb{H}P^{N},g_{0}) \rightarrow \mathbb{H}P^{N}$ be the bundle projection of the unit tangent bundle of $\mathbb{H}P^{N}$, we would then have:
\begin{equation*}
\displaystyle \int\limits_{\mathbb{H}P^{N}} \varphi(x)^{2} dVol_{g_{0}} = \frac{1}{\sigma(4N-1)} \int\limits_{U(\mathbb{H}P^{N},g_{0})} \varphi(\zeta(\vec{u}))^{2} d\vec{u} \bigskip = \frac{1}{\sigma(4N-1)} \int\limits_{\mathcal{H}(\mathbb{H}P^{N},g_{0})} \int\limits_{U(\mathcal{Q},g_{0})} \varphi(\zeta(\vec{u}))^{2} d\vec{u} d\mathcal{Q} \bigskip
\end{equation*}
\begin{equation}
\label{hpn_thm_pf_eqn_5}
\displaystyle = \frac{\sigma(3)}{\sigma(4N-1)} \int\limits_{\mathcal{H}(\mathbb{H}P^{N},g_{0})} \int\limits_{\mathcal{Q}} \varphi(x)^{2} dx d\mathcal{Q} = \left(\frac{6\pi^{2N-1}}{(2N+1)!}\right) B^{\star}. \bigskip
\end{equation}
On the other hand, we would also have:
\begin{equation*}
\displaystyle \int\limits_{\mathbb{H}P^{N}} \varphi(x)^{2} dVol_{g_{0}} = \frac{1}{\pi} \int\limits_{\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}} \varphi(\Psi(\tau))^{2} d\tau = \frac{256\pi^{4(N+1)}}{(2N-2)!(2N-3)!} \int\limits_{\mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}} \int\limits_{\mathcal{C}_{2}(\tau)} \varphi(\Psi(\tau))^{2} dV d\tau \bigskip
\end{equation*}
\begin{equation}
\label{hpn_thm_pf_eqn_6}
\displaystyle = \frac{256\pi^{4(N+1)}}{(2N-2)!(2N-3)!} \int\limits_{\mathcal{C}(\mathbb{H}P^{N},g_{0})} \int\limits_{X} \varphi(x)^{2} dxdX = \left( \frac{\pi^{2N-2}}{(2N-2)!} \right) B^{\star}. \bigskip
\end{equation}
The right-hand sides of (\ref{hpn_thm_pf_eqn_5}) and (\ref{hpn_thm_pf_eqn_6}) can only be equal if $B^{\star} = 0$. Therefore, equality is only possible in Theorem \ref{hpn_thm} if $F$ is a constant map. \end{proof}
A theorem of White \cite{Wh2} shows that if $F:(W^{d},g) \rightarrow (N^{n},h)$ is a mapping of compact, connected, oriented Riemannian manifolds with $F_{*}: \pi_{1}(W) \rightarrow \pi_{1}(N)$ surjective and $\dim(W) \geq 3$, then the infimum of the areas of mappings homotopic to $F$ is equal to the minimum mass of an integral current $T$ representing $F([W])$ in $H_{d}(N;\mathbb Z)$. If $M$ in Theorem \ref{hpn_thm} is simply connected, $B^{\star}$ is therefore equal to the infimum of the areas of mappings $f:S^{4} \rightarrow M$ in the free homotopy class of $F_{*}(\mathbb{H}P^{1})$, as in Theorems \ref{rpn_thm} for $\mathbb R} \newcommand{\T}{\mathbb T P^{n}$ and \ref{cpn_thm} for $\mathbb C} \newcommand{\N}{\mathbb N P^{N}$. \\
The proof of Theorem \ref{hpn_thm} implies that the identity mapping of $(\mathbb{H}P^{N},g_{0})$ minimizes energy among maps $F$ in its homotopy class such that the average volume of $F(X)$ for $X \in \mathcal{C}(\mathbb{H}P^{N},g_{0})$ is at least $Vol(\mathbb C} \newcommand{\N}{\mathbb N P^{2},g_{0}) = \frac{\pi^{2}}{2}$. More generally, with more precise information about the minima or averages of the areas of $F(\mathbb{H}P^{1})$ and $F(\mathbb C} \newcommand{\N}{\mathbb N P^{2})$, one can deduce stronger lower bounds for energy functionals of the mapping $F$. \\
One possible approach to investigating the $p$-energy of mappings homotopic to the identity of $\mathbb{H} P^{N}$ is to use the homotopy lifting property to consider the family of mappings $\widetilde{F}: \mathbb C} \newcommand{\N}{\mathbb N P^{2N+1} \rightarrow \mathbb C} \newcommand{\N}{\mathbb N P^{2N+1}$ which cover a mapping $F: \mathbb{H} P^{N} \rightarrow \mathbb{H} P^{N}$ via the twistor fibration. In particular, this may be helpful in finding optimal lower bound for the $p$-energy of mappings homotopic to the identity mapping of $\mathbb{H} P^{N}$ and determining when the identity is $p$-energy minimizing in its homotopy class.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,073 |
Народна партија Канаде (, ), је десничарска до крајње десницасавезна политичка партија у Канади. Странку је основао Максим Берније у септембру 2018, убрзо након његовог иступања из Конзервативне партије Канаде. Берније, бивши кандидат за изборе за руководство Конзервативне партије Канаде 2017. године и министар у кабинету, био је једини члан парламента (МП) странке од њеног оснивања 2018. године, до свог пораза на канадским савезним изборима 2019. године. ПиПиСи је формирала удружења изборних округа (ЕДА) у 326 изборних места и кандидовала своје кандидате у 315 изборних целина . У Канади су на савизним изборима 2019. године постојале 338 изборних јединица, међутим, ниједан кандидат из његове странке није изабран па је тиме Берније изгубио своју шансу за реизбор у изборном дистрикту Беусе. Странка је кандидовала 312 кандидата на канадским савезним изборима 2021. године, али опет ниједан није изабран у парламент, упркос томе што је повећао свој удео у гласовима народа на скоро пет процената.
Партија се залаже за смањење имиграције у Канаду на 150.000 годишње, укидање канадског закона о мултикултурализму, повлачење из Париског споразума и окончање управљања снабдевањем. На савезним изборима 2021. године, ПиПиСи се такође супротставио затварању и мерама ограничењима због ковида 19, пасошима за вакцину и обавезним вакцинацијама. Описана као популистичка странка по узору на радикално-десничарске европске странке, али са десним економским либералним, или економски либералним у европском контексту, политикама, такође је на различите начине означена као класично либерална, конзервативна, икао десни популиста. Партија се сматра да је десничарска и да је на десном политичком спектру.
Историја
Оснивање партије
ПиПиСи је формирана неколико недеља након оставке Максима Бернијеа, кандидата на изборима за руководство Конзервативне партије Канаде 2017. и бившег министра из Конзервативне партије Канаде. У свом говору о оставци, Берније је изјавио да одлази зато што "сам схватио да је ова странка превише интелектуално и морално корумпирана да би се реформисала". Берније је такође навео да је под опозиционим лидером Ендрјуом Широм, од кога је Берније изгубио на изборима за руководство странке 2017. Конзервативна партија напустила своје принципе, укључујући политичку коректност, корпоративно благостање, реформу уједначавања исплата за провинције и управљање снабдевањем. У часопису Натионал пост-а, Берније је навео да је његов мотив за формирање странке био да преокрене динамику јавног избора у канадском политичком систему што је резултирало куповином гласова и додворавање другим политичким партијама. Он је поновио своје уверење да се Конзервативна партија не може реформисати да би се окончала ова пракса и да је потребна нова политичка партија.
Бернијеа су угледни конзервативни политичари, попут бивших премијера Стивена Харпера и Брајана Малрунија, оптужили да покушава да подели политичку десницу. Он је у емисији Снага и политике на телевизији СиБиСи одговорио да жели да се фокусира на незадовољне гласаче и као пример навео политички успон француског председника Емануела Макрона. Берније је касније навео пробој Народне алијансе Њу Брансвика на општим изборима у Њу Брансвику 2018. и победу Коалиције Авенир Квебек на општим изборима у Квебеку 2018. као примере презира бирача према традиционалним политичким партијама и изражавања жеље за променом гласањем за нове странке.
Пре изласка из Конзервативне партије, Берније је почео поново да успоставља контакт са појединцима који су подржали његову кандидатуру за лидерство конзервативаца 2017, веровали су да има неопходну подршку да региструје странку у Елекнс Канада. Ле Дево је известио да су чланови седам асоцијација конзервативних бирачких јединица прешли у странку. Неколико дана након што је објавио назив странке, лидер Либертаријанске партије Канаде Тим Моен, који је претходно понудио вођство те партије Бернијеу, изјавио је да је отворен за идеју спајања са Народном партијом. На питање Глобал Њуза о овоме предлогу спајања, Берније је одговорио да није заинтересован за тако нешто. На питање о организацији странке, поменуо је да ће користити алате који нису постојали у прошлости, као што је коришћење друштвених медија.
Берније је планирао да кандидује кандидате у свих 338 канадских савезних избора на канадским савезним изборима 2019. године. Документи за регистрацију странке су званично достављени Елекшн Канади 10. октобра 2018. године. Поред тога, он је навео да ће удружења изборних округа (ЕДА) бити успостављена до 31. децембра 2018. и да ће ЕДА почети да се фокусирају на проналажење кандидата почевши од јануара 2019. године. Првог новембра 2018. странка је открила да има преко 30.000 "чланова оснивача". Новинске куће су касније откриле да је један од оснивача ПиПиСија био бивши амерички бели националиста, а да су друга два имала везе са анти-имигрантским групама. Бивши бели националиста је уклоњен из странке 29. августа 2019, након што је његова прошлост изашла на видело. Портпарол странке је изјавио да се његова прошлост није појавила током процеса провере пошто је дошао из Сједињених Држава. Друга два члана негирали су да имају расистичке ставове, а странка је касније рекла да за Ле Девоара нису имали довољно ресурса да их провере на почетку формирања ПиПиСија.
У новембру 2018, министарка демократских институција Карина Гулд рекла је да ће Берније бити квалификован за дебате које води Комисија за дебате лидера ако странка номинује кандидате у 90 одсто изборних јединица. Партија је одржала скупове у Ванкуверу, Калгарију, Торонту, Отави-Гатиноу, Винипегу, Саскатуну и Квебек Ситију. Године 2019. одржала је митинге у Хамилтону, Сент Џону и Халифаксу.] Партија је 21. децембра 2018. године основала ЕДА у свих 338 изборних округа.
Регистрација
Странка је добила квалификовани статус 11. новембра 2018. године. Регистрована је од стране Елекшн Канада 19. јануара 2019. године, након номиновања кандидата за допунске изборе у Оутремонту, Јорк—Симкоу, Бурнаби Сауту и Нанаимо—Ледисмит. Избори су били расписани за 25. фебруар 2019. На допунским изборима 25. фебруара, странка је добила 10,9 одсто гласова у Бурнаби Сауту и 1,5 одсто у оба Јорка—Симко и Аутремонту.
Изборни резултати
Референце
Спољашње везе
званични веб-сајт
Политичке партије у Канади | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 14 |
{"url":"https:\/\/juliaintervals.github.io\/pages\/apiDocs\/apiRootFinding\/index.html","text":"# IntervalRootFinding.jl API\n\n## Externals\n\n### Types\n\nRoot\u00a0 - Type\nRoot\n\nObject representing a possible root inside a given region. The field status is either :unknown or :unique. If status is :unique then we know that there is a unique root of the function in question inside the given region.\n\nInternally the status may also be :empty for region guaranteed to contain no root, however such Roots are discarded by default and thus never returned by the roots function.\n\n### Fields\n\n\u2022 interval: a region (either Interval of IntervalBox) searched for roots.\n\n\u2022 status: the status of the region, valid values are :empty, unkown and :unique.\n\n### Methods\n\nisunique\u00a0 - Function\nisunique(rt)\n\nReturn whether a Root is unique.\n\nroots\u00a0 - Function\nroots(f, X, contractor=Newton, strategy=BreadthFirstSearch, tol=1e-15)\nroots(f, deriv, X, contractor=Newton, strategy=BreadthFirstSearch, tol=1e-15)\nroots(f, X, contractor, tol)\nroots(f, deriv, X, contractor, tol)\n\nUses a generic branch and prune routine to find in principle all isolated roots of a function f:R^n \u2192 R^n in a region X, if the number of roots is finite.\n\nInputs:\n\n\u2022 f: function whose roots will be found\n\n\u2022 X: Interval or IntervalBox in which roots are searched\n\n\u2022 contractor: function that, when applied to the function f, determines the status of a given box X. It returns the new box and a symbol indicating the status. Current possible values are Bisection, Newton and Krawczyk\n\n\u2022 deriv: explicit derivative of f for Newton and Krawczyk\n\n\u2022 strategy: SearchStrategy determining the order in which regions are processed.\n\n\u2022 tol: Absolute tolerance. If a region has a diameter smaller than tol, it is returned with status :unknown.\n\nFunction to solve a quadratic equation where the coefficients are intervals. Returns an array of intervals of the roots. Arguments a, b and c are interval coefficients of x\u00b2, x and 1 respectively. The interval case differs from the non-interval case in that there might be three disjoint interval roots. In the third case, one interval root extends to \u2212\u221e and another extends to +\u221e. This algorithm finds the set of points where F.lo(x) \u2265 0 and the set of points where F.hi(x) \u2264 0 and takes the intersection of these two sets. Eldon Hansen and G. William Walster : Global Optimization Using Interval Analysis - Chapter 8\n\nroot_status\u00a0 - Function\nroot_status(rt)\n\nReturn the status of a Root.\n\n## Internals\n\n### Types\n\nBBSearch\u00a0 - Type\nBBSearch{DATA}\n\nBranch and bound search interface in element of type DATA.\n\nThis interface provide an iterable that perform the search.\n\nThere is currently three types of search supported BreadFirstBBSearch, DepthFirstBBSearch and KeyBBSearch, each one processing the element of the tree in a different order. When subtyping one of these, the following methods must be implemented:\n\n\u2022 root_element(::BBSearch): return the element with which the search is started\n\n\u2022 process(::BBSearch, elem::DATA): return a symbol representing the action to perform with the element elem and an object of type DATA reprensenting the state of the element after processing (may return elem unchanged).\n\n\u2022 bisect(::BBSearch, elem::DATA): return two elements of type DATA build by bisecting elem\n\nSubtyping BBSearch directly allows to have control over the order in which the elements are process. To do this the following methods must be implemented:\n\n\u2022 root_element(::BBSearch): return the first element to be processed. Use to build the initial tree.\n\n\u2022 get_leaf_id!(::BBSearch, wt::BBTree): return the id of the next leaf that will be processed and remove it from the list of working leaves of wt.\n\n\u2022 insert_leaf!(::BBSearch, wt::BBTree, leaf::BBLeaf): insert a leaf in the list of working leaves.\n\n### Valid symbols returned by the process function\n\n\u2022 :store: the element is considered as final and is stored, it will not be further processed\n\n\u2022 :bisect: the element is bisected and each of the two resulting part will be processed\n\n\u2022 :discard: the element is discarded from the tree, allowing to free memory\n\nBisection\u00a0 - Type\nBisection{F} <: Contractor{F}\n\nContractor type for the bisection method.\n\nBreadthFirstSearch <: BreadthFirstBBSearch\n\nType implementing the BreadthFirstBBSearch interface for interval roots finding.\n\n### Fields\n\n\u2022 initial: region (as a Root object) in which roots are searched.\n\n\u2022 contractor: contractor to use (Bisection, Newton or Krawczyk)\n\n\u2022 tol: tolerance of the search\n\nContractor\u00a0 - Type\nContractor{F}\n\nAbstract type for contractors.\n\nDepthFirstSearch <: DepthFirstBBSearch\n\nType implementing the DepthFirstBBSearch interface for interval roots finding.\n\n### Fields\n\n\u2022 initial: region (as a Root object) in which roots are searched.\n\n\u2022 contractor: contractor to use (Bisection, Newton or Krawczyk)\n\n\u2022 tol: tolerance of the search\n\nKrawczyk\u00a0 - Type\nKrawczyk{F, FP} <: Contractor{F}\n\nContractor type for the Krawczyk method.\n\n### Fields\n\n\u2022 f::F: function whose roots are searched\n\n\u2022 f::FP: derivative or jacobian of f\n\n(C::Krawczyk)(X, tol, \u03b1=where_bisect)\n\nContract an interval X using Krawczyk operator and return the contracted interval together with its status.\n\n### Inputs\n\n\u2022 R: Root object containing the interval to contract.\n\n\u2022 tol: Precision to which unique solutions are refined.\n\n\u2022 \u03b1: Point of bisection of intervals.\n\nNewton\u00a0 - Type\nNewton{F, FP} <: Contractor{F}\n\nContractor type for the Newton method.\n\n### Fields\n\n\u2022 f::F: function whose roots are searched\n\n\u2022 f::FP: derivative or jacobian of f\n\n(C::Newton)(X, tol, \u03b1=where_bisect)\n\nContract an interval X using Newton operator and return the contracted interval together with its status.\n\n### Inputs\n\n\u2022 R: Root object containing the interval to contract.\n\n\u2022 tol: Precision to which unique solutions are refined.\n\n\u2022 \u03b1: Point of bisection of intervals.\n\n### Methods\n\nbranch_and_prune\u00a0 - Function\nbranch_and_prune(X, contractor, strategy, tol)\n\nGeneric branch and prune routine for finding isolated roots using the given contractor to determine the status of a given box X.\n\nSee the documentation of the roots function for explanation of the other arguments.\n\ndata\u00a0 - Function\ndata(leaf::BBLeaf)\n\nReturn the data stored in the leaf.\n\ndata(wt::BBTree)\n\nReturn all the data stored in a BBTree as a list. The ordering of the elements is arbitrary.\n\nSolves the system of linear equations using Gaussian Elimination. Preconditioning is used when the precondition keyword argument is true.\n\nREF: Luc Jaulin et al., Applied Interval Analysis, pg. 72\n\nIteratively solves the system of interval linear equations and returns the solution set. Uses the Gauss-Seidel method (Hansen-Sengupta version) to solve the system. Keyword precondition to turn preconditioning off. Eldon Hansen and G. William Walster : Global Optimization Using Interval Analysis - Chapter 5 - Page 115\n\nnewton1d\u00a0 - Function\n\nnewton1d performs the interval Newton method on the given function f with its derivative f\u2032 and initial interval x. 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\section{Introduction}\label{sec1}
The epidemic-type aftershock sequence (ETAS) model
[\citeauthor{OGA85} (\citeyear{OGA85,OGA86,OGA88,OGA89})] is one of the
earliest point-process
models created for clustered events. It is defined in terms of a
conditional intensity [\citet{Haw71}, \citet{HAWADA73},
\citeauthor{Oga78}
(\citeyear{Oga78,OGA81})], is equivalent to epidemic branching
processes [\citet{Ken49}, \citet{HawOak74}], and allows each
earthquake to generate
(or trigger) offspring earthquakes. Besides being used in seismology, the
ETAS model has been applied to various fields in the social and natural
sciences [e.g., \citet{Baletal12}, \citet{CHAMCG12},
\citet{HAS}, \citet{HERSCH09}, \citet{Mohetal11}, \citet{PenSchWoo05}, \citet{SCHPENWOO03}].
Similar magnitude-dependent point-process models have been applied to
seismological studies [\citet{VERDAV66}, \citet{LOM74}, \citet{KagKno87}] and statistical studies [\citet{Ver70}]. The ETAS
model is stationary if the immigration rate (background seismicity
rate) of
an earthquake remains constant and the branching ratio is subcritical
[\citet{Haw71}, \citet{HawOak74}, \citet{ZHUOGA06}].
The history-dependent form of the ETAS model on occurrence times and sizes
(magnitudes) lends itself to the accumulated empirical studies by
\citeauthor{UTS61} (\citeyear{UTS61,UTS62,UTS69,UTS70,UTS71,UTS72}) and
\citet{UTSSEK55}, and its establishing
history is detailed by \citet{UTSOGAMAT95}. ETAS model parameters can be
estimated from earthquake occurrence data by maximizing the log-likelihood
function to provide estimates for predicting seismic activity (i.e., number
of earthquakes per unit time). The model has been frequently used and cited
in seismological studies, especially to compare the features of simulated
seismicity with those of real seismicity data. The model is also
recommended for use in short-term predictions [\citet{JORCHEGAS12}] in the
report of the International Commission on Earthquake Forecasting for Civil
Protection. It is planned to be adopted for operational forecasts of
earthquakes in \mbox{California} (The Uniform California Earthquake Rupture
Forecast, Version 3, URL:
\url{http://www.wgcep.org/sites/wgcep.org/files/UCERF3\_Project\_Plan\_v55.pdf}).
The ETAS model has also been used to detect anomalies such as
quiescence in
seismicity. Methods and applications are detailed in
\citeauthor{OGA88} (\citeyear
{OGA88,OGA89,OGA92,OGA99,OGA,OGA06N1,OGA07,OGA10,OGA11N1,OGA12}),
\citet{OGAJONTOD03},
\citet{KUMOGATOD10}, and \citet{BANOGA13}. A change-point
analysis examines a simple hypothesis that specific parameters change after
a certain time. The misfit of occurrence rate prediction after a change
point is then \mbox{preliminarily} shown by the deviation of the empirical
cumulative counts of the earthquake occurrences from the predicted
cumulative function. The predicted function is the extrapolation of the
model fitted before the change point. A~downward and upward deviation
corresponds to relative quiescence and activation, respectively.
This study considers a number of nonstationary extensions of the ETAS model
to examine more detailed nonstandard transient features of earthquake
series. The extended models take various forms for comparison with the
reference ETAS model, which represents the preceding normal activity in a
given focal region. Because changing stresses in the crust are not directly
observable, it is necessary to infer relevant quantitative characteristics
from seismic activity data. For example, \citet{HAIOGA05} and
\citet{LOMCOCMAR10} estimated time-dependent background rates
(immigration rates) in a moving time window by removing the triggering
effect in the ETAS model.
In Section~\ref{sec2}, time-dependent parameters for both background rates and
productive rates are simultaneously estimated. There, the penalized
log-likelihood is considered for the trade-off between a better fit of the
nonstationary models and the roughness penalties against overfitting. Then,
not only is an optimal strength adjusted for each penalty but also a better
penalty function form is selected using the Akaike Bayesian Information
Criterion (\textit{ABIC}) [\citet{Aka80}]. These parameter constraints
together with
the existence of a change point are further examined to determine if they
improve the model fit. One benefit of this model is that it allows varying
parameters to have sharp changes or discontinuous jumps at the change point
while sustaining the smoothness constraints in the rest of the period.
In Section~\ref{sec3}, the methods are demonstrated by applying the model to a swarm
activity. The target activity started after the March 11, 2011 Tohoku-Oki
earthquake of magnitude (M) 9.0, induced at a distance from the M9.0
rupture source. Section~\ref{sec4} concludes and discusses the models and methods.
The reproducibility of the inversion results is demonstrated in the
\hyperref[sec6]{Appendix} by synthesizing the data and re-estimating it using the same
procedure.
\section{Methods}\label{sec2}
\subsection{The ETAS model}\label{sec21} A conditional intensity
function characterizes
a point (or counting) process $N(t)$ [\citet{DALVER03}]. The
conditional intensity $\lambda(t|H_t)$ is defined as follows:
\begin{equation}
\label{equ1} \Pr \bigl\{ N(t,t + dt) = 1|H_{t} \bigr\} = \lambda
(t|H_{t})\,dt + o(dt),
\end{equation}
where $H_{t}$ represents the history of occurrence times of marked events
up to time~$t$. The conditional intensity function is useful for the
probability forecasting of earthquakes, which is obtained by integrating
over a time interval.
The ETAS model, developed by \citeauthor{OGA85} (\citeyear
{OGA85,OGA86,OGA88,OGA89}), is a special
case of the marked Hawkes-type self-exciting process, and has the following
specific expression for conditional intensity:
\begin{equation}
\label{equ2} \lambda_{\theta} (t|H_{t}) = \mu+ \sum
_{ \{ i\dvtx S < t_{i} < t
\}} \frac{K_{0}e^{\alpha( M_{i} - M_{z} )}}{ ( t -
t_{i} + c )^{p}},
\end{equation}
where $S$ is the starting time of earthquake observation and $M_{z}$
represents the smallest magnitude (threshold magnitude) of earthquakes to
be treated in the data set. $M_{i}$ and $t_{i}$ represent the magnitude and
the occurrence time of the $i$th earthquake, respectively, and $H_{t}$
represents the occurrence series of the set ($t_{i}$, $M_{i}$) before time~$t$. The parameter set $\theta$ thus consists of five elements ($\mu$,
$K_{0}, c, \alpha, p$). In fact, the second term of equation~(\ref{equ2}) is a
weighted superposition of the Omori--Utsu empirical function
[\citet{UTS61}]
for aftershock decay rates,
\begin{equation}
\label{equ3} \lambda_{\theta} (t) = \frac{K}{ ( t + c )^{p}},
\end{equation}
where $t$ is the elapsed time since the main shock. It is important to note
that, while the concept of a main shock and its aftershocks is intuitively
classified by seismologists sometime after the largest earthquake occurs,
there is no clear discrimination between them in equation (\ref{equ2}). That is,
each earthquake can trigger aftershocks, and the expected cluster size
depends on the magnitude of the triggering earthquake with the parameter
$\alpha$.
The parameter $K_{0}$ (earthquakes/day) is sometimes called the
``aftershock productivity.'' As the name explains, the parameter controls
the overall triggering intensity. The factor $c$ (day) is a scaling
time to
establish the power-law decay rate and allows a finite number of
aftershocks at the origin time of a triggering earthquake (a main shock).
In practice, the fitted values for $c$ are more likely to be caused by the
under-reporting of small earthquakes hidden in the overlapping wave trains
of large earthquakes [\citet{UTSOGAMAT95}]. The exponent $p$ is the
power-law decay rate of the earthquake rate in equation (\ref{equ3}). The magnitude
sensitivity parameter $\alpha$ (magnitude $^{-1}$) accounts for the
efficiency of an earthquake of a given magnitude in generating aftershocks.
A small $\alpha$ value allows a small earthquake to trigger a larger
earthquake more often. Finally, the background (spontaneous) seismicity
rate $\mu$ represents sustaining external effects and superposed occurrence
rates of long-range decays from unobserved past large earthquakes. It also
accounts for the triggering effects by external earthquakes.
The FORTRAN program package associated with manuals regarding\break ETAS analysis
is available to calculate the maximum likelihood estimates (MLEs)
of~$\theta$ and to visualize model performances [\citet{OGA06N2}]. See
also \url{http://www.ism.ac.jp/\textasciitilde ogata/Ssg/ssg\_softwaresE.html}.
\subsection{Theoretical cumulative intensity function and time transformation}\label{sec22} Suppose that the parameter values $\theta
= (\mu, K, c,
\alpha, p)$ of the ETAS, equation (\ref{equ2}), are given. The integral of the
conditional intensity function,
\begin{equation}
\label{equ4} \Lambda_{\theta} (t|H_{t}) = \int
_{S}^{t} \lambda_{\theta}
(u|H_{u}) \,du,
\end{equation}
provides the expected cumulative number of earthquakes in the time interval
$[0, t]$. The time transformation from $t$ to $\tau$ is based on the
cumulative intensity,
\begin{equation}
\label{equ5} \tau= \Lambda(t|H_{t}),
\end{equation}
which transforms the original earthquake occurrence time $(t_{1}, t_{2},\ldots, t_{N})$ into the sequence $(\tau_{1}, \tau_{2},\ldots, \tau
_{N})$ in
the time interval $[0,\Lambda(T)]$. If the model represents a good
approximation of the real seismicity, it is expected that the integrated
function [equation~(\ref{equ4})] and the empirical cumulative counts $N(t)$ of the
observed earthquakes are similar. This implies that the transformed
sequence appears to be a \mbox{stationary} Poisson process (uniformly distributed
occurrence times) if the model is sufficiently correct, and appears to be
heterogeneous otherwise.
\subsection{Two-stage ETAS model and the change-point problem}\label
{sec23} In
change-point analysis, the whole period is divided into two disjointed
periods to fit the ETAS models separately, and is therefore called a
two-stage ETAS model. This is one of the easiest ways to treat
nonstationary data, and is best applied to cases in which parameters are
suspected to change at a specific time. Such a change point is observed
when a notably large earthquake or slow slip event (regardless of observed
or unobserved) occurs in or near a focal region. Many preceding studies
[e.g., \citet{OGAJONTOD03}, \citeauthor{OGA} (\citeyear
{OGA,OGA06N1,OGA07,OGA10}), \citet{KUMOGATOD10}] have adopted this
method to their case studies, and details can
be found therein.
The question of whether the seismicity changes at some time $T_{0}$ in a
given period $[S, T]$ is reduced to a problem of model selection. In this
analysis, the ETAS models are separately fitted to the divided periods $[S,
T_{0}]$ and $[T_0, T]$, and their total performance is compared
to an ETAS model fitted over the whole period $[S, T]$ by the Akaike Information Criterion (\textit{AIC}) [Akaike (\citeyear{Ak73,Ak74,Ak77})]. The
\textit{AIC} is described as follows:
\begin{equation}
\label{equ6} \mathit{AIC} = - 2\max\log L(\theta) + 2k,
\end{equation}
where ln $L(\theta)$ represents the log-likelihood of the ETAS model,
\begin{equation}
\label{equ7} \log L(\theta) = \sum_{ \{ i\dvtx S < t_{i} < T \}} \log
\lambda_{\theta} ( t_{i}| H_{t_{i}} ) - \int
_{S}^{T} \log\lambda_{\theta} ( t|
H_{t} )\,dt,
\end{equation}
and $k$ is the number of parameters to be estimated. The variables $t_{i}$
and $H_{t_i}$ are the same as those in equation (\ref{equ2}).
Under this criterion, the model with a smaller \textit{AIC} value performs
better. It is useful to keep in mind that $\exp\{- \Delta\mathit{AIC}/2\}$
can be interpreted as the relative probability of how a model with a
smaller \textit{AIC} value is superior to others [e.g., \citet{Aka80}].
Let \textit{AIC}$_{0}$ be the \textit{AIC} of the ETAS model estimated for
the whole period $[S, T]$, $\mathit{AIC}_{1}$ be that of the first period
$[S, T_{0}]$, and $\mathit{AIC}_{2}$ be that of the second period $[T_0,
T]$, therefore,
\begin{eqnarray}
\label{equ8} \mathit{AIC}_{0} &=& - 2\max_{\theta_{0}}\log L(
\theta_{0};S,T) + 2k_{0},\nonumber
\\
\mathit{AIC}_{1} &=& - 2\max_{\theta_{1}}\log L(
\theta_{1};S,T_{0}) + 2k_{1},
\\
\mathit{AIC}_{2} &=& - 2\max_{\theta_{2}}\log L(
\theta_{2};T_{0},T) + 2k_{2}.\nonumber
\end{eqnarray}
Let $\mathit{AIC}_{12}$ represent the total $\mathit{AIC}$ from the divided periods,
such that
\begin{equation}
\label{equ9} \mathit{AIC}_{12} = \mathit{AIC}_{1} + \mathit{AIC}_{2} + 2q,
\end{equation}
with $q$ being the degrees of freedom to search for the best change-point
candidate $T_{0}$. Next, $\mathit{AIC}_{12}$ is compared against
\textit{AIC}$_{0}$. If $\mathit{AIC}_{12}$ is smaller, the two-stage ETAS
model with the change point $T_{0}$ fits better than the ETAS model applied
to the whole interval. The quantity $q$ monotonically depends on sample
size (number of earthquakes in the whole period $[S, T])$ when searching
for the maximum likelihood estimate of the change point [\citeauthor
{OGA92}
(\citeyear{OGA92,OGA99}), \citet{KUMOGATOD10}, \citet
{BANOGA13}]. This penalty term
$q$, as well as an increased number of estimated parameters, imposes a
hurdle for a change point to be significant, and it is usually rejected
when the one-stage ETAS model fits sufficiently well. If the change point
$T_{0}$ is predetermined from some information other than the data,
then $q
= 0$. This is often the case when a conspicuously large earthquake occurs
within swarm activity, and will be discussed below. Also, even in this
case, the overfitting by the change point is avoided by the $\mathit{AIC}_{12}$
of a
two-stage ETAS model, which has two times as many parameters of a single
stationary ETAS model throughout the whole period.
\subsection{Anomaly factor functions for nonstationary ETAS models}\label{sec24}
Assume
that the ETAS model fits the data well for a period of ordinary seismic
activity. Then, the concern is whether this model shows a good fit to the
seismicity in a forward extended period. If there are misfits,
time-dependent compensating factors are introduced to the parameters to be
made time-dependent. These factors are termed ``anomaly factor functions''
and, thus, the transient changes in parameters are tracked. If earthquake
activity is very low in and near a target region preceding the transient
activity, data from a wider region, such as the polygonal region in
Figure~\ref{fig1}, is used to obtain a reference stationary ETAS model
(Figure~\ref{fig2}). Such a
model is stable against small local anomalies, and is therefore a good
reference model. The reference ETAS model, coupled with the corresponding
anomaly factor functions, becomes the nonstationary ETAS model in this
study.
\begin{figure
\includegraphics{759f01.eps}
\caption{Epicenters of earthquakes of magnitude (M)${}\geq 3.0$ in
the Northern Honshu region, Japan, with depths shallower than 40 km, from
1997 to 2012, selected from the JMA Hypocenter catalog. The gray and black
dots represent the earthquakes that occurred before and after the M9.0
Tohoku-Oki earthquake, respectively. The rectangular regions A and B
include the aftershocks of the 2008 Iwate-Miyagi Prefectures Inland
Earthquake of M7.2 and the swarm near Lake Inawashiro, respectively. Their
inset panels magnify the epicenter distribution with M${}\geq 2.0$
and M${}\geq 2.5$, respectively. The polygonal region indicates the
Tohoku inland and its western offshore region; the earthquakes in this
region are used in the reference stationary ETAS model. The closed star
represents the epicenter of the 2004 Chuetsu earthquake of M6.8, and the
open star represents the 2007 Chuetsu-Oki earthquake of M6.8.}\label{fig1}
\end{figure}
\begin{figure
\includegraphics{759f02.eps}
\caption{Cumulative number and magnitude of earthquakes of
M${}\geq 3$ against the ordinary time and transformed time by the
ETAS model from the polygonal region in Figure~\protect\ref{fig1}. The fitted period of the
model is from October 1997 to the M9.0 March 2011 Tohoku-Oki earthquake
(indicated by vertical dashed lines). Red curves in the top and bottom
panels represent the theoretical cumulative numbers against the ordinary
time (\protect\ref{equ4}) and the transformed time, respectively. The dashed black ellipses
and dashed rectangles highlight the anomalies around 2008 and after the
Tohoku-Oki earthquake, respectively.}\vspace*{-2pt}\label{fig2}
\end{figure}
Among the parameters of the ETAS model, the background rate $\mu$ and the
aftershock productivity $K_{0}$ are sensitive to nonstationarity. We
therefore introduce the anomaly factor functions as the nonstationary
components to modify the reference stationary ETAS model in such a way
that
\begin{equation}
\label{equ10} \lambda_{\theta} (t|H_{t}) = \mu q_{\mu}
(t) + \sum_{\{ i\dvtx S < t_{i} <
t\}} \frac{K_{0}q_{K}(t_{i})e^{\alpha( M_{i} - M_{z} )}}{ (
t - t_{i} + c )^{p}}.
\end{equation}
Here $k_\mu(t)$ and $q_K (t)$ are
referred to as anomaly factor functions of the parameters $\mu$ and
$K_{0}$, respectively. Because of technical reasons to avoid further model
complexity, we did not consider the case in which the other three
parameters $c$, $\alpha$ and $p$ in equation (\ref{equ2}) also are time-varying.
One structural problem of the ETAS model is that $K_{0}$ is correlated with
the parameter $\alpha$. The trade-off is not negligible, especially
when the
range of magnitudes in the data set is small. See Section~\ref{sec4} for additional
discussion of this issue.
We use the first-order spline function of the ordinary time $t$. This
is a
broken line interpolated by the coordinates $\{(t_{i}, q_{i}); i = 0,
1, 2,\ldots, N+1\}$, where $t_{i}$ is the occurrence time of the $i$th
earthquake, and $t_{0}$ and $t_{N+1}$ are the start and end of the period,
respectively. Then, the spline functions are defined as follows:
\begin{equation}
\label{equ11} \quad q_{\mu} (t) = \sum_{i = 1}^{N}
I_{(t_{i},t_{i + 1})}(t) \biggl\{ \frac{q_{\mu,i + 1} - q_{\mu,
i}}{t_{i + 1} - t_{i}}(t - t_{i}) +
q_{\mu,
i} \biggr\} = \sum_{i = 1}^{N}
q_{\mu, i}F_{i} (t)
\end{equation}
and
\begin{equation}
\label{equ12} \quad q_{K}(t) = \sum_{i = 1}^{N}
I_{(t_{i},t_{i + 1})}(t) \biggl\{ \frac{q_{K, i + 1} - q_{K, i}}{t_{i +
1} - t_{i}}(t - t_{i}) +
q_{K, i} \biggr\} = \sum_{i = 1}^{N}
q_{K, i}F_{i} (t),
\end{equation}
where $I_{(t_{i},t_{i + 1})}(t)$ is the indicator function, with the explicit
form of $F_{i} (t)$ given as
\begin{equation}
\label{equ13} F_{i}(t) = \frac{t - t_{i - 1}}{t_{i} - t_{i - 1}}I_{(t_{i -
1},t_{i})}(t) +
\frac{t_{i + 1} - t}{t_{i + 1} - t_{i}}I_{(t_{i},t_{i +
1})}(t).
\end{equation}
The log-likelihood function of the nonstationary point process can be
written as follows:
\begin{equation}
\label{equ14} \log L(q) = \sum_{\{ i; S < t_{i} < T\}} \log
\lambda_{q}(t_{i}|H_{t_{i}}) - \int
_{S}^{T} \lambda_{q} (t|H_{t})
\,dt,
\end{equation}
where $q = (q_{\mu}, q_{K})$.
\subsection{Penalties against rough anomaly factor functions}\label{sec25}
Since these
ano\-maly functions have many coefficients representing flexible variations,
coefficients are estimated under an imposed smoothness constraint to avoid
their overfitting. This study uses the penalized log-likelihood
[\citet{GooGas71}] described below. With the roughness penalty functions,
\begin{eqnarray}
\label{equ15} \Phi_{\mu} &=& \sum_{i = 0}^{N}
\biggl( \frac{q_{\mu,i + 1} - q_{\mu,
i}}{t_{i + 1} - t_{i}} \biggr)^{2}(t_{i + 1} -
t_{i})\quad\mbox{and}
\nonumber
\\[-8pt]
\\[-8pt]
\Phi_{K} &=& \sum_{i = 0}^{N}
\biggl( \frac{q_{K,i + 1} - q_{K, i}}{t_{i + 1}
- t_{i}} \biggr)^{2}(t_{i + 1} -
t_{i}),
\nonumber
\end{eqnarray}
and the penalized log-likelihood against the roughness becomes
\begin{equation}
\label{equ16} Q ( q| w_{\mu},w_{K} ) = \log L ( q ) -
w_{\mu} \Phi_{\mu} - w_{K}\Phi_{K},
\end{equation}
where each ``$w$'' represents weight parameters that tune the smoothness
constraints of the anomaly factors. The roughness penalty, equation (\ref{equ15}),
imposes penalties to the log-likelihood according to parameter
differentials at successive event occurrence times.
Furthermore, the degree of the smoothness constraints may not be
homogeneous in ordinary time because earthquake series are often highly
clustered. In other words, it is expected that more detailed or rapid
changes of the anomaly factors appear during dense event periods rather
than during sparse periods [\citet{OGA89}, \citet{AdeOga10}].
Hence, for the same model, alternative constraints are considered by
replacing $\{t_{i}\}$ in equation (\ref{equ15}) with $\{ \tau_{i}\}$ on the
transformed time $\tau$ in equation (\ref{equ5}) of the reference ETAS model.
The following restricted cases of the nonstationary model in equation (\ref{equ10}),
together with different types of the aforementioned parameter constraints,
are examined and summarized in Table~\ref{tab1}. Model~1 restricts the parameter
$K_{0}$ to be constant and unchanged from the reference model, leaving
$q_{\mu} (t)$ to be unrestricted. Model~2 restricts the parameters $\mu$
and $K_{0}$ to have the same factor. In other words, model~2 estimates the
anomaly factor for the total intensity $\lambda_{\theta} (t|H_{t})$ in
equation (\ref{equ10}). This restriction is assumed in \citet{AdeOga10}.
Model~3 has no restriction.
\begin{table}[t]
\tabcolsep=0pt
\caption{Summary of the competing nonstationary ETAS models. The
numbers index the models. The row headers explain the model
restrictions of
anomaly factors $q_{\mu}(t)$ and $q_{K}(t)$. The
first column \textup{(a)} uses smoothing on ordinary time, the second column \textup{(b)} on
the transformed time}\label{tab1}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}@{}lcc@{}}
\hline
\textbf{Restrictions} & \textbf{(a) Smoothing on ordinary time} & \textbf{(b) Smoothing on transformed time}\\
\hline
$q_{K} (t) = 1$ & Model 1(a) & Model 1(b)\\
$q_{\mu} (t) = q_{K}(t)$ & Model 2(a) & Model 2(b)\\
No restriction & Model 3(a) & Model 3(b)\\
\hline
\end{tabular*}
\end{table}
Here, from a statistical modeling viewpoint, it should be noted that
$\mu$
and $K_{0}$ are linearly parameterized regarding the conditional intensity
[equation~(\ref{equ2})], and likewise the linearly parameterized coefficients of the
functions $q_{\mu}$ and $q_{K}$ in equation (\ref{equ10}). Together, they force the
penalized log-likelihood function [equation~(\ref{equ16})] to be strictly concave
regardless of the dimensions of the coefficients' space [\citeauthor
{Oga78} (\citeyear{Oga78,Oga01}),
\citet{OGAKAT93}]. Therefore, the maximizing solutions of the
penalized log-likelihood function can be obtained uniquely and stably under
a suitable numerical optimization algorithm [e.g., appendices of
\citeauthor{OGA04} (\citeyear{OGA04,OGA11N2})]. The reproducibility of
the inversion results of $\mu(t)$ and
$K_{0} (t)$ are demonstrated in the \hyperref[sec6]{Appendix}.
\subsection{Tuning smoothness constraints, model selection and error evaluation}\label{sec26}
In a Bayesian context, given the weights, the solution of the parameters
$q$ that minimize the penalized log-likelihood $Q$ in (\ref{equ16}) is termed the
maximum a posteriori (MAP) estimate. In the following section, we describe
how to determine the optimal MAP (OMAP) estimate. To obtain the optimal
weights in the penalty \mbox{functions} in equation (\ref{equ16}), this study uses a
Bayesian interpretation of penalized log-likelihood as suggested by
\citet{Aka80}. Specifically, the exponential of each penalty
function is
proportional to a prior Gaussian distribution of the forms
\begin{equation}
\label{equ17} \pi( q_{\mu} | w_{\mu} ) \propto e^{ -
w_{\mu} q_{\mu} \Sigma_{\mu} q_{\mu}^{t}/2}
\quad \mbox{and}\quad\pi( q_{K}| w_{K} ) \propto
e^{ -
w_{K}q_{K}\Sigma_{K}q_{K}^{t}/2},
\end{equation}
since the coefficients of the function $q_{.}(\cdot)$ in the penalty term
$\Phi$ take a quadratic form with a symmetric $(N+1) \times(N+1)$
nonnegative definite matrix $\Sigma$. Since each matrix $\Sigma$ is
degenerate and has $\operatorname{rank} (\Sigma) = N$, above each prior
distribution becomes improper [\citet{OGAKAT93}]. To avoid such improper
priors, we divide each of the vectors $q$ into $(q^{c}, q^{(N+1)})$ so that
each of the priors becomes a probability density function with respect to
$q^{c}$:
\begin{equation}
\label{equ18} \pi \bigl( q^{c}| w, q_{N + 1} \bigr) =
\frac{ (
w^{N}\det\Sigma^{c} )^{1 / 2}}{\sqrt{2\pi}^{N}}\exp \biggl( - \frac{1}{2}w^{N}q^{c}
\Sigma^{c} {}^{t}q^{c} \biggr),
\end{equation}
where $\Sigma^{c}$ is the cofactor of the last diagonal element of
$\Sigma$, and $w$ and $q^{(N+1)}$ are considered hyperparameters to
maximize the integral of the posterior distribution with respect to
$q^{c}$,
\begin{eqnarray}\label{equ19}
&& \Psi \bigl( w_{\mu},w_{K};
q_{\mu}^{(N + 1)}, q_{K}^{(N + 1)} \bigr)
\nonumber\\[-8pt]\\[-8pt]\nonumber
&&\qquad = \int L
( q_{\mu}, q_{K} ) \pi( q_{\mu} | w_{\mu} )
\pi( q_{K}| w _{K} )\,dq_{\mu}^{c}\,dq_{K}^{c},
\end{eqnarray}
which refers to the likelihood of a Bayesian model. \citet{Goo65} suggests
the maximization of equation (\ref{equ19}) with respect to the hyperparameters and
termed this the Type II maximum likelihood procedure.
By applying Laplace's method [\citet{Sti86}, pages 366--367], the posterior
distribution is approximated by a Gaussian distribution, by which the
integral in equation (\ref{equ19}) becomes
\begin{eqnarray}\label{equ20}
&& \Psi \bigl( w_{\mu},w_{K};
q_{\mu}^{(N + 1)}, q_{K}^{(N + 1)} \bigr)\nonumber
\\
&&\qquad = Q
\bigl( \hat{q}_{\mu}^{c}, \hat{q}_{K}^{c}
| w_{\mu}, w_{K}; q_{\mu}^{(N + 1)},
q_{K}^{(N + 1)} \bigr)
\\
&&\quad\qquad{} - \tfrac{1}{2}\log( \det H_{\mu} ) - \tfrac{1}{2}\log( \det
H_{K} ) + MN\log2\pi,\nonumber
\end{eqnarray}
where $\hat{q}$ is the maximum of the penalized log-likelihood $Q$ in
equation (\ref{equ16}) and
\begin{equation}
\label{equ21} H \bigl( \hat{q}{}^{c}| w, q^{(N + 1)} \bigr) =
\frac{\partial^{2}\log L ( \hat{q}^{c}\llvert w, q^{(N + 1)}
)}{\partial q^{c}\,\partial(q^{c})^{t}} - \Sigma^{c} \bigl(w, q^{(N +
1)} \bigr),
\end{equation}
for a fixed weight $w$ for either $w_{\mu}$ or $w_{K}$.
Thus,\vspace*{1pt} maximizing equation (\ref{equ16}) with respect to $q^{c}$ and equation (\ref{equ20})
with respect to $ ( w_{\mu},w_{K}; q_{\mu}^{(N + 1)}, q_{K}^{(N + 1)}
)$, in turn, achieves our objective. In the former maximization, a
quasi-Newton method using the gradients $\partial\log L ( q ) /
\partial q$ and the Newton method making use of the Hessian matrices,
equation (\ref{equ21}), endure a fast convergence regardless of high dimensions. For
the latter maximization, a direct search such as the simplex method is used.
A flowchart of numerical algorithms is described in the appendices of
\citeauthor{OGA04}
(\citeyear{OGA04,OGA11N2}).
Anomaly factor functions under the optimal roughness penalty result in
suitably smooth curves throughout the period. Furthermore, there may be a
change point that results in sudden changes in parameters $\mu$ or $K$. To
examine such a discontinuity, a sufficiently small weight is put into the
interval that includes a change point (e.g., $w = 10^{-5}$), and the
goodness-of-fit by \textit{ABIC} is compared with that of the smooth model
with the optimal weights for all intervals.
It is useful to obtain the estimation error bounds of the MAP estimate
$\hat q$ at each time of an observed earthquake. The
joint error distribution of the parameters
at $\hat q$ is nearly a $2N$-dimensional\vspace*{1pt} normal
distribution $N(0, H^{- 1})$, where $H^{- 1} = (h^{ i, j})$, and $H =
(h_{ i, j})$ is the Hessian matrix in equation (\ref{equ21}). Hence, the covariance
function of the error process becomes
\begin{equation}
\label{equ22} c ( u,v ) = \sum_{i = 1}^{2N}
\sum_{j = 1}^{2N} F_{i} ( u
)h^{i,j}F_{j} ( v ),
\end{equation}
where $F_{i} = F_{ N+ i}$ for $i = 1, 2,\ldots, N$, which is defined in
equation (\ref{equ13}). Thus, the standard error of $q$ is provided by
\begin{equation}
\label{equ23} \varepsilon( t ) = \bigl[ \varepsilon_{\mu} ( t ),
\varepsilon_{K} ( t ) \bigr] = \sqrt{C ( t,t )}.
\end{equation}
\subsection{Bayesian model comparison}\label{sec27} It is necessary to
compare the
goodness of fit among the competing models. From equation (\ref{equ20}), the \textit{ABIC}
[\citet{Aka80}] can be obtained as
\begin{eqnarray}
\label{equ24} \mathit{ABIC} &=& ( - 2 )\max_{w_{\mu},w_{K}; q_{\mu}^{(N + 1)},
q_{K}^{(N + 1)}}\log\Psi \bigl(
w_{\mu},w_{K}; q_{\mu}^{(N + 1)},
q_{K}^{(N + 1)} \bigr)
\nonumber
\\[-8pt]
\\[-8pt]
&&{} + 2 \times( \# \mathrm{hyperparameter} ).
\nonumber
\end{eqnarray}
Specifically, models 1 and 2 [1(a)~and~(b), 2(a)~and~(b) in Table~\ref{tab1}] have four
hyperparameters, and model 3 [3(a)~and~(b)] has eight. A Bayesian model with
the smallest \textit{ABIC} value provides the best fit to the data.
Since there are various constraints in the different setups, the resulting
\textit{ABIC} values cannot be simply compared because of unknown different
constants, mainly due to the approximations in equation (\ref{equ20}).
Alternatively, the difference of \textit{ABIC} values relative to those
corresponding to the reference model are used. In other words, the
reduction amount of the \textit{ABIC} value from a very heavily constrained
case,
\begin{equation}
\label{equ25} \Delta \mathit{ABIC} = \mathit{ABIC} - \mathit{ABIC}_{0},
\end{equation}
where \textit{ABIC} is that of equation (\ref{equ24}) and $\mathit{ABIC}_{0}$ is
the \textit{ABIC} value with very heavy fixed weights, which constrain the
function to be almost constant. Therefore, the $\Delta\mathit{ABIC}$
approximates the \textit{ABIC} improvement from the flat anomaly functions
[$q(t) = 1$ for all $t$] to the optimal functions.
Likewise in \textit{AIC}, it is useful to keep in mind that exp$\{-
\Delta
\mathit{ABIC}/2\}$ can be interpreted as the relative probability of how
the model with the smallest \textit{ABIC} value is superior to others
[e.g., \citet{Aka80}].
\section{Applications}\label{sec3}
\subsection{The stationary ETAS model versus the two-stage ETAS model}\label{sec31}
First, we estimate the stationary ETAS model that has been applied to a
series of earthquakes of magnitude (M) 3.0 and larger contained in the
polygonal region highlighted in Figure~\ref{fig1}, from October 1997 to the M9.0
Tohoku-Oki earthquake on March 11, 2011. Specifically, the MLE has been
obtained for the stationary ETAS model [equation~(\ref{equ2})] by applying a normal
activity for earthquakes of M3.0 and larger from October 1997 to March 10,
2011 (Figure~\ref{fig2}). According to the estimated theoretical cumulative
curve in
ordinary time [equation~(\ref{equ4})] and transformed time [equation(\ref{equ5})] in Figure~\ref{fig2},
the ETAS model appears to fit very well except for a period near 2008
and a
period after the Tohoku-Oki earthquake, which is in good accordance with
\citet{OGA12}. These anomalies are highlighted by dashed ellipses
and dashed
rectangles in Figure~\ref{fig2}.
The former is the apparent lowering due to substantially small productivity
in the aftershock activity of the 2007 Chuetsu-Oki earthquake (open
star in
Figure~\ref{fig1}). Interestingly enough, the 2004 Chuetsu earthquake (closed star)
and the 2007 Chuetsu-Oki earthquake, which are about 40 km apart, have the
same magnitude (M6.8), but the number of aftershocks of $M\geq4.0$
differs by 6--7 times [\citet{Age}].
The latter is due to the activation relative to the predicted ETAS model.
The March 11, 2011 M9.0 Tohoku-Oki earthquake induces this activation. On
the other hand, a series of aftershocks (located in region~A, Figure~\ref{fig1}) of
the 2008 M7.2 Iwate-Miyagi Prefecture inland earthquake is quiet relative
to the occurrence rate predicted by the ETAS model estimated from the
aftershock data before the M9.0 earthquake.
\begin{figure
\includegraphics{759f03.eps}
\caption{Cumulative number and magnitude of the aftershock sequence
with $M\geq 1.5$, following the 2008 Iwate-Miyagi earthquake of
M7.2, from the region A against ordinary time. The ETAS model is fitted to
the sequence for the period from one day after the main shock ($S=1.0$
day) to the Tohoku-Oki earthquake (March 11, 2011; dashed line). The almost
overlapping red curve indicates the theoretical ETAS cumulative function,
equation (\protect\ref{equ4}), and the extension to the rest of the period until April 2012.
The inset rectangle magnifies the cumulative curve for the extrapolated
period.}\label{fig3}
\end{figure}
An analysis of the 2008 earthquake aftershock sequence is shown in
Figure~\ref{fig3}. Here the ETAS model is fitted to the period from one day
after the main
shock until the M9.0 earthquake. The estimated intensity is then
extrapolated to span an additional year. The change point at the M9.0
earthquake is substantial, decreasing the total \textit{AIC} by $28.5$, showing a
relative quiescence afterward. The penalty quantity $q$ in the
$\mathit{AIC}_{12}$ of equation (\ref{equ9}) equals zero because the change
point is
given by the information outside of the aftershock data, hence, $\Delta
\mathit{AIC} = -28.5$. Therefore, the occurrence of the Tohoku-Oki
earthquake is a significant change point.
Hereafter, the data set becomes very difficult for conventional ETAS
analysis. The earthquake swarm near Lake Inawashiro began March 18,
2011, a
week after the M9.0 earthquake in region~B (Figure~\ref{fig1}). Seismic activity in
this area was very low before the M9.0 event. The swarm mostly
consisted of
small earthquakes with magnitudes less than $3.0$. The largest earthquake in
this cluster, an earthquake of~M4.6, occurred $50$ days after the M9.0
earthquake, and its aftershock sequence seemed to decay normally.
\begin{figure
\includegraphics{759f04.eps}
\caption{Stationary and two-stage ETAS models fitted to region B.
The ETAS model is fitted to the entire period from March 18, 2011, to the
end of 2012 with the preliminary period of the first 0.1 days (blue line),
the period before the M4.6 event ($t = 49.8$ days) (green solid line) then
extrapolated forward (green dashed line), and the period after the M4.6
event (red solid line) then extrapolated backward (red dashed line). The
black curve shows the cumulative number of observed earthquakes. The left
panel plots these against ordinary time, whereas the right panel plots
these against the number of earthquakes.}\label{fig4}
\end{figure}
\begin{table
\tabcolsep=0pt
\caption{The ETAS parameters of region B fitted to \textup{(a)} the entire
period, \textup{(b)} and \textup{(c)} before the change point, and \textup{(d)} after the change
point. Their standard errors are in parentheses. The improvement of the
two-stage ETAS model relative to the stationary ETAS model is $\Delta \mathit{AIC}
= (422.9 - 118.3) - 442.8 = -138.2$. The MLE for the change point is
$t = 49.8$, which coincides with the time just before the M4.6.
The threshold magnitude is $M_{z} = 2.5$. Numbers are
rounded to three significant digits}\label{tab2}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}@{}ld{2.9}d{2.9}d{2.9}d{2.9}d{2.9}d{4.1}@{}}
\hline
\textbf{Period} & \multicolumn{1}{c}{$\bolds{\mu}$} & \multicolumn{1}{c}{$\bolds{K_{0}}$}
& \multicolumn{1}{c}{$\bolds{c}$} & \multicolumn{1}{c}{$\bolds{\alpha}$}
& \multicolumn{1}{c}{$\bolds{p}$} & \multicolumn{1}{c@{}}{$\bolds{\mathit{AIC}}$}\\
\hline
(a) The whole & 9.77 \times10^{-2} & 6.54 \times10^{-2} & 9.64 \times10^{-4} & 0.215 & 0.900 & 442.8 \\
period & (7.81 \times10^{-2}) & (2.37 \times10^{-2}) & (6.35\times10^{-4}) & (9.77 \times10^{-2}) & (9.84 \times10^{-3}) &
\\[3pt]
(b) Before & 1.41 & 1.05 \times10^{-1} & 8.52 \times10^{-2} & 3.06 \times10^{-15} & 1.00 & -103.0\\
change point & (3.39 \times10^{-1}) & (6.99 \times10^{-2}) & (1.09 \times10^{-1}) & (9.35 \times10^{-1}) & & \\
with fixed\\
$p = 1.0$
\\[3pt]
(c) Before & 1.27 & 2.12 \times10^{+ 11} & 1.04 \times 10^{+1} & 2.25 \times10^{-12} & 1.13 \times10^{+1} & -118.3\\
change point & (5.52 \times10^{-1}) & (4.71) & (3.81 \times 10^{-1}) & (1.03) & (2.31 \times10^{-1}) & \\
without\\
fixed $p$\\[3pt]
(d) After & 6.58 \times10^{-2} & 3.58 \times10^{-2} & 7.11 \times10^{-5} & 0.912 & 0.945 & 422.9\\
change point & (1.43 \times10^{-1}) & (1.90 \times10^{-2}) & (1.01\times10^{-3}) & (1.10 \times10^{-1}) & (1.87 \times10^{-1}) &\\
\hline
\end{tabular*}
\end{table}
First, the stationary ETAS model is applied to the whole period. The
theoretical cumulative function (solid light blue curves, Figure~\ref{fig4}) is
biased below from the empirical cumulative function, indicating a
substantial misfit. Hence, the two-stage ETAS model is applied to the data
to search the MLE for a change point. Table~\ref{tab2} lists the estimated
parameters and \textit{AIC} values. The change-point analysis (cf. Section~\ref{sec22})
implies that the MLE of the change point is at $t = 49.8$ days from the
beginning of this cluster, which coincides with the time just before the
M4.6 earthquake occurred. The two-stage ETAS model with this change point
improves the \textit{AIC} by $138.2$ (see Table~\ref{tab2}). The first-stage ETAS model before
the change point, with a fixed parameter $p = 1.0$, still displays a large
deviation from the ideal fit (cf. the solid green curve in Figure~\ref{fig4}). The
magnitude sensitivity parameter $\alpha$ becomes very small relative to
that of the second-stage ETAS model. Such a small value implies that almost
all earthquakes in the first stage occurred independently to preceding
magnitudes (i.e., close to a Poisson process), and can be mostly attributed
to the average $\mu$ rate of the background seismicity. The first stage
$\mu$ rate is two orders of magnitude higher than the second stage rate.
If $p$ is not fixed, the estimated $K_{0}$, $c$ and $p$ have extremely
large values for a normal earthquake sequence while $\alpha$ approaches
zero. Consequently, the model is again approximate to a nonstationary
Poisson process, characterizing the sequence as a swarm, with an \textit{AIC}
smaller than that of the $p = 1.0$ scenario. The large discrepancies
between the estimated parameter values between (b) and (c) in Table~\ref{tab2}
suggest that the stationary ETAS model is not well defined for this
particular earthquake sequence in the first period before the change point.
The standard errors for the parameter $\alpha$ are multiple orders of
magnitude greater than those of the estimates themselves. The narrow
magnitude range makes it difficult for the model to distinguish the effects
of $K_{0}$ and $\alpha$, causing a trade-off between these two parameters,
thus providing inaccurate estimations. For the case without a fixed $p$,
the aftershock productivity $K_{0}$ becomes extremely small in compensation
for the small $\alpha$ estimate.
After the change-point time of the M4.6 earthquake, the ETAS model fits
considerably well for several months. Then, a deviation becomes noticeable
relative to the solid red cumulative curve in Figure~\ref{fig4}. From these
observations, it is concluded that the M4.6 earthquake has reduced swarm
activity and that decaying normal aftershock type activity has dominated.
\subsection{Comparison of the nonstationary models}\label{sec32} In
this section the
proposed nonstationary models and methods outlined in Sections~\ref{sec24}--\ref{sec26} are
applied to the same data from region B near Lake Inawashiro. To replicate
the transient nonstationary activities in this particular region, we use
the seismic activity in the larger polygonal region in Figure~\ref{fig1} for the
period before the M9.0 earthquake (MLEs are shown in Figure~\ref{fig2}). Such a
reference model represents a typical seismicity pattern over a wide region
throughout the period, and therefore represents a robust estimate against
the inclusion of local and transient anomalies.
By fixing the reference parameters $c, \alpha$ and $p$, both in the
stationary and two-stage ETAS models, $\mu$ and $K_{0}$ are estimated for
events from region B after the M9.0 event, with a magnitude $M\geq 2.5$. Table~\ref{tab3} summarizes the re-estimated parameters, together with
the corresponding \textit{AIC} values. The \textit{AIC} improvement of the two-stage ETAS
model is $126.2$.
\begin{table}
\tabcolsep=0pt
\caption{Reference parameters adjusted to the data from region B
and the parameters of the present two-stage ETAS model (standard errors in
parentheses) with fixed $c$, $\alpha$ and $p$ of the reference model (standard
errors in brackets), with their \textit{AIC} values. The improvement of the
two-stage ETAS model relative to the present stationary ETAS model is
$\Delta \mathit{AIC} = 434.7 - 95.4 - 465.5 = -126.2$. Also, the improvement of the
present two-stage ETAS model relative to the stationary ETAS model in
Table~\protect\ref{tab2} is as follows: $\Delta \mathit{AIC} = 434.7 - 95.4 - 442.8 = -103.5$.
The change point is at $t = 49.8$, corresponding to the time just before the
M4.6 earthquake. The threshold magnitude $M_{z} = 2.5$.
Numbers are rounded to three significant digits}\label{tab3}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}@{}ld{2.9}d{2.9}d{3.9}d{2.9}d{2.9}d{3.1}@{}}
\hline
\textbf{Period} & \multicolumn{1}{c}{$\bolds{\mu}$} & \multicolumn{1}{c}{$\bolds{K_{0}}$}
& \multicolumn{1}{c}{$\bolds{c}$} & \multicolumn{1}{c}{$\bolds{\alpha}$}
& \multicolumn{1}{c}{$\bolds{p}$} & \multicolumn{1}{c@{}}{$\bolds{\mathit{AIC}}$}\\
\hline
(a) The whole & 1.92 \times10^{-1} & 2.49 \times10^{-2} & 6.02 \times10^{-3} & 2.03 & 1.11 & 465.5\\
period & (3.58 \times10^{-2}) & (5.82 \times10^{-3}) & {[}2.50 \times10^{-3}{]} & {[}1.27 \times10^{-2}{]} & {[}5.44 \times10^{-3}{]} &
\\[3pt]
(b) Before the & 3.31 & 6.77 \times10^{-3} & \multicolumn{1}{c}{$602 \times10^{-3}$\phantom{.00}} & 2.03 & 1.11 & -95.4\\
change point & (1.04 \times10^{-1}) & (3.27 \times10^{-3}) & & &&
\\[3pt]
(c) After the & 1.95 \times10^{-1} & 1.56 \times10^{-2} & 6.02 \times10^{-3} & 2.03 & 1.11 & 434.7\\
change point & (2.99 \times10^{-2}) & (6.41 \times10^{-3}) & & & \\
\hline
\end{tabular*}
\end{table}
Next, we have applied the nonstationary ETAS models listed in Table~\ref{tab1}, with
and without a change point taken into consideration, using the reference
parameters in the first row of Table~\ref{tab3}. Here, if a change point of M4.6 at
the time $t = 49.8$ days occurs, we propose a very small fixed value such
as that described in Section~\ref{sec25}.
\begin{figure}[t]
\includegraphics{759f05.eps}
\caption{Various inversion results of all considered models for the
data from region B. The model numbers correspond to those of Table~\protect\ref{tab1}, and
the models with prime ($'$) correspond to those that include a change
point. The background rates $\mu (t)$ are shown in red connected lines, and
the productivity $K_{0}(t)$ is shown in blue dots at earthquake
occurrence times. The gray spiky curves represent the conditional intensity
rates $\lambda (t | H_{t})$. The above three functions
are plotted on a logarithmic scale. The upper and lower gray horizontal
lines represent the reference parameters $\mu$ and $K_{0}$,
respectively (see Table~\protect\ref{tab2}). The vertical dashed line shows the change-point
time, $t = 4$ days elapsed from March~18, 2011. The horizontal
axis indicates days elapsed.}\label{fig5}
\end{figure}
Figure~\ref{fig5} shows all of the inversion results (maximum posterior estimates)
for a total of 12 models. The $\Delta\mathit{ABIC}$ values of the
corresponding models are given in Table~\ref{tab4}. Models with the change point
outperform corresponding models without the change point. This highlights
the significance of jumps at the change point. Such improvements via jumps
are smaller between corresponding models with constraints on the
transformed time. This is because those models already present jumps or
sharp changes to some extent in the target parameters even without setting
change points, due to the expanded transformed time during the dense event
period after the M4.6 event in ordinary time. Results also show that models
with constraints on ordinary time yield better results than those with the
transformed time. This is probably because the data set only contains
gradually changing parameters except at the change point.
The smallest $\Delta\mathit{ABIC}$ is achieved by model 3(a$'$) in
which both $q_{\mu} (t)$ and $q_{K}(t)$ are nonstationary on the smoothness
constraints under ordinary time, with a jump at the time of the M4.6
earthquake. Figure~\ref{fig6} shows variations of the background and productivity
rates in the selected nonstationary model. These variations suggest that
the intensity of aftershock productivity $K_{0} (t) ( = K_{0}
q_{K}(t))$ is
extremely low during early periods of earthquake swarms until the M4.6
earthquake occurs; meanwhile, the background seismicity $\mu(t) ( = \mu
q_{\mu} (t))$ changes at a high rate. Therefore, the total seismicity
$\lambda_{\theta} (t | H_{t})$ in that period is similar to a nonstationary
Poisson process with intensity rates $\mu(t)$ of the background activity.
After the M4.6 earthquake occurred, the $\mu(t)$ rate gradually decreased
while $K_{0} (t)$ increased. These changes are roughly approximated by the
estimated two-stage ETAS model in Table~\ref{tab3}, in which $\mu$ before the change
point is higher, while $K_{0}$ is lower than those after the change
point.
\begin{table}[b]
\tabcolsep=0pt
\caption{$\Delta\mathit{ABIC}$ value of each model defined in equation
(\protect\ref{equ25}). The underlined model has the smallest value. The prime ($'$)
indicates the models that further assume a change point at $t = 49.8$, the
time when the M4.6 earthquake occurred}\label{tab4}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}@{}lcccc@{}}
\hline
\textbf{Models} & \textbf{a} & \textbf{a$\bolds{'}$} & \textbf{b} & \textbf{b}$\bolds{'}$\\
\hline
{1} & $-170.0$ & $-177.2$ & $-132.4$ & $-$134.1\\
{2} & $-175.3$ & $-180.1$ & $-136.1$ & $-$137.2\\
{3} & $-250.1$ & $-$\underline{260.8} & $-148.1$ & $-$151.5\\
\hline
\end{tabular*}
\end{table}
\begin{figure
\includegraphics{759f06.eps}
\caption{The selected best-fitted model 3(a$'$) and errors of the
inversion solutions. The background rate $\mu$(t) is shown in solid red,
with $\mbox{one}-\sigma$ error bounds in red dashed lines. $K_{0}(t)$ is
shown in blue dots with $\mbox{one}-\sigma$ error bars at the occurrence times.
The gray spiky curve represents the variation of the intensity rates
$\lambda(t | H_{t})$. All of the above estimates are
plotted on a logarithmic scale. The solid gray horizontal line represents
the reference $\mu$ value, and the horizontal dashed line represents the
reference $K_{0}$ value (see Table~\protect\ref{tab2}). The horizontal axis is the
elapsed days from March 18, 2011. The vertical dashed line shows the change
point $t = 49.8$ elapsed days. The middle panel displays the
longitudes versus the elapsed times of the earthquake occurrences in region
B. The diameters of the circles are proportional to the earthquake
magnitudes. The bottom panel shows magnitudes of earthquakes versus the
ordinary elapsed times in days.}\label{fig6}
\end{figure}
If the $\Delta\mathit{ABIC}$ of model 3(a$'$) in Table~\ref{tab4} and
$\Delta
\mathit{AIC}$ of the two-stage ETAS models in Tables~\ref{tab2} and \ref{tab3} are compared
[\citeauthor{Aka85} (\citeyear{Aka85,Aka87})], the former model
displays a much better fit,
with a
difference of more than 130. This indicates that the specific details of
transient variations in model~3(a$'$) appear to be substantial. Model
3(a$'$) further shows that the background $\mu(t)$ rate decreased
after about $t = 400$ days, indicating that the swarm component of the
seismicity decreased. To demonstrate the reproducibility of the detailed
variations with the similar data sets, Figure~\ref{fig7} shows the re-estimated model
3(a$'$) utilizing the same optimization procedure from simulated data
in the estimated model 3(a$'$) in Figure~\ref{fig6}. See the \hyperref[sec6]{Appendix} for more
details.
\begin{figure
\includegraphics{759f07.eps}
\caption{The maximum a posteriori (MAP) solution of a synthesized
data set by the estimated model~3(\textup{a}$'$) (shown in Figure~\protect\ref{fig6}) with the same
reference parameters in Table~\protect\ref{tab2}. The re-estimated parameters $\mu(t)$ and
$K_{0}(t)$ are shown in red and blue curves, respectively, with
two-fold error bounds. The upper and lower dashed black curves represent
the true $\mu(t)$ and $K_{0}(t)$ (same as those in Figure~\protect\ref{fig6}),
respectively.}\label{fig7}
\end{figure}
The model's performance is graphically examined by plotting the estimated
cumulative number of events (\ref{equ4}) to compare with the observed events in
Figure~\ref{fig8}, which shows that the observed events become almost a stationary
Poisson process, although a few clustering features remain.
\begin{figure
\includegraphics{759f08.eps}
\caption{Estimated cumulative number of events by model 3(\textup{a}$'$) (red
curves) and the observed number of events (black curve) for the ordinary
time (top panel) and residual time (bottom panel). Gray circles show the
depths of the swarm events versus the corresponding time.}\label{fig8}
\end{figure}
It is worthwhile to discuss why model 3(b$'$) with constraints under
the transformed time has a poorer fit than model 3(a$'$) with
constraints under ordinary time. The MAP estimate of model 3(b$'$) is
shown in Figure~\ref{fig9}, where the transformed time $\tau$ in this case is
defined in equations (\ref{equ4}) and (\ref{equ5}) using the reference ETAS model, the
parameter value of which is listed in the first row of Table~\ref{tab3}. Parameter
variations in the period after the change point are similar to those of the
overall best model 3(a$'$). Variations during the period before the
change point are different with higher $K_{0} (t)$ and lower $\mu(t)$.
However, in this particular application, the performance of model
3(b$'$) on the whole is inferior in terms of $\Delta\mathit
{ABIC}$ by
a difference of greater than $100$. This may be because the above mentioned
reference ETAS-based transformed time of the former period worked poorly,
unlike during the latter period.
\begin{figure
\includegraphics{759f09.eps}
\caption{Variations of conditional intensity rates $\lambda(\tau|
H_{\tau})$, background rate $\mu(\tau)$ and aftershock
productivity $K_{0}(\tau)$ of model 3(\textup{b}$'$) versus the transformed
time $\tau$ of the reference ETAS model (the first row of Table~\protect\ref{tab3}). The
other details are the same as those in Figure~\protect\ref{fig6}.}\label{fig9}
\end{figure}
Although the goodness of fit of model 3(b$'$) over the whole period
(particularly during the former period) is not quite satisfactory, it is
worthwhile to examine the changes of $\mu( \tau)$ and $K_{0} (\tau)$
during the latter period in Figure~\ref{fig9}. The conditional intensity rate
$\lambda_{\theta} (t | H_{t})$, background rate $\mu( \tau)$ and
aftershock productivity rate $K_{0} (\tau)$ rapidly decrease not only after
the M4.6 earthquake but also after relatively large earthquakes. On such
sharp drops, there is a technical but simple explanation. Models in
Table~\ref{tab4}
with smoothness constraints on the transformed time are sensitive to
catalog incompleteness during small time intervals after large earthquakes.
In other words, a substantial number of small earthquakes that occur
immediately after a large earthquake are missing in the earthquake catalog
[e.g., \citet{OGAKAT06}; \citet{OMIetal13}]. Present results
suggest that the smoothing on the transformed time can be used as a
supplemental tool to check catalog completeness. The time transformation
stretches out ordinary time where the intensity rate is high and, hence,
transforming the smoothed parameters back to ordinary time can result in
sharp changes. This type of constraint can be useful for different
applications in which occasional rapid changes are expected.\looseness=-1
\subsection{Seismological complements and implications of the results}\label{sec33} Used
as a reference model, the polygonal region in Figure~\ref{fig1} is known to have a
similar seismicity pattern with similar focal mechanisms under the
west--east compressional tectonic field, as described in \citet{TERMAT}
and \citeauthor{TODLIAROS11} (\citeyear{TODLIAROS11,TODSTEJIA11}). For example, earthquakes have
mostly north--south strike angles and west--east directional thrust faults
in this region. This pattern can also be seen in the configurations of
active fault systems on the surface.
In the above sections, the estimation procedures of the models presented
here have been illustrated with a data set that includes a cluster of swarm
earthquakes triggered by the March 11, 2011 M9.0 Tohoku-Oki earthquake.
Swarm activity in this region seems to be triggered by surface waves
emitted from the M9.0 source, and has been studied by \citet{TERHASMAT13}
using the seismological theory and methods used in \citet{TERMILDEI12}. Here they attribute swarm activity to the weakening of the fault
via an increase of pore fluid pressure caused by the dynamic triggering
effect due to surface waves of the Tohoku-Oki rapture. Thus, the initially
very high and then decreasing rate of $\mu(t)$ reflects changes in fault
strength, probably due to the intrusion and decrease in pore fluid
pressure. The analyses presented here support the quantitative,
phenomenological evidence of fault weakening via the intrusion of water
into the fault system in earlier periods [\citeauthor{TERMILDEI12} (\citeyear{TERMILDEI12,TERHASMAT13})].
Similarly, by monitoring swarm activity, this nonstationary model can be
expected to make quantitative inferences of magma intrusions and draining
during volcanic activity.
The background seismicity parameter in the ETAS model is sensitive to
transient aseismic phenomena such as slow slips (quiet earthquakes) on and
around tectonic plate boundaries [\citet{LLEMCGOGA09}, \citet{OKUIDE11}]. This could possibly link a given swarm activity to the weakening
of interfaces. Changes in the pore fluid pressure, for example, alter the
friction rate of fault interfaces, thereby changing the fault strength.
Hence, monitoring the changes in background seismicity has the
potential to
detect such aseismic events.
Changes in the aftershock productivity $K_{0}$, on the other hand, appear
to depend on the locations of earthquake clusters and appear to vary among
clusters where secondary aftershocks are conspicuous. The aftershock
productivity $K_{0}$ therefore reflects the geology around faults rather
than the changes in stress rate. The application of the space--time ETAS
model with location-dependent parameters [e.g., \citet{OgaKatTan03},
\citeauthor{OGA04}
(\citeyear{OGA04,OGA11N2})] reveals that the $K_{0}$ function varies
(i.e., location
sensitive) unlike other parameters. Still, the task remains to confirm the
link between the changes in ETAS parameters and physical processes
happening on and around faults.
\section{Conclusions and discussion}\label{sec4} There are many
examples in seismology
in which different authors have obtained differing inversion results for
the same scientific phenomenon. These differences are attributed to the
adoption of different priors for the parameters of a given model. Scenarios
in this study have the same problem and are highlighted in Figure~\ref{fig5}. Model
parameters in this study are estimated by maximizing the penalized log
likelihood, which is intrinsically nonlinear. Besides adjusting the weights
in the penalty (namely, hyperparameters of a prior distribution), it is
necessary to compare the adequacy of different penalties (prior
distributions) associated with the same likelihood function. For these
purposes, we have proposed the objective procedure using $\Delta \mathit{ABIC}$ and $\Delta \mathit{AIC}$.
A suitable ETAS model [equation~(\ref{equ2})] is first established with MLE as the
reference predictive model to monitor future seismic activity and to detect
anomalous seismic activity. Sometimes, transient activity starts in a
region with very low seismicity. In such a case, it is both practical and
applicable to use a data set from a wider region to estimate the stable and
robust parameter values of $c$, $\alpha$ and $p$ in the ETAS model
[equation~(\ref{equ2})]. Then, the competing nonstationary ETAS models in equation
(\ref{equ10}) are fitted together with constraint functions in equations (\ref{equ11})~and~(\ref{equ12}) using either ordinary time or transformed time to penalize the
time-dependent parameters in the models. The corresponding Bayesian models
include a different prior distribution of the anomaly factor coefficients
$q_{\mu} (\cdot)$ and $q_{K}(\cdot)$, which are functions of either the ordinary
time $t$ [models 1(a)--3(a) in Table~\ref{tab1}] or the transformed
time $\tau$ in the reference ETAS model [models~\mbox{1(b)--3(b)}].
Furthermore, models in which the anomaly functions involve a discontinuity
[models 1(a$'$)--3(a$'$) and 1(b$'$)--3(b$'$)] are considered. Using the
$\Delta \mathit{ABIC}$ value, the goodness-of-fit performances of all of the
different models are summarized in Table~\ref{tab4}. Among the competing models,
model~3(a$'$) attained the smallest $\Delta \mathit{ABIC}$ value, and it is
therefore concluded that this model provides the best inversion result for
this particular data set.
Thus, changes in background seismicity $\mu$ and/or aftershock productivity
$K_{0}$ of the ETAS model can be monitored. The background seismicity rate
in the ETAS models represents a portion of the occurrence rate due to
external effects that are not included in the observed earthquake
occurrence history in the focal region of interest. Therefore, changes in
the background rate have been attracting the interest of many researchers
because such changes are sometimes precursors to large earthquakes. The
declustering algorithms [e.g., \citet{REA85}, \citeauthor{ZhuOgaVer02} (\citeyear{ZhuOgaVer02,ZHUOGAVER04})] have been adopted to determine the background seismicity by
stochastically removing the clustering components depending on the
ratio of
the background rate to the whole intensity at each occurrence time. The
change-point analysis and nonstationary models presented in this study,
however, objectively serve a more quantitatively explicit way to approach
this task.
The case where the other three parameters $c, \alpha$ and $p$ in equation
(\ref{equ2}) also vary with time was not examined in this study. For example, in
Figure~\ref{fig8}, we have seen that the best model in our framework does not
capture all of the clustering events but misses a few small clusters, which
suggests the time dependency of the parameters. For another example, we
have seen the effect of missing earthquakes in Figure~\ref{fig7}, suggesting that
parameter $c$ may depend on the magnitude of the earthquake, leading to a
significant correlation between $c$ and $p$. Furthermore, in Section~\ref{sec23},
it is mentioned that $K_{0}$ is correlated with the parameter $\alpha$.
Unstable estimations of $K_{0}$ and the $\alpha$ value in the swarm period
before the M4.6 earthquake can be seen in Table~\ref{tab2}, during which period most
of the magnitudes are between 2.5 and 3. This is another reason why the
$\alpha$ value is fixed by the corresponding reference parameter $\alpha$
when the nonstationary models are applied. Owing to the linearly
parameterized coefficients of the functions $q_{\mu}$ and $q_{K}$ in
equation (\ref{equ10}), the maximizing solutions of the penalized log-likelihood
function [equation~(\ref{equ16})], in spite of the high dimension, can be obtained
uniquely and stably by fixing the three parameters $c, \alpha$ and $p$.
\begin{appendix}
\section*{Appendix: Synthetic test of reproducibility of nonstationary patterns}\label{sec6}
We tested our method with synthetic data sets to check if both $\mu(t)$
and $K_{0} (t)$ can be reproduced by simulated data sets that are similar
to observed data sets. We used the reference parameter set (Table~\ref{tab2}) with
the best estimated $\mu(t)$ and $K_{0} (t)$ of model 3(a$'$).
The magnitude sequence of the synthetic data was generated on the basis of
the Gutenberg--Richter law with a $b$-value of the original data set
($b =
1.273$). In other words, the magnitude of each earthquake will
independently obey an exponential distribution such that $f (M) = \beta\exp\{- \beta(M - M_{c})$, $M \geq M_{c}$, where $\beta= b\ln10$, and
$M_{c} = 2.5$ is the magnitude value above which all earthquakes are
detected.
The thinning method [\citeauthor{OGA81} (\citeyear{OGA81,OGA98})] is
adopted for data
simulation. A~total of 470 events were simulated with a threshold magnitude of 2.5. Model
3(a$'$) was then fitted to the simulated data sets, with a change point
at the same time as the original data (between the 182nd and 183rd event).
Results are shown in Figure~\ref{fig7}; the estimated $\mu(t)$ and $K_{0} (t)$
appear to be similar to the original $\mu(t)$ and $K_{0} (t)$ in
Figure~\ref{fig6},
respectively, within a $2 \sigma$ error.
\end{appendix}
\section*{Acknowledgments}\label{sec5}
We are grateful to the Japan Meteorological Agency (JMA), the National
Research Institute for Earth Science and Disaster Prevention (NIED) and
the universities for the hypocenter data. We used the TSEIS visualization
program package [\citet{TSU96}] for the study of hypocenter data.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,978 |
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Interview with MTV Lebanon News on US policy on Syria and Hezbollah, and US actions against ISIS http://ow.ly/Zee850HR108
Interview with Syria TV on D-ISIS Campaign, Feb 3, 2022
Short interview with opposition channel Syria TV, discussing the future of the U.S. campaign to defeat ISIS, following the operation against the leader of the group in Syria. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,374 |
# About Island Press
Island Press is the only nonprofit organization in the United States whose principal purpose is the publication of books on environmental issues and natural resource management. We provide solutions-oriented information to professionals, public officials, business and community leaders, and concerned citizens who are shaping responses to environmental problems.
In 1998, Island Press celebrates its fourteenth anniversary as the leading provider of timely and practical books that take a multidisciplinary approach to critical environmental concerns. Our growing list of titles reflects our commitment to bringing the best of an expanding body of literature to the environmental community throughout North America and the world.
Support for Island Press is provided by The Jenifer Altman Foundation, The Bullitt Foundation, The Mary Flagler Cary Charitable Trust, The Nathan Cummings Foundation, The Geraldine R. Dodge Foundation, The Ford Foundation, The Vira I. Heinz Endowment, The W. Alton Jones Foundation, The John D. and Catherine T. MacArthur Foundation, The Andrew W. Mellon Foundation, The Charles Stewart Mott Foundation, The Curtis and Edith Munson Foundation, The National Fish and Wildlife Foundation, The National Science Foundation, The New-Land Foundation, The David and Lucile Packard Foundation, The Surdna Foundation, The Winslow Foundation, The Pew Charitable Trusts, and individual donors.
Copyright © 1998 by Island Press
All rights reserved under International and Pan-American Copyright Conventions. No part of this book may be reproduced in any form or by any means without permission in writing from the publisher: Island Press, 1718 Connecticut Avenue, N.W., Suite 300, Washington, DC 20009.
Acknowledgment of permission to reprint previously published material appears on pages 361-362.
ISLAND PRESS is a trademark of The Center for Resource Economics.
Library of Congress Cataloging-in-Publication Data
McHarg, Ian L.
[Selections. 1998]
To heal the earth: the selected writings of Ian L. McHarg / edited by Ian L. McHarg and Frederick R. Steiner: foreword by Bob Yaro.
p. cm.
Includes bibliographical references (p. ) and index.
9781597269223
1. Landscape ecology. 2. Land use—Planning—Environmental aspects. 3. Landscape design—Environmental aspects. I. Steiner, Frederick R. II. Title.
QH541.15.L35M38 1998
98-3337
577.5'5—dc21
CIP
Printed on recycled, acid-free paper
Manufactured in the United States of America
10 9 8 7 6 5 4 3 2 1
# Table of Contents
About Island Press
Title Page
Copyright Page
Foreword
Preface
Introduction
Part I \- Changing the Nature of Design and Planning: Theoretical Writings
References
1 \- Man and Environment (1963)
2 \- The Place of Nature in the City of Man (1964)
3 \- Ecological Determinism (1966)
4 \- Values, Process and Form (1968)
5 \- Natural Factors in Planning (1997)
Part II \- Planning the Ecological Region
References
6 \- Regional Landscape Planning (1963)
7 \- Open Space from Natural Processes (1970)
8 \- Must We Sacrifice the West?(1975)
9 \- Ecological Planning: The Planner as Catalyst (1978)
10 \- Human Ecological Planning at Pennsylvania (1981)
Part III \- Form and Function Are Indivisible
References
11 \- The Court House Concept (1957)
12 \- Architecture in an Ecological View of the World (1970)
13 \- Nature Is More Than a Garden (1990)
14 \- Landscape Architecture (1997)
15 \- Ecology and Design (1997)
Part IV \- Revealing the Genius of the Place: Methods and Techniques for Ecological Planning
References
16 \- An Ecological Method for Landscape Architecture (1967)
17 \- A Comprehensive Highway Route Selection Method (1968)
18 \- Biological Alternatives to Water Pollution (1976)
19 \- A Case Study in Ecological Planning: The Woodlands, Texas (1979)
Part V \- Linking Knowledge to Action
References
20 \- Plan for the Valleys vs. Spectre of Uncontrolled Growth (1965)
21 \- An Ecological Planning Study for Wilmington and Dover, Vermont (1972)
22 \- Ecological Plumbing for the Texas Coastal Plain (1975)
23 \- A Strategy for a National Ecological Inventory (1992)
Prospectus (1998)
Acknowledgment of Sources
Index
Island Press Board of Directors
# Foreword
When historians in the next millennium look back at the second half of the twentieth century, in all likelihood the wars, revolutions, and internecine struggles of this violent period will fade in significance. But one revolutionary movement that emerged in this century, the global environmental movement, will continue to shape events in the next.
The products of this international movement and its underlying ethic have already profoundly changed the way the world manages its environment, as witnessed by the 1992 Rio conference, Agenda 21, and the 1997 Kyoto climate accords. In the United States and other developed countries, national environmental protection agencies and environmental impact review procedures have been established. And in a growing number of countries, effective national, regional, state, and local plans, as well as a rapidly expanding legion of environmental advocacy groups now protect landscapes and communities from the destructive effects of urban sprawl. Perhaps most important, polls in the United States and other developed countries show environmental protection to be immensely popular with the general public. Although much remains to be done to heal the planet, at least now we understand the extent of the environmental threats we face, and we have a growing number of tools to protect them.
How did this movement and these tools emerge? They certainly are not inevitable. In 1949 Aldo Leopold proposed a new environmental ethic and a new relationship between people and the land in A Sand County Almanac. In the 1960s Rachel Carson popularized scientific concerns about the destruction of the environment in her 1962 classic, Silent Spring.
But it was Ian McHarg, a University of Pennsylvania professor and practicing landscape architect who, hearing the environmental alarm sounded by Carson and others, brought it home to the general public, and in particular, to the design and planning professions. McHarg made it clear that as a terrestrial species, we homo sapiens have to be at least as concerned about destruction of the land as we were about the destruction of the atmosphere, the seas, and the habitats of other species.
McHarg popularized his views through his classic 1969 book Design with Nature and through dozens of professional articles. His CBS television series The House We Live In reached a mass audience and was surely the most verdant oasis in the "vast wasteland" that television was termed at the time. McHarg further promoted his vision for a new relationship between humans and their environment through hundreds of lectures to professional and lay audiences. As a result, since the 1960s, every design professional, environmentalist, and most thinking adults in the United States and much of the industrialized world has at some point had a first-hand opportunity to experience Ian McHarg's passion for combating the threats that face the planet.
McHarg had a particularly profound impact on the nearly two generations of students he taught at the University of Pennsylvania. Many of them became leaders in the design professions as government officials, consultants, and teachers, and most have put Ian's environmental dogma and practices to work in their own careers. And, probably more than any other group, McHarg's Penn graduates have for decades dominated the planning and landscape architecture faculties at leading universities in the United States and abroad.
But the direct impact of McHarg's teaching and writing has reached far beyond Penn. He made a particularly deep and lasting impression on the outlooks and careers of the generation of environmentalists, planners, and landscape architects who came of age in the heady period following the first Earth Day which took place on April 22, 1970, and which McHarg helped instigate.
I can write from personal experience on his impact on my own life and career. Like thousands of others, I helped organize Earth Day observances and activities. While McHarg was keynoting the Earth Day gathering at Fairmount Park in Philadelphia, I was helping to organize and lead a clean-up along the Connecticut River in Middletown, Connecticut, where I was a student at Wesleyan University. As a newly minted environmentalist, I knew I wanted to save my piece of the planet, but it was only when I brought my own copy of Design with Nature that spring that I realized how I was going to do it.
Within a year I was working as a professional city planner, first in Connecticut and then in Boston, trying to put Ian's concepts and techniques to work. Among other things, I wrote some of the first iteration of environmental assessments under the National Environmental Policy Act (NEPA) and MEPA, Massachusett's own environmental impact statement process.
Although I was already a disciple, it was not until 1973 that I experienced McHarg in person. That year he lectured at the Harvard Graduate School of Design, where Ian had received degrees in city planning and landscape architecture two decades earlier and where I would receive a master's degree in city and regional planning. I will always remember the intensity and passion of Ian's presentation that day, as well as his persistent optimism and sardonic wit. These combined with his thick Scottish accent, tweedy suit, and sometimes bawdy humor to make his presentation even more enthralling, like a shot of single-malt scotch. (Indeed, as a life-long chain smoker he had the smoky aroma of Islay scotch, and he probably did loosen up his presentation with a "wee dram" prior to his talk.) Ian's colorful characterization of suburban sprawl as a "cancerous excrescence on the landscape" was particularly memorable.
Not since Frederick Law Olmsted, the great nineteenth-century pioneering city planner and landscape architect, has one person had such a profound effect on the theory and practice of city planning and city building. And like Olmsted, McHarg has moved seamlessly and effectively between theory, advocacy, and practice.
As a theorist, McHarg is part of the tradition of environmental thinking that relates human progress to the quality of its environment, one that begins with George Perkins Marsh and builds upon the writings of Aldo Leopold and Lewis Mumford. It is worthwhile to note that McHarg promoted his view of the planet and all its species as representing a single living organism a decade before James Lovelock and Lynn Margulis advanced the Gaia theory.
Through dozens of professional projects around the world, McHarg has demonstrated the techniques needed to transform our relationship with the natural world. His plans for the valleys outside Baltimore, New York's Battery Park City and West Side Waterfronts, the new town of The Woodlands in Texas, and others set the standard for community and environmental planning in the 1960s and 1970s and beyond. Consequently, McHarg not only helped instigate and lead, but also shaped the outcome of the great environmental revolution of the late twentieth century.
What is perhaps his most lasting impact, however, came as a result of his advisory role to the federal government during the Johnson administration. In this capacity, McHarg successfully promoted the establishment of the National Environmental Policy Act and the world's first institutionalized system of environmental impact reviews. McHarg also advocated the creation of a biological survey at the federal level that could parallel the work of the U.S. Geological Survey. In the mid-1990s Interior Secretary Bruce Babbitt finally established the U.S. Biological Survey, although its power as an analytical tool led to its near-destruction at the hands of a growth-at-any-cost-Congress.
McHarg's plans for the federal highway administration in the 1960s demonstrated the value of suitability analyses, in which layers of environmental data were overlayed on each other to identify suitable sites or corridors for highway projects. This overlay mapping process established a procedure for the siting of major facilities that has been the standard in the United States and abroad since the 1970s. This process also inspired the early geographic information systems, laying the foundations for today's advanced computerized GIS analyses.
It is ironic that, despite his enormous contribution to the fields of landscape architecture and urban and environmental planning, few of McHarg's many scholarly articles, book chapters, or presentation papers, have had a wide audience. Many of the pieces were written for obscure professional journals or for public clients and did not receive much attention after their initial publication. Other papers were prepared for presentations at conferences, and although they may have had an important impact on the audience at the time, were not published except in ephemeral proceedings. This book serves to remedy that situation. For the first time many of McHarg's theoretical and applied papers are compiled and edited in book form. This allows the reader to see how the author's thinking evolved over time, and to see how his theories were later applied in McHarg's professional practice. Frederick Steiner's thoughtful introductory essays provide the reader with the context for each article or presentation, creating a continuity that pulls you through the book.
Robert D. Yaro
# Preface
This book is a result of a relationship which developed into a friendship between teacher and student that began in 1972 and stemmed from interwoven concerns about civil rights, war and peace, and the environment. The environmental interests of the student, Frederick Steiner, prompted him to read Design with Nature, one of the clarion works of those times. This reading led the student to attend a speech by the author in a standing-room-only Cincinnati conference hall, then to graduate school at the University of Pennsylvania (called Penn) in the department that the teacher then chaired.
The teacher, Ian McHarg, had begun his quest much earlier; for the student Penn was just the beginning of the journey. Years passed and their relationship matured and evolved. Eventually, the student assisted the teacher with his autobiography. In the process, the student discovered that the teacher had produced a body of published work that had become eclipsed by his most famous publication, Design with Nature.
This book is a collection of those works of the teacher, connected and introduced by essays by the student. To create a useful retrospective of McHarg's work, we (teacher and student, and please excuse us for drifting in and out of the third person, but it seems the only comfortable and honest way to introduce this collaboration) set ourselves the task of selecting writings from McHarg's extensive oeuvre that do not duplicate his major publications Design with Nature and A Quest for Life. Some overlap, of course, is inevitable, as ideas explored in papers or lectures found full expression years later in a book. We included those papers that exhibit the evolution of key ideas, while attempting to keep redundancies to a minimum. The selections span much of McHarg's career from the 1950s to the present. An original essay, "Landscape Architecture," as well as section introductions, and a "Prospectus" were written for this collection.
The essays in this selection have been edited for consistency. For example, we have adopted consistent systems for citations and headings. A few errors that occurred in the originals have been corrected and some information has been updated. Several illustrations from the originals have been replaced or omitted, but otherwise the essays have not been changed.
We owe several debts of gratitude to those who assisted with this book. First, a grant from the Graham Foundation allowed us to prepare the manuscript. We appreciate the support for the Graham proposal which we received from Deborah Dalton, William McDonough, and Dan Sayre.
The University of Pennsylvania provides Professor Emeritus Ian McHarg office space, which has been essential to the completion of this work. The Arizona State University School of Planning and Landscape Architecture has provided considerable support. In particular, we are thankful for the contribution of Chris Duplissa, who was responsible for scanning the original publications and organizing them in a consistent format. She also typed original material which required many revisions. John Meunier, Gary Hack, and John Dixon Hunt are responsible for the institutional support at ASU and Penn. Their ongoing encouragement and support is valued.
We greatly appreciate the comments on draft manuscripts by Lynn Miller, Meto Vroom, Danilo Palazzo, Ward Brady, Bill Miller, Sung-Kyun Kim, and Joochul Kim. Michael Clarke, Ignacio San Martin, Kim Shetter, Michael Rushman, Paul Smith, and Dan Sirois provided helpful advice and information. We are grateful to the publishers and co-authors who granted us permission to reprint the papers here. They are listed in the acknowledgments on pages 361-362. Co-authors are identified in the table of contents as well as with each paper. Jonathan Sutton was especially helpful and provided original slides from The Woodlands project.
We thank Bob Yaro of the Regional Plan Association for his insightful foreword. Heather Boyer, Cecilia González, Christine McGowan, and Dan Sayre of Island Press have been wonderful collaborators with this effort. We value their professionalism and attention to detail. We highly value the support and critical advice of our families and thank Anna, Halina, and Andrew Steiner, and Carol, Ian, and Andrew McHarg for their love and encouragement.
Frederick Steiner
Tempe, Arizona
Ian L. McHarg
Philadelphia, Pennsylvania
# Introduction
Two landscape architects/city planners stand alone in their contribution to American culture and to how we view landscapes. One was Frederick Law Olmsted, who pioneered both landscape architecture and city planning in the last half of the nineteenth century. Olmsted was responsible for, among other things, New York City's Central Park, the preservation of Yosemite State Park and of Niagara Falls, the design of many college campuses and city parks, and the layout of the Chicago Columbian Exposition of 1893. The other is Ian L. McHarg, who advanced landscape architecture and planning in the last half of the twentieth century. Both Olmsted and McHarg sought to generate healthy, creative environments.
Frederick Law Olmsted initiated the American professions of landscape architecture and city planning in the nineteenth century. He was influenced by the leading American writings of his time. The transcendentalists Emerson and Thoreau advanced the concepts of self-awareness and freedom through interaction with Nature. Whitman sang of himself, rewriting that song over and over to clarify his place in the world. Whitman pondered the uniqueness of our Nation of teeming nations. Olmsted was a writer before he was a planner or landscape architect. Like the transcendentalists, he believed interaction with nature was a prerequisite for freedom. Like Whitman, Olmsted was concerned about nation building, especially in a country divided by war.
Influenced by his visit to Britain's Birkenhead Park in 1850, Olmsted saw the concept of the urban park as an antidote for the demoralizing aspects of nineteenth-century American city life brought about by the industrial revolution. The eighteenth-century English landscape gardeners (or landscape improvers, as they called themselves) had developed a rehabilitation concept for the estates of the rich and eventually began to apply these same design principles to parks. This English landscape concept is often referred to as the pastoral aesthetic. Olmsted imported the pastoral vision, first to urban parks, then to suburbs and campuses. From Central Park in Manhattan to the Biltmore Estate in rural North Carolina, Olmsted imprinted the pastoral aesthetic into the American consciousness.
Olmsted's vision became institutionalized through the efforts of his son and nephew as well as several protégés and followers, most notably Charles Eliot and John Nolen. These individuals established organizations, such as the American Society of Landscape Architects, the Trustees of Public Reservation, and the American Planning and Civic Association, as well as academic programs in landscape architecture and planning at leading American universities, such as Harvard and the University of Illinois. They also influenced American policy for national parks and national forests, wrote numerous plans and zoning ordinances for towns and cities, and published books and articles.
Although Americans enjoyed the pastoral aesthetic in grassy lawns and massed tree clumps, the institutions fractured in the decades following the deaths of the senior Olmsted and of Eliot. City planning separated from landscape architecture and after World War II, began promoting itself as an applied social science rather than an environmental design art. Planning grew in prominence, but eventually lost its effectiveness in improving human communities. Planners became engaged in many of the most pressing political issues of our day, from social equity to environmental quality. However, they distanced themselves from the creation and rehabilitation of the places where people live. An emphasis on process, quantitative analysis, and policy improved the capacities of planners in many ways, but this improvement came at the expense of physical plan-making. While policies for engaging citizens in public decisions proliferated, the quality of communities deteriorated.
Meanwhile, landscape architecture retreated from the social advocacy of the Olmsteds. Notable exceptions existed, such as Jens Jensen's promotion of native plants in park design in Chicago, Alfred Caldwell's vision for a living city, also in Chicago, and Frank Waugh's concern for rural communities and for national forests and parks in the American West from his Amherst, Massachusetts, base. But, largely landscape architects were engaged in applying the pastoral aesthetic to the country homes of the wealthy, to golf courses, and to exclusive subdivisions (that frequently excluded people because of their race or religion). During and after World War II, academic programs in landscape architecture began to disappear or become marginalized.
In the 1960s a landscape architect-planner emerged to challenge the status quo and to establish a new direction. Ian McHarg revived, redirected, and re-created the professions of landscape architecture and planning in the late twentieth century. Like Olmsted, McHarg was influenced by the leading American environmental writings of his time. Aldo Leopold and Rachel Carson had related the science of ecology to the quality of the world we live in. Lewis Mumford had advanced the notion that planning and design could be viewed as forms of social criticism and that the regional, ecological city could improve our quality of life.
McHarg was a landscape architect and planner before he became a writer and theorist. Like Carson and Leopold, he believed ecology could be used to understand complex interactions between people and their environments and that the science could be employed to guide actions. Like his mentor Mumford, McHarg came to see planning and design as means for criticism of the environments we had created. McHarg was also profoundly influenced by the new view of Earth that was provided from space. A veteran of war, he sought means for peacefully inhabiting the planet by greening and healing it.
Like the English landscape gardeners before him, McHarg went beyond the garden wall and discovered through his work all nature to be a garden. McHarg transcended disciplinary boundaries, and by making this leap the limits of his own fields were extended. A new theory was advanced:that we should plan and design with nature, that we should use ecology to inform the environmental design arts, that we should follow nature's lead. McHarg presented this theory at a time when the nation and the world were receptive to creating better environments—a time when we were beginning to recognize the limits to growth and our dependence on our surroundings for our well-being.
McHarg's Design with Nature was not published in a vacuum. Compatible ideas about the use of ecology were being presented by others in the 1960s, most notably by the Israeli-Dutch planner Artur Glikson (another protégé of Mumford), the Canadian Angus Hills, and the American landscape architect Philip Lewis.
But McHarg emerged as the principal spokesman. He was more colourful, blunter, and wittier than his peers. He had hosted his own network television program, produced a documentary for PBS, and was a frequent guest on that American institution—the talk show. The nation was looking for representatives for the environment and at the same time had become enamored by the culture of the British Isles, from the Beatles and the Rolling Stones, to the Scot Sean Connery and Ian Fleming's 007, on to Twiggy and Carnaby Street. McHarg, a Scot like Connery, was a former commando and paratrooper, a British red beret who showed no fear either to the Fortune 500 or to thousands of hippies at an Earth Day rally.
Design with Nature was not the only channel through which McHarg relayed his message. As an American academic, he was required to publish, and publish he did in scholarly and professional journals and in edited volumes. McHarg was also an academic practitioner. He founded a company with David Wallace which became an international business. Wallace-McHarg Associates grew into Wallace, McHarg, Roberts, and Todd and, then after McHarg's departure, the present Wallace Roberts and Todd. McHarg also pursued numerous design and planning projects under the auspices of his home institution, the University of Pennsylvania (Penn). At Penn, he promoted a multidisciplinary, collaborative approach to planning and design. The team-based studio became a vehicle to combat the reductionism of the science, while drawing on scientific knowledge, to create new syntheses.
McHarg, the scholar and the academic practitioner, produced a body of writings, selections of which are gathered here. For readers familiar with Design with Nature, this collection will provide a context for the work that preceded that classic text and for the writings that flowed after it. For others unfamiliar with Ian McHarg's work, this collection will provide a comprehensive introduction to his ideas.
Although all of his ideas were certainly not new, McHarg cultivated theories of ecology and, in doing so, was able to present them in a way that changed the very nature of landscape architecture and planning. This collection also relates the study, practice, and theory of landscape architecture to the broader conceptions of ecology, architecture, and the built environment.
The book is divided into five parts, representing five different themes from McHarg's work: First, McHarg's theoretical writings and their influence on the basic nature of design and planning; second, planning the ecological region; third, the interrelationship between ecology and design; fourth, methods and techniques related to design and planning; and, fifth, links between theory and action through McHarg's innovative planning projects.
Each part begins with a short essay that provides the "connective tissue" for the book, context for the works represented, and continuity for the book as a whole. A short comment introduces each work. The comments attempt to frame each piece and to provide further context.
A lesson from the pragmatism of John Dewey is that philosophy works best when eternal questions are connected to everyday practice. An eternal question facing us is how best to live on earth in a sustaining manner. The writings of Ian McHarg can help us heal the planet by making better decisions about how we plan and design our surroundings.
Frederick R. Steiner
# Part I
# Changing the Nature of Design and Planning: Theoretical Writings
Design with nature is an elegant theory. Both simple and direct, it is as much a proposition as a principle. Design with nature is a normative theory, an ideal to be achieved. A process is suggested to reach that goal. The conception that we should design with nature is deeply rooted in the Western arts and sciences; some would argue it is a universal theme underlying all cultures. Certainly nature as represented by our material surroundings and our own human character underlies all art and science. In Western societies knowledge about our surroundings has too often been used "to multiply and to subdue" nature. A grand canyon exists between the values espoused and the reality created.
A better fit between the ideal and the actual, between our surroundings and our interventions, has long been promoted in design theory. From ancient to modern times, architects have worked to fit buildings to a given site. In the first century B.C., the Roman architect and engineer Marcus Vitruvius Pollio devoted much of his ten books on architecture to understanding sites and to the primordial elements of air, fire, earth, and water. In the planning of a city, he noted the need to "consider and observe the natures of birds, fishes, and land animals" and suggested that the designs of houses should "conform to the nature of the country and to diversities of climate." Two thousand years later, the American architect Frank Lloyd Wright advocated an "organic" approach to architecture, seeking to blur the distinction between the inside and the outside of a building. Such wisdom should be used in the planning of groups of houses that form communities as well as communities that comprise cities and their regions.
Landscape architects and planners have also promoted environmental understanding to guide their arts. Jens Jensen advocated the use of native plants in park design. Patrick Geddes promoted the idea of a regional survey of environmental factors to precede planning, a concept embraced by Lewis Mumford, Benton MacKaye, and others.
With his contribution of connecting the science of ecology to the environmental design and planning arts, McHarg is an important part of this tradition. He linked the Vitruvius–Wright–Jensen–Mumford tradition with that of Aldo Leopold, Paul Sears, and Rachel Carson.
The following five papers represent the core of McHarg's theoretical ideas. They overlap with the topics of the subsequent sections of the book on planning and design, but focus more on the theoretical than the applied aspects (although McHarg has always jumped back and forth). The earliest essay was published in 1963 and the most recent in 1997. There is a gap between the four theoretical papers from the 1960s and the 1997 article in this section. This gap is filled with the four subsequent sections that illustrate how his theories were transformed into actions.
McHarg has identified "Man and Environment" as his first serious theoretical writing that set the stage for those that followed, including the themes of religion, science, and creativity that emerge and reappear throughout his work. What may surprise some readers and some of the critics who lump McHarg in the American antiurban tradition is the attention McHarg gives to city life and his desire to seek out an alternative urban morphology. His interest in, and knowledge of, urban history is also impressive and due in part to the mentorship of Mumford.
McHarg's 1964 "The Place of Nature in the City of Man" explicitly addresses "the place of nature" in our urban habitat. The prose exhibits McHarg's often sardonic humor as well as his clever way with words, for example, "that anarchy which constitutes urban growth" and "the place where man and nature are in closest harmony in the city is the cemetery." The essay clearly defines the urban environmental agenda that still dominates planning debates: the loss of prime farmland "by the most scabrous housing," the pollution of air and water, the paving over of precious green spaces within the city, the filling in of marshes that we now call wetlands, and the gradual uglification of everything. These processes occur worldwide from Phoenix to Madrid, from Mexico City to Seoul. McHarg lays the blame of the "urban growth anarchy" on economic determinism and he provides a theoretical antidote.
In the 1966 article, "Ecological Determinism," McHarg proposed the use of ecology in planning and design to avert the necropolis predicted by Mumford. The Mumford influence on McHarg's 1960s writings is evident. So too is the thinking of the great minds he invited to his The House We Live In television program and his "Man and Environment" course at Penn, especially, I think, Paul Sears, Paul Shepard, and Ruth Patrick. He also learned much from his interactions with his Penn colleagues, such as David Goddard and the wonderful Loren Eiseley, whom McHarg once described as "a large, wise, round, magnificent man."
In "Ecological Determinism" McHarg pushes his ideas for a new urban morphology, one determined by ecology to counter the prevailing economic determinism, which McHarg masterfully critiques. Ecological determinism differs from environmental determinism as it was developed by geographers and other social scientists early in the twentieth century. These social scientists were strongly influenced by biological ideas, and briefly the prospects for a synthetic human ecology were bright. Unfortunately these ideas were appropriated by individuals seeking to advance racist notions about the influence of surroundings on human physiology. These theories were rightly debunked, but an unfortunate drifting began of the social scientists away from the natural scientists that is only recently being bridged. McHarg's ecological determinism focuses on the importance of interactions and rather exclusively on the role of surroundings.
Another contribution of the "Ecological Determinism" paper is McHarg's treatment of the English landscape movement. Before McHarg's analysis, the English landscape school was viewed mainly as a period of garden history populated by funny, eccentric people doing bold things. He observed that "Nature itself produced the esthetic" of the landscapes designed by William Kent, Capability Brown, Humphry Repton, and others. McHarg recognized the complexity of their work, which contrasted the simplicity (or "simple-mindedness") of French Renaissance garden design. Since McHarg's 1966 observations, the English landscape school has been viewed as an applied ecology that formed the basis for creating functional landscapes with a new aesthetic.
McHarg was quite clear early on, even before the publication of Design with Nature about his quest. In the 1968 "Values, Process and Form," he wrote: "We need a general theory which encompasses physical, biological, and cultural evolution; which contains an intrinsic value system; which includes criteria of creativity and destruction and, not least, principles by which we can measure adaptations and their form." The theory was to be based on an understanding of ecology: "The place, the plants, the animals, and man and the orderings which they have accomplished over time, are revealed in form." Our role, then, was to be creative agents of change, that is, "The role of man is to understand nature, which is to say man, and to intervene to enhance its creative processes."
Creativity was one of McHarg's central and recurring themes. His persistent optimism is impressive. McHarg became mildly disillusioned about the state of the earth without being embittered. His optimism provides a counterbalance to the pessimism, the necropolis of Mumford. "There are the challenges. What are the opportunities?" is a question posed by McHarg frequently.
The writings from the 1960s are as contemporary as if they were written today, except for his use of "man" instead of "human" or "people," an indication of his time when "man" was used in its Greek sense to refer to humanity. They address contemporary, perhaps timeless, topics. That we should "understand nature, which is to say man" is a response to critics of environmentalists who claim ecological designers and planners ignore people. In fact, McHarg's themes from the 1960s foreshadow Neil Evernden's ideas in his insightful The Social Creation of Nature (1992) as well as Daniel Botkin's in Discordant Harmonies (1990). Like Evernden, McHarg indeed recognizes that nature is a social creation. Like Botkin in his "New Ecology for the Twenty-First Century," McHarg has long acknowledged people as part of ecology, as active agents who interact with and bring about change in their environments. Similar, too, are McHarg's ideas to those of the Dutch geobiochemist Peter Westbroek (1991) that life, including human life, is a geological force.
He proposed the concept of the biosphere as a superorganism a decade before James Lovelock (1979) and Lynn Margulis put forth their Gaia hypothesis. Although his focus in the 1960s was more on local and regional landscapes, even then McHarg offered a global view, graphically displayed in the early use of the portrait of the Earth from space on the cover of Design with Nature. Early on, he provocatively compared people to "a global pathogen, an agent of planetary disease." Simultaneously he recognized the potential for people to become the Earth's physicians.
His reflective 1997 "Natural Factors in Planning" challenges the human race to transform itself from being a "global pathogen" to that of a catalyst for maintaining crucial processes. After presenting the consequences for not becoming such catalysts, McHarg summarizes the challenges to more effective ecological planning. First, there is the fragmentation of knowledge. "Integration requires bridging between separate sciences," McHarg observes.
A second challenge is the fragmentation of government. "There are redundant and often conflicting policies, evidence of cross purposes" that handicap governmental environmental management efforts. McHarg also identifies the inadequacies of planning initiatives to respond to the environmental challenge, but, he provides hope.
McHarg observes the emergence of the environment in public policy since the 1970s. He gives several reasons for optimism, including the environmental literacy of today's children. Scientific knowledge about the environment, although still fragmented, has grown. He urges us to "direct our energies toward synthesis." Such synthesis of environmental knowledge is necessary "to improve the human condition."
Design the nature of the planet: heal it, restore its health. This is our challenge, this is our opportunity. In these five chapters, McHarg provides a foundation. Ecology is the basis for that foundation—ecology, the subversive science, as Paul Sears (1964) called it. Sears speculated that if ecology was "taken seriously for the long-run welfare of mankind, [then it would] endanger the assumptions and practices accepted by modern societies, whatever their doctrinal commitments" (1964, p. 11). Ian McHarg has indeed taken ecology seriously and, in doing so, changed how we approach planning and design.
# References
Botkin, Daniel. 1990. Discordant Harmonies, A New Ecology for the Twenty-First Century. New York: Oxford University Press.
Evernden, Neil. 1992. The Social Creation of Nature. Baltimore: The Johns Hopkins University Press.
Lovelock, J. E. 1979. Gaia, A New Look at Life on Earth. Oxford: Oxford University Press.
Sears, Paul. 1964. "Ecology—A Subversive Subject:" BioScience 14 (7, July):11.
Westbroek, Peter. 1991. Life as a Geological Force. New York: Norton.
# 1
# Man and Environment (1963)
Ian McHarg considers the writing of this paper, published in The Urban Condition edited by Leonard Duhl, as "a threshold in my professional life and . . . the first summation of my perceptions and intentions." It began when McHarg was invited by Duhl to join his Committee on Environmental Variables and Mental Health. Duhl, a medical doctor, was director of research for the National Institute of Mental Health. He selected the members of the committee, which included Herbert Gans, J. B. Jackson, and Melvin Webber.
For McHarg the paper represented a "tremendous leap in scale." He changed his focus from small-scale urban concerns to a targer regional vision. He wrote "Man and Environment" at the time when he was organizing his The House We Live In television program for CBS. The influence of the guests from that program is evident in this paper. Not only did the scale of McHarg's concerns change, but also the nature of his audience. Prior to 1962, his lectures outside of Penn had been limited to state associations of garden clubs, where he agreed to devote half his speech to garden design history if he could spend the other half speaking about the environment. This paper is a "coming out," where the half garden designer is shed for the complete environmentalist. It was, according to McHarg, "my most embracing address on the subject of the environment to that point."
The nature and scale of this enquiry can be simply introduced through an image conceived by Loren Eiseley. Man, far out in space, looks back to the distant earth, a celestial orb, blue-green oceans, green of verdant land, a celestial fruit. Examination discloses blemishes on the fruit, dispersed circles from which extend dynamic tentacles. The man concludes that these cankers are the works of man and asks, "Is man but a planetary disease?"
There are at least two conceptions within this image. Perhaps the most important is the view of a unity of life covering the earth, land and oceans, interacting as a single superorganism, the biosphere. A direct analogy can be found in man, composed of billion upon billion of cells, but all of these operating as a single organism. From this the full relevance of the second conception emerges, the possibility that man is but a dispersed disease in the world-life body.
The conception of all life interacting as a single superorganism is as novel as is the conception of man as a planetary disease. The suggestion of man the destroyer, or rather brain the destroyer, is salutary to society which has traditionally abstracted brain from body, man from nature, and vaunted the rational process. This, too, is a recent view. Yet the problems are only of yesterday. Pre-atomic man was an inconsequential geological, biological, and ecological force; his major power was the threat of power. Now, in an instant, post-atomic man is the agent of evolutionary regression, a species now empowered to destroy all life.
In the history of human development, man has long been puny in the face of overwhelmingly powerful nature. His religions, philosophies, ethics, and acts have tended to reflect a slave mentality, alternately submissive or arrogant toward nature. Judaism, Christianity, Humanism tend to assert outrageously the separateness and dominance of man over nature, while animism and nature worship tend to assert total submission to an arbitrary nature. These attitudes are not urgent when human societies lack the power to make any serious impact on environment. These same attitudes become of first importance when man holds the power to cause evolutionary regressions of unimaginable effect or even to destroy all life.
Modern man is confronted with the awful problem of comprehending the role of man in nature. He must immediately find a modus vivendi, he must seek beyond for his role in nature, a role of unlimited potential yet governed by laws which he shares with all physical and organic systems. The primacy of man today is based more upon his power to destroy than to create. He is like an aboriginal, confronted with the necessity of operating a vast and complex machine, whose only tool is a hammer. Can modern man aspire to the role of agent in creation, creative participant in a total, unitary, evolving environment? If the pre-atomic past is dominated by the refinement of concern for man's acts towards man, the inauguration of the atomic age increases the dimension of this ancient concern and now adds the new and urgent necessity of understanding and resolving the interdependence of man and nature.
While the atomic threat overwhelms all other considerations, this is by no means the only specter. The population implosion may well be as cataclysmic as the nuclear explosion. Should both of these threats be averted there remain the lesser processes of destruction which have gathered momentum since the nineteenth century. In this period we have seen the despoliation of continental resources accumulated over aeons of geological time, primeval forests destroyed, ancient resources of soil mined and sped to the sea, marching deserts, great deposits of fossil fuel dissipated into the atmosphere. In the country, man has ravaged nature; in the city, nature has been erased and man assaults man with insalubrity, ugliness, and disorder. In short, man has evolved and proliferated by exploiting historic accumulations of inert and organic resources, historic climaxes of plants and animals. His products are reserved for himself, his mark on the environment is most often despoliation and wreckage.
# The Duality of Man and Nature
Conceptions of man and nature range between two wide extremes. The first, central to the Western tradition, is man-oriented. The cosmos is but a pyramid erected to support man on its pinnacle, reality exists only because man can observe it, indeed God is made in the image of man. The opposing view, identified with the Orient, postulates a unitary and all-encompassing nature within which man exists, man in nature.
These opposing views are the central duality, man and nature, West and East, white and black, brains and testicles, Classicism and Romanticism, orthodoxy and transnaturalism in Judaism, St. Thomas and St. Francis, Calvin and Luther, anthropomorphism and naturalism. The Western tradition vaunts the individual and the man-brain, and denigrates nature, animal, non-brain. In the Orient nature is omnipotent, revered, and man is but an aspect of nature. It would be as unwise to deny the affirmative aspects of either view as to diminish their negative effects. Yet today this duality demands urgent attention. The adequacy of the Western view of man and nature deserves to be questioned. Further, one must ask if these two views are mutually exclusive.
The opposition of these attitudes is itself testimony to an underlying unity, the unity of opposites. Do our defining skin and nerve ends divide us from environment or unite us to it? Is the perfectibility of man self-realizable? Is the earth a storeroom awaiting plunder? Is the cosmos a pyramid erected to support man?
The inheritors of the Judaic-Christian-H umanist tradition have received their injunction from Genesis, a man-oriented universe, man exclusively made in the image of God, given dominion over all life and non-life, enjoined to subdue the earth. The naturalist tradition in the West has no comparable identifiable text. It may be described as holding that the cosmos is unitary, that all systems are subject to common physical laws yet having unlimited potential; that in this world man is simply an inhabitant, free to develop his own potential. This view questions anthropocentrism and anthropomorphism; it does not diminish either man's uniqueness or his potential, only his claims to primacy and exclusive divinity. This view assumes that the precursor of man, plant and animal, his co-tenant contemporaries, share a cosmic role and potential.
From its origin in Judaism, extension in Classicism, reinforcement in Christianity, inflation in the Renaissance, and absorption into the nineteenth and twentieth centuries, the anthropomorphic-anthropocentric view has become the tacit view of man versus nature.
# Evolution of Power
The primate precursors of man, like their contemporary descendants, support neither a notably constructive, nor a notably destructive role in their ecological community. The primates live within a complex community which has continued to exist; no deleterious changes can be attributed to the primate nor does his existence appear to be essential for the support of his niche and habitat. When the primates abandoned instinct for reason and man emerged, new patterns of behavior emerged and new techniques were developed. Man acquired powers which increased his negative and destructive effect upon environment, but which left unchanged the possibility of a creative role in the environment. Aboriginal peoples survive today: Australian aborigines, Dravidians and Birbory in India, South African Bushmen, Veda in Ceylon, Ainu in Japan, Indians of Tierra del Fuego; none of these play a significantly destructive role in the environment. Hunters, primitive farmers, fishermen—their ecological role has changed little from that of the primate. Yet from aboriginal people there developed several new techniques which gave man a significantly destructive role within his environment. The prime destructive human tool was fire. The consequences of fire, originated by man, upon the ecology of the world cannot be measured, but there is reason to believe that its significance was very great indeed.
Perhaps the next most important device was that of animal husbandry, the domestication of grazing animals. These sheep, goats, and cattle, have been very significant agents historically in modifying the ecology in large areas of the world. This modification is uniformly deleterious to the original environment. Deforestation is perhaps the third human system which has made considerable impact upon the physical environment. Whether involuntary, that is, as an unconscious product of fire, or as a consequence of goat and sheep herding, or as an economic policy, this process of razing forests has wrought great changes upon climate and microclimate, flora and fauna. However, the regenerative powers of nature are great; and while fire, domestic animals, and deforestation have denuded great areas of world surface, this retrogression can often be minimized or reversed by the natural processes of regeneration. Perhaps the next consequential act of man in modifying the natural environment was large-scale agriculture. We know that in many areas of the world agriculture can be sustained for many centuries without depletion of the soil. Man can create a new ecology in which he is the prime agent, in which the original ecological community has been changed, but which is nevertheless self-perpetuating. This condition is the exception. More typically agriculture has been, and is today, an extractive process in which the soil is mined and left depleted. Many areas of the world, once productive, are no longer capable of producing crops. Extractive agriculture has been historically a retrogressive process sustained by man.
The next important agent for modifying the physical environment is the human settlement: hamlet, village, town, city. It is hard to believe that any of the pre-classical, medieval, Renaissance, or even eighteenth-century cities were able to achieve a transformation of the physical environment comparable to the agents mentioned before—fire, animal husbandry, deforestation, or extensive agriculture. But with the emergence of the nineteenth-century industrial city, there arose an agent certainly of comparable consequence, perhaps even of greater consequence, even more destructive of the physical environment and the balances of ecological communities in which man exists, than any of the prior human processes.
The large modern metropolis may be thirty miles in diameter. Much, if not all, of the land which it covers is sterilized. The micro-organisms in the soil no longer exist; the original animal inhabitants have largely been banished. Only a few members of the plant kingdom represent the original members of the initial ecology. The rivers are foul; the atmosphere is polluted; the original configuration of the land is only rarely in evidence; climate and microclimate have retrogressed so that the external microclimate is more violent than was the case before the establishment of the city. Atmospheric pollution may be so severe as to account for 4,000 deaths in a single week of intense "fog," as was the case in London. Floods alternate with drought. Hydrocarbons, lead, carcinogenic agents, carbon dioxide, carbon monoxide concentrations, deteriorating conditions of atmospheric electricity—all of these represent retrogressive processes introduced and supported by man. The epidemiologist speaks of neuroses, lung cancer, heart and renal disease, ulcers, the stress diseases, as the badges of urban conditions. There has also arisen the specter of the effects of density and social pressure upon the incidence of disease and upon reproduction. The modern city contains other life-inhibiting aspects whose effects are present but which are difficult to measure: disorder, squalor, ugliness, noise.
In its effect upon the atmosphere, soil as a living process, the water cycle, climate and micro-climate, the modern city represents a transformation of the original physical environment certainly greater over the area of the city than the changes achieved by earlier man through fire, animal husbandry, deforestation, and extensive agriculture.
Indeed, one can certainly say that the city is at least an ecological regression, although as a human institution it may represent a triumph. Whatever triumphs there are to be seen in the modern city as an institution, it is only with great difficulty that one can see any vestige of triumph in the modern city as a physical environment. One might ask of the modern city that it be humane; that is, capable of supporting human organisms. This might well be a minimum requirement. In order for this term to be fully appropriate—that is, that the city be compassionate and elevating—it should not only be able to support physiological man, but also should give meaning and expression to man as an individual and as a member of an urban society. I contend that far from meeting the full requirements of this criterion, the modern city inhibits life, that it inhibits man as an organism, man as a social being, man as a spiritual being, and that it does not even offer adequate minimum conditions for physiological man; that indeed the modern city offers the least humane physical environment known to history.
Assuredly, the last and most awful agent held by man to modify the physical environment is atomic power. Here we find post-atomic man able to cause evolutionary regressions of unimaginable effect and even able to destroy all life. In this, man holds the ultimate destructive weapon; with this, he can become the agent of destruction in the ecological community, of all communities, of all life. For any ecological community to survive, no single member can support a destructive role. Man's role historically has been destructive; today or tomorrow it can be totally, and for all life existent, irrevocably destructive.
Now, wild nature, save a few exceptions, is not a satisfactory physical environment. Where primitive peoples exist in a wild nature little adapted by man, their susceptibility to disease, life expectancy, vulnerability to climatic vagaries, and to the phenomena of drought and starvation is hardly ideal. Yet the certainty that man must adapt nature and himself does not diminish his dependence upon natural, non-human processes. These two observations set limits upon conceptions of man and nature. Man must adapt through both biological and cultural innovation but these adaptations occur within a context of natural, non-human processes. It is not inevitable that adapting nature to support human congregations must of necessity diminish the quality of the physical environment.
Creation of a physical environment by organisms as individuals and as communities is not exclusively a human skill. The chambered nautilus, the beehive, and the coral formation are all efforts by organisms to take inert materials and dispose them to create a physical environment. In these examples the environments created are complementary to the organisms. They are constructed with great economy of means; they are expressive; they have, in human eyes, great beauty; and they have survived periods of evolutionary time vastly longer than the human span. Can we hope that man will be able to change the physical environment to create a new ecology in which he is primary agent, but which will be a self-perpetuating and not a retrogressive process? We hope that man will be able at least to equal the chambered nautilus, the bee, and the coral—that he will be able to build a physical environment indispensable to life, constructed with economy of means, having lucid expression, and containing great beauty. When man learns this single lesson he will be enabled to create by natural process an environment appropriate for survival—the minimum requirement of a humane environment. When this view is believed, the artist will make it vivid and manifest. Medieval faith, interpreted by artists, made the Gothic cathedral ring with holiness. Here again we confront the paradox of man in nature and man transcendent. The vernacular architecture and urbanism of earlier societies and primitive cultures today, the Italian hill town, medieval village, the Dogon community, express the first view, a human correspondence to the nautilus, the bee, and the coral. Yet this excludes the Parthenon, Hagia Sofia, Beauvais, statements which speak of the uniqueness of man and his aspirations. Neither of these postures is complete, the vernacular speaks too little of the consciousness of man, yet the shrillness of transcendence asks for the muting of other, older voices.
Perhaps when the achievements of the past century are appraised, there will be advanced as the most impressive accomplishment of this period the great extension of social justice. The majority of the population of the Western world moved from an endemic condition of threatening starvation, near desperation, and serfdom, to relative abundance, security, and growing democratic freedoms. Human values utilized the benison of science, technology, and industry, to increase wealth absolutely and to distribute it more equitably. In the process, responsibility and individual freedom increased, brute hunger, bare suppression, and uncontrolled disease were diminished. It is a paradox that in this period of vastly increased wealth, the quality of the physical environment has not only failed to improve commensurately, but has actually retrogressed. If this is true, and I believe that there is more than ample evidence to support this hypothesis, then it represents an extraordinary failure on the part of Western society. The failure is the more inexplicable as the product of a society distinguished by its concern for social justice; for surely the physical environment is an important component of wealth and social justice. The modern city wears the badges which distinguish it as a product of the nineteenth and twentieth centuries. Polluted rivers, polluted atmosphere, squalid industry, vulgarity of commerce, diners, hot dog stands, second-hand car lots, gas stations, sagging wire and billboards, the whole anarchy united by ugliness—at best neutral, at worst offensive and insalubrious. The product of a century's concern for social justice, a century with unequaled wealth and technology, is the least humane physical environment known to history. It is a problem of major importance to understand why the nineteenth and twentieth centuries have failed in the creation of a physical environment; why the physical environment has not been, and is not now, considered as a significant aspect of wealth and social justice.
# Renaissance and Eighteenth Century
If we consider all the views in our Western heritage having an anti-environmental content, we find they represent a very impressive list. The first of these is the anthropomorphic view that man exclusively is made in the image of God (widely interpreted to mean that God is made in the image of man). The second assumption is that man has absolute dominion over all life and nonlife. The third assumption is that man is licensed to subdue the earth. To this we add the medieval Christian concept of other-worldliness, within which life on earth is only a probation for the life hereafter, so that only the acts of man to man are of consequence to his own soul. To this we add the view of the Reformation that beauty is a vanity; and the Celtic, and perhaps Calvinistic, view that the only beauty is natural beauty, that any intent to create beauty by man is an assumption of God's role, is a vanity, and is sacrilegious. The total of these views represents one which can only destroy and which cannot possibly create. The degree to which there has been retention of great natural beauty, creation of beauty and order, recognition of aspects of natural order, and particularly recognition of these aspects of order as a manifestation of God, would seem to exist independently of the Judaic-Christian view. They may be animist and animitist residues that have originated from many different sources; but it would appear, whether or not they are espoused by Christian and Jews, that they do not have their origins in Judaism or Christianity. It would also appear that they do not have their origins in classical or humanist thought, or even in eighteenth-century rationalist views.
These two opposed views of man's role in the natural world are reflected in two concepts of nature and the imposition of man's idea of order upon nature. The first of these is the Renaissance view most vividly manifest in the gardens of the French Renaissance and the projects of André le Nôtre for Louis XIV. The second is the eighteenth-century English picturesque tradition. The gardens of the Renaissance clearly show the imprint of humanist thought. A rigid symmetrical pattern is imposed relentlessly upon a reluctant landscape. If this pattern was, as is claimed, some image of a perfect paradisiac order, it was a human image which derived nothing from the manifest order and expression of wild nature. It was rather, I suggest, an image of flexed muscles, a cock's crow of power, and an arrogant presumption of human dominance over nature. Le Roi Soleil usurped none of the sun's power by so claiming. Art was perverted to express a superficial pattern while claiming this as an expression of a fundamental order.
If the Renaissance sought to imprint upon nature a human order, the eighteenth-century English tradition sought to idealize wild nature, in order to provide a sense of the sublime. The form of estates in the eighteenth century was of an idealized nature, replacing the symmetrical patterns of the Renaissance. The form of ideal nature had been garnered from the landscape painting of Nicolas Poussin and Salvator Rosa; developed through the senses of the poets and writers, such as Alexander Pope, Abraham Cowley, James Thomson, Joseph Addison, Thomas Gray, the third earl of Shaftesbury, and the Orientalist William Temple—a eulogy of the campagna from the painters; a eulogy of the natural countryside and its order from the writers; and from Temple, the occult balance discovered in the Orient. However, the essential distinction between the concept of the Renaissance, with its patterning of the landscape, and that of eighteenth-century England was the sense that the order of nature itself existed and represented a prime determinant, a prime discipline for man in his efforts to modify nature. The search in the eighteenth century was for creation of a natural environment which would evoke a sense of the sublime. The impulse of design in the Renaissance was to demonstrate man's power over nature; man's power to order nature; man's power to make nature in his human image. With so inadequate an understanding of the process of man relating to nature, his designs could not be self perpetuating. Where the basis for design was only the creation of a superficial order, inevitably the consequence was decoration, decay, sterility, and demise. Within the concepts of eighteenth-century England, in contrast, the motivating idea was to idealize the laws of nature. The interdependence of micro-organisms—plants, insects, and animals, the association of particular ecological groupings with particular areas and particular climates—this was the underlying discipline within which the aristocrat-landscape architect worked. The aim was to create an idealized nature which spoke to man of the tranquillity, contemplation, and calm which nature brought, which spoke of nature as the arena of sublime and religious experience, essentially speaking to man of God. This represents, I believe, one of the most healthy manifestations of the Western attitude toward nature. To this eighteenth-century attitude one must add a succession of men who are aberrants in the Western tradition, but whose views represent an extension of the eighteenth-century view—among them, Wordsworth and Coleridge, Thoreau and Emerson, Jonathan Edwards, Jonathan Marsh, Gerald Manley Hopkins, and many more.
# Natural Science and Naturalism
It might be productive to examine the natural scientist's view of the evolution of nature and certain aspects of this order. The astronomer gives us some idea of immensity of scale, a hundred billion galaxies receding from us at the speed of light. Of these hundred billion galaxies is one which is our own, the Milky Way. Eccentric within the immensity of the Milky Way, the inconspicuous solar system exists. Within the immensity of the solar system, revolves the minute planet Earth. The astronomer and geologist together give us some sense of the process during which the whirling, burning gases increased in density, coalesced with cooling, condensed, gave off steam, and finally produced a glassy sphere, the Earth. This sphere with land and oceans had an atmosphere with abundant carbon dioxide, with abundant methane, and little or no free oxygen. A glassy sphere with great climatic ranges in temperature, diurnal and seasonal, comparable to an alternation between Arctic and Equatorial conditions. From the biologist, we learn of the origins of life. The first great miracle of life was this plant-animal in the sea; the emergence of life on land, the succession of fungi, mosses, liverworts, ferns. The miracle beyond life is photosynthesis, the power by which plants, absorbing carbon dioxide, give out oxygen and use the sun's energy to transform light into substance. The substance becomes the source of food and fuel for all other forms of life. There seems to be good reason to believe that the Earth's atmosphere, with abundant oxygen, is a product of the great evolutionary succession of plants. On them we depend for all food and fossil fuels. From the botanist we learn of the slow colonization of the Earth's surface by plants, the degree to which the surface of the Earth was stabilized, and, even more significantly, how plants modified the climatic extremes to support the amphibian, reptilian, and subsequent mammalian evolutionary sequence.
The transcendental view of man's relation to nature implicit in Western philosophies is dependent upon the presumption that man does in fact exist outside of nature, that he is not dependent upon it. In contemporary urban society the sense of absolute dependence and interdependence is not apparent, and it is an extraordinary experience to see a reasonably intelligent man become aware of the fact that his survival is dependent upon natural processes, not the least of which are based upon the continued existence of plants. This relationship can be demonstrated by experiment with three major characters: light, man, and algae. The theater is a cylinder in which is a man, a certain quantity of algae, a given quantity of water, a given quantity of air, and a single input, a source of light corresponding to sunlight (in this case a fluorescent tube). The man breathes the air, utilizes the oxygen, and exhales carbon dioxide. The algae utilize the carbon dioxide and exhale oxygen. There is a closed cycle of carbon dioxide and oxygen. The man consumes water, passes the water, the algae consume the water, the water is transpired, collected. and the man consumes the water. There is a closed water cycle. The man eats the algae, the man passes excrement, the algae consume the excrement, the man consumes the algae. There is a closed cycle of food. The only input is light. In this particular experiment the algae is as dependent upon the man as the man is upon the algae. In nature this is obviously not true. For some two billion years nature did exist without man. There can, however, be absolutely no doubt about the indispensability of the algae or plant photosynthesis to the man. It is the single agent able to utilize radiant energy from the sun and make it available as products to support life. This experiment very clearly shows the absolute dependence of man on nature.
Man has claimed to be unique. Social anthropologists have supported this claim on the ground that he alone has the gift of communication, and again that he alone has values. It might be worthwhile considering this viewpoint. A very famous biologist, David Goddard, said that a single human sperm, weighing one billionth of a gram, contains more information coded into its microscopic size than all of the information contained in all of the libraries of all men in all time. This same statement can be made for the seed of other animals or plants. This is a system of communication which is not rational, but which is extraordinarily delicate, elegant, and powerful, and which is capable of transmitting unimaginable quantities of information in microscopic volume.
This system of communication has enabled all species to survive the evolutionary time span. All forms of extant life share this system of communication; man's participation in it is in no sense exceptional.
Man also claims a uniqueness for himself on the grounds that he alone, of all of the animals, has values from which cultural objectives are derived. It would appear that the same genetic system of communication also contains a value system. Were this not so, those systems of organic life which do persist today would not have persisted; the genetic information transmitted is the information essential for survival. That information ensures the persistence of the organism within its own ecological community. The genetic value system also contains the essential mutation; that imperfection essential for evolution and survival. This system of communication is elegant, beautiful, and powerful, capable of sifting enormous numbers of conflicting choices. Man participates in and shares this system, but his participation is in no sense exceptional.
Yet another aspect of man's assumption that he is independent of natural processes is the anthropomorphic attitude which implies a finite man who is born, grows, and dies, but who during his life is made of the same unchanging stuff—himself. Not so. If we simply measure that which man ingests, uses, and rejects, we begin to doubt this premise. Hair, nails, skin, and chemical constituents are replaced regularly. He replaces several billion cells daily. The essential stuff of man is changed very regularly indeed. In a much more fundamental way, however, man is a creature of environment. We have learned that he is absolutely dependent upon stimuli—light, shadow, color, sound, texture, gravity; and upon his sense of smell, taste, touch, vision, and hearing. These constantly changing environmental conditions are his references. Without them there would be hallucination, hysteria, perhaps mental disintegration, certainly loss of reality.
# The Ecological View
It remains for the biologist and ecologist to point out the interdependence which characterizes all relationships, organic and inorganic, in nature. It is the ecologist who points out that an ecological community is only able to survive as a result of interdependent activity between all of the species which constitute the community. To the basic environment (geology, climate) is added an extraordinary complexity of inert materials, their reactions, and the interaction of the organic members of the community with climate, inert materials, and other organisms. The characteristic of life is interdependence of all of the elements of the community upon each other. Each one of these is a source of stimulus; each performs work; each is part of a pattern, a system, a working cycle; each one is to some lesser or greater degree a participant and contributor in a thermodynamic system. This interdependence common to nature—common to all systems—is in my own view the final refutation of man's assumption of independence. It appears impossible to separate man from this system. It would appear that there is a system, the order of which we partly observe. Where we observe it, we see interdependence, not independence, as a key. This interdependence is in absolute opposition to Western man's presumption of transcendence, his presumption of independence, and, of course, his presumption of superiority, dominion, and license to subdue the earth.
A tirade on the theme of dependence is necessary only to a society which views man as independent. Truly there is in nature no independence. Energy is the basis for all life; further, no organism has, does, or will live without an environment. All systems are depletive. There can be no enduring system occupied by a single organism. The minimum, in a laboratory experiment, requires the presence of at least two complementary organisms. These conceptions of independence and anthropocentrism are baseless.
The view of organisms and environment widely held by natural scientists is that of interdependence—symbiosis. Paul Sears of Yale University has written:
Any species survives by virtue of its niche, the opportunity afforded it by environment. But in occupying this niche, it also assumes a role in relation to its surroundings. For further survival it is necessary that its role at least be not a disruptive one. Thus, one generally finds in nature that each component of a highly organized community serves a constructive, or, at any rate, a stabilizing role. The habitat furnishes the niche, and if any species breaks up the habitat, the niche goes with it. . . . That is, to persist they [ecological communities] must be able to utilize radiant energy not merely to perform work, but to maintain the working system in reasonably good order. This requires the presence of organisms adjusted to the habitat and to each other, so organized as to make the fullest use of the influent radiation and to conserve for use and re-use the materials which the system requires. The degree to which a living community meets these conditions is therefore a test of its efficiency and stability (Sears 1956).
Man, too, must meet this test. Sears states:
Man is clearly the beneficiary of a very special environment which has been a great while in the making. This environment is more than a mere inert stockroom. It is an active system, a pattern and a process as well. Its value can be threatened by disruption no less than by depletion.
The natural scientist states that no species can exist without an environment, no species can exist in an environment of its exclusive creations, no species can survive, save as a non-disruptive member of an ecological community. Every member must adjust to other members of the community and to the environment in order to survive. Man is not excluded from this test.
Man must learn this prime ecological lesson of interdependence. He must see himself linked as a living organism to all living and all preceding life. This sense may impel him to understand his interdependence with the micro-organisms of the soil, the diatoms in the sea, the whooping crane, the grizzly bear, sand, rocks, grass, trees, sun, rain, moon, and stars. When man learns this he will have learned that when he destroys he also destroys himself; that when he creates, he also adds to himself. When man learns the single lesson of interdependence he may be enabled to create by natural process an environment appropriate for survival. This is a fundamental precondition for the emergence of man's role as a constructive and creative agent in the evolutionary process. Yet this view of interdependence as a basis for survival, this view of man as a participant species in an ecological community and environment, is quite contrary to the Western view.
I have reminded the reader that the creation of a physical environment by organisms, as individuals and as communities, is not exclusively a human skill; it is shared with the bee, the coral, and the chambered nautilus, which take inert materials and dispose them to create a physical environment, complementary to—indeed, indispensable to—the organism.
When man abandoned instinct for rational thought, he abandoned the powers that permitted him to emulate such organisms; if rationality alone sufficed, man should at least be able to equal these humble organisms. But thereby hangs a parable:
The nuclear cataclysm is over. The earth is covered with gray dust. In the vast silence no life exists, save for a little colony of algae hidden deep in a leaden cleft long inured to radiation. The algae perceive their isolation; they reflect upon the strivings of all life, so recently ended, and on the strenuous task of evolution to be begun anew. Out of their reflection could emerge a firm conclusion: "Next time, no brains."
# Reference
Sears, Paul B. 1956. "The Process of Environmental Change by Man." In W L. Thomas, Jr. ed., Man's Role in Changing the Face of the Earth. Chicago: University of Chicago Press.
# 2
# The Place of Nature in the City of Man (1964)
In the early 1960s McHarg's interest in values toward nature and the physical environment increased. Published in a special issue on urban revival in The Annals of the American Academy of Political and Social Science edited by Robert Mitchell, this article directs that interest toward the topic of nature in the city. McHarg proposes a theory for a "simple working method for open space." Essentially, he suggests that environmentally sensitive areas be used as open space. Such areas usually have multiple benefits. For example, by protecting wetland areas, floods can be controlled and safe drinking water supplies may be ensured. This theory for a "simple working method for open space" has had considerable influence since McHarg presented it in 1964.
# Abstract
Unparalleled urban growth is pre-empting a million acres of rural lands each year and transforming them into the sad emblems of contemporary urbanism. In that anarchy which constitutes urban growth, wherein the major prevailing values are short-term economic determinism, the image of nature is attributed little or no value. In existing cities, the instincts of eighteenth-and nineteenth-century city builders, reflected in the pattern of existing urban open space, have been superseded by a modem process which disdains nature and seems motivated by a belief in salvation through stone alone. Yet there is a need and place for nature in the city of man. An understanding of natural processes should be reflected in the attribution of value to the constituents of these natural processes. Such an understanding, reflected in city building, will provide a major structure for urban and metropolitan form, an environment capable of supporting physiological man, and the basis for an art of city building, which will enhance life and reflect meaning, order, and purpose.
# Introduction
"Before we convert our rocks and rills and templed hills into one spreading mass of low grade urban tissue under the delusion that, because we accomplish this degradation with the aid of bulldozers and atomic piles and electronic computers, we are advancing civilization, we might ask what all this implies in terms of the historic nature of man" (Lewis Mumford 1956, p. 142).
The subject of this essay is an inquiry into the place of nature in the city of man. The inquiry is neither ironic nor facetious but of the utmost urgency and seriousness. Today it is necessary to justify the presence of nature in the city of man; the burden of proof lies with nature, or so it seems. Look at the modern city, that most human of all environments, observe what image of nature exists there—precious little indeed and that beleaguered, succumbing to slow attrition.
William Penn effectively said, Let us build a fair city between two noble rivers; let there be five noble squares, let each house have a fine garden, and let us reserve territories for farming. But that was before rivers were discovered to be convenient repositories for sewage, parks the best locus for expressways, squares the appropriate sites for public monuments, farmland best suited for buildings, and small parks best transformed into asphalted, fenced playgrounds.
Charles Eliot once said, in essence, This is our city, these are our hills, these are our rivers, these our beaches, these our farms and forests. I will make a plan to cherish this beauty and wealth for all those who do or will live here. And the plan was good but largely disdained. So here, as elsewhere, man assaulted nature disinterestedly, man assaulted man with the city; nature in the city remains precariously as residues of accident, rare acts of personal conscience, or rarer testimony to municipal wisdom, the subject of continuous assault and attrition while the countryside recedes before the annular rings of suburbanization, unresponsive to any perception beyond simple economic determinism.
Once upon a time, nature lay outside the city gates a fair prospect from the city walls, but no longer. Climb the highest office tower in the city, when atmospheric pollution is only normal, and nature may be seen as a green rim on the horizon. But this is hardly a common condition and so nature lies outside of workaday experience for most urban people.
Long ago, homes were built in the country and remained rural during the lives of persons and generations. Not so today, when a country house of yesterday is within the rural-urban fringe today, in a suburb tomorrow, and in a renewal area of the not-too-distant future.
When the basis for wealth lay in the heart of the land and the farms upon it, then the valleys were verdant and beautiful, the farmer steward of the landscape, but that was before the American dream of a single house on a quarter acre, the automobile, crop surpluses, and the discovery that a farmer could profit more by selling land than crops.
Once men in simple cabins saw only wild nature, silent, implacable, lonely. They cut down the forests to banish Indians, animals, and shadows. Today, Indians, animals, and forests have gone and wild nature, silence, and loneliness are hard to find.
When a man's experience was limited by his home, village, and environs, he lived with his handiworks. Today, the automobile permits temporary escapes from urban squalor, and suburbanization gives the illusion of permanent escape.
Once upon a time, when primeval forests covered Pennsylvania, its original inhabitants experienced a North Temperate climate, but, when the forests were felled, the climate became, in summer, intemperately hot and humid.
Long ago, floods were described as Acts of God. Today, these are known quite often to be consequences of the acts of man.
As long ago, droughts were thought to be Acts of God, too, but these, it is now known, are exacerbated by the acts of man.
In times past, pure air and clean abundant water were commonplaces. Today, "pollution" is the word most often associated with the word "atmosphere," drinking water is often a dilute soup of dead bacteria in a chlorine solution, and the only peoples who enjoy pure air and clean water are rural societies who do not recognize these for the luxuries they are.
Not more than two hundred years ago, the city existed in a surround of farmland, the sustenance of the city. The farmers tended the lands which were the garden of the city. Now, the finest crops are abject fruits compared to the land values created by the most scabrous housing, and the farms are defenseless.
In days gone by, marshes were lonely and wild, habitat of duck and goose, heron and egret, muskrat and beaver, but that was before marshes became the prime sites for incinerator wastes, rubbish, and garbage—marshes are made to be filled, it is said.
When growth was slow and people spent a lifetime on a single place, the flood plains were known and left unbuilt. But, now, who knows the flood plain? Caveat emptor.
Forests and woodlands once had their own justification as sources of timber and game, but second-growth timber has little value today, and the game has long fled. Who will defend forests and woods?
Once upon a time, the shad in hundreds of thousands ran strong up the river to the city. But, today, when they do so, there is no oxygen, and their bodies are cast upon the shores.
# The Modern Metropolis
Today, the modern metropolis covers thousands of square miles, much of the land is sterilized and waterproofed, the original animals have long gone, as have primeval plants, rivers are foul, the atmosphere is polluted, climate and microclimate have retrogressed to increased violence, a million acres of land are transformed annually from farmland to suburban housing and shopping centers, asphalt and concrete, parking lots and car cemeteries, yet slums accrue faster than new buildings, which seek to replace them. The epidemiologist can speak of urban epidemics—heart and arterial disease, renal disease, cancer, and, not least, neuroses and psychoses. A serious proposition has been advanced to the effect that the modern city would be in serious jeopardy without the safeguards of modern medicine and social legislation. Lewis Mumford can describe cities as dysgenic. There has arisen the recent specter, described as "pathological togetherness," under which density and social pressure are being linked to the distribution of disease and limitations upon reproduction. We record stress from sensory overload and the response of negative hallucination to urban anarchy. When one considers that New York may well add 1,500 square miles of new "low-grade tissue" to its perimeter in the next twenty years, then one recalls Loren Eiseley's image and sees the cities of man as gray, black, and brown blemishes upon the green earth with dynamic tentacles extending from them and asks: "Are these the evidence of man, the planetary disease?"
# Western Views: Man and Nature
Yet how can nature be justified in the city? Does one invoke dappled sunlight filtered through trees of ecosystems, the shad run or water treatment, the garden in the city or negative entropy? Although at first glance an unthinkable necessity, the task of justifying nature in the city of man is, with prevailing values and process, both necessary and difficult. The realities of cities now and the plans for their renewal and extension offer incontrovertible evidence of the absence of nature present and future. Should Philadelphia realize its comprehensive plan, then $20 billion and twenty years later there will be less open space than there is today. (A prediction which, indeed, came to pass, with the single sad exceptions of the numerous "brownfield" sites of abandoned industry as well as deserted and derelict houses and businesses.) Cities are artifacts becoming ever more artificial—as though medieval views prevailed that nature was defiled, that living systems shared original sin with man, that only the artifice was free of sin. The motto for the city of man seems to be: salvation by stone alone.
Within the Western tradition exists a contrary view of man and nature which has a close correspondence to the Oriental attitude of an aspiration to harmony of man in nature, a sense of a unitary and encompassing natural order within which man exists. Among others, the naturalist tradition in the West includes Duns Scotus, Joannes Scotus Erigena, Francis of Assisi, Wordsworth, Goethe, Thoreau, Gerald Manley Hopkins, and the nineteenth-and twentieth-century naturalists. Their insistence upon nature being at least the sensible order within which man exists or a manifestation of God demanding deference and reverence is persuasive to many but not to the city builders.
Are the statements of scientists likely to be more persuasive?
David R. Goddard:
No organism lives without an environment. As all organisms are depletive, no organism can survive in an environment of its exclusive creation (1960).
F. R. Fosberg:
An ecosystem is a functioning, interacting system composed of one or more organisms and their effective environment, both physical and biological. All ecosystems are open systems. Ecosystems may be stable or unstable. The stable system is in a steady state. The entropy in an unstable system is more likely to increase than decrease. There is a tendency towards diversity in natural ecosystems. There is a tendency towards uniformity in artificial ecosystems or those strongly influenced by man (1958).
Paul Sears:
Any species survives by virtue of its niche—the opportunity afforded it by environment. But in occupying this niche, it also assumes a role in relation to its surroundings. For further survival it is necessary that its role at least be not a disruptive one. Thus, one generally finds in nature that each component of highly organized community serves a constructive, or at any rate a stabilizing, role. The habitat furnishes the niche, and, if any species breaks up the habitat, the niche goes with it . . . to persist [organic systems] must be able to utilize radiant energy not merely to perform work, but to maintain the working system in reasonably good order. This requires the presence of organisms adjusted to the habitat and to each other so organized to make the fullest use of the influent radiation and to conserve for use and reuse the materials which the system requires (1956, p. 472).
Complex creatures consist of billions of cells, each of which, like any single-celled creature, is unique, experiences life, metabolism, reproduction, and death. The complex animal exists through the operation of symbiotic relationships between cells as tissues and organs integrated as a single organism. Hans Selyé describes this symbiosis as intercellular altruism, the situation under which the cell concedes some part of its autonomy towards the operation of the organism and the organism responds to cellular processes.
Aldo Leopold has been concerned with the ethical content of symbiosis:
Ethics so far studied by philosophers are actually a process in ecological as well as philosophical terms. They are also a process in ecological evolution. An ethic, ecologically, is a limitation on freedom of action in the struggle for existence. An ethic, philosophically, is a differentiation of social from antisocial conduct. These are two definitions of one thing which has its origin in the tendency of interdependent individuals and groups to evolve modes of cooperation. The ecologist calls these symbioses. There is as yet no ethic dealing with man's relation to the environment and the animals and plants which grow upon it. The extension of ethics to include man's relation to environment is, if I read the evidence correctly, an evolutionary possibility and an ecological necessity. All ethics so far evolved rest upon a single premise that the individual is a member of a community of interdependent parts. His instincts prompt him to compete for his place in the community, but his ethics prompt him to cooperate, perhaps in order that there may be a place to compete for (1949).
The most important inference from this body of information is that interdependence, not independence, characterizes natural systems. Thus, man-nature interdependence presumably holds true for urban man as for his rural contemporaries. We await the discovery of an appropriate physical and symbolic form for the urban man–nature relationship.
# Natural and Artificial Environments
From the foregoing statements by natural scientists, we can examine certain extreme positions. First, there can be no conception of a completely "natural" environment. Wild nature, save a few exceptions, is not a satisfactory physical environment. Yet the certainty that man must adapt nature and himself does not diminish his dependence upon natural, nonhuman processes. These two observations set limits upon conceptions of man and nature. Man must adapt through both biological and cultural innovation, but these adaptations occur within a context of natural, nonhuman processes. It is not inevitable that adapting, nature to support human congregations must of necessity diminish the quality of the physical environment. Indeed, all of preindustrial urbanism was based upon the opposite premise, that only in the city could the best conjunction of social and physical environment be achieved. This major exercise of power to adapt nature for human ends, the city, need not be a diminution of physiological, psychological, and aesthetic experience.
While there can be no completely natural environments inhabited by man, completely artificial environments are equally unlikely. Man in common with all organisms is a persistent configuration of matter through which the environment ebbs and flows continuously. Mechanically, he exchanges his substance at a very rapid rate while, additionally, his conceptions of reality are dependent upon the attribution of meaning to myriads of environmental stimuli which impinge upon him continuously. The materials of his being are natural, as are many of the stimuli which he perceives; his utilization of the materials and of many stimuli is involuntary. Man makes artifices, but galactic and solar energy, gases of hydrosphere and atmosphere, the substance of the lithosphere, and all organic systems remain elusive of human artificers.
Yet the necessity to adapt natural environments to sustain life is common to many organisms other than man. Creation of a physical environment by organisms as individuals and as communities is not exclusively a human skill. The chambered nautilus, the beehive, the coral formation, to select but a few examples, are all efforts by organisms to take inert materials and dispose them to create a physical environment. In these examples, the environments created are complementary to the organisms. They are constructed with great economy of means; they are expressive; they have, in human eyes, great beauty; and they have survived periods of evolutionary time vastly longer than the human span.
Simple organisms utilize inert materials to create physical environments which sustain life. Man also confronts this necessity. Man, too, is natural in that he responds to the same laws as do all physical and biological systems. He is a plant parasite, dependent upon the plant kingdom and its associated microorganisms, insects, birds, and animals for all atmospheric oxygen, all food, all fossil fuel, natural fibers, and cellulose, for the stability of the water cycle and amelioration of climate and microclimate. His dependence upon the plant and photosynthesis establishes his dependence upon the microorganisms of the soil, particularly the decomposers which are essential to the recycling of essential nutrients, the insects, birds, and animals which are in turn linked to survival of plant systems. He is equally dependent upon the natural process of water purification by microorganisms. The operation of these nonhuman physical and biological processes is essential for human survival.
Having concluded that there can be neither a completely artificial nor a completely natural environment, our attention is directed to some determinants of optimal proportions. Some indication may be inferred from man's evolutionary history. His physiology and some significant part of his psychology derive from the billions of years of his biological history. During the most recent human phase of a million or so years, he has been preponderantly food gatherer, hunter, and, only recently, farmer. His urban experience is very recent indeed. Thus, the overwhelming proportion of his biological history has involved experience in vastly more natural environments than he now experiences. It is to these that he is physiologically adapted.
According to F. R. Fosberg:
It is entirely possible that man will not survive the changed environment that he is creating, either because of failure of resources, war over their dwindling supply, or failure of his nervous system to evolve as rapidly as the change in environment will require. Or he may only survive in small numbers, suffering the drastic reduction that is periodically the lot of pioneer species, or he may change beyond our recognition.... Management and utilization of the environment on a true sustaining yield basis must be achieved. And all this must be accomplished without altering the environment beyond the capacity of the human organism, as we know it, to live in it.(1957, p. 160).
# Human Ecosystems
There are several examples where ecosystems, dominated by man, have endured for long periods of time; the example of traditional Japanese agriculture is perhaps the most spectacular. Here an agriculture of unequaled intensity and productivity has been sustained for over a thousand years, the land is not impoverished but enriched by human intervention: the ecosystem, wild lands, and farmlands are complex, stable, highly productive, and beautiful. The pervasive effect of this harmony of man-nature is reflected in a language remarkable in its descriptive power of nature, a poetry succinct yet capable of the finest shades of meaning, a superb painting tradition in which nature is the icon, an architecture and town building of astonishing skill and beauty, and, not least, an unparalleled garden art in which nature and the garden are the final metaphysical symbol.
In the Western tradition, farming in Denmark and England has sustained high productivity for two or more centuries, appears stable, and is very beautiful; in the United States, comparable examples exist in Amish, Mennonite, and Pennsylvania Dutch farming.
Understanding of the relationship of man to nature is more pervasive and operative among farmers than any other laymen. The farmer perceives the source of his food in his crops of cereal, vegetables, roots, beef, fish, or game. He understands that, given a soil fertility, his crop is directly related to inputs of organic material, fertilizer, water, and sunlight. If he grows cotton or flax or tends sheep, he is likely to know the source of the fibers of his clothes. He recognizes timber, peat, and hydroelectric power as sources of fuel; he may well know of the organic source of coal and petroleum. Experience has taught him to ensure a functional separation between septic tank and well, to recognize the process of erosion, runoff, flood and drought, the differences of altitude and orientation. As a consequence of this acuity, the farmer has developed a formal expression which reflects an understanding of the major natural processes. Characteristically, high ground and steep slopes are given over to forest and woodland as a source of timber, habitat for game, element in erosion control, and water supply. The more gently sloping meadows below are planted to orchards, above the spring frost line, or in pasture. Here a seep, spring, or well is often the source of water supply. In the valley bottom, where floods have deposited rich alluvium over time, is the area of intensive cultivation. The farm buildings are related to conditions of climate and microclimate, above the flood plain, sheltered and shaded by the farm woodland. The septic tank is located in soils suitable for this purpose and below the elevation of the water source.
Here, at the level of the farm, can be observed the operation of certain simple, empirical rules and a formal expression which derives from them. The land is rich, and we find it beautiful.
Clearly, a comparable set of simple rules is urgently required for the city and the metropolis. The city dweller is commonly unaware of these natural processes, ignorant of his dependence upon them. Yet the problem of the place of nature in the city is more difficult than that of the farmer. Nature, as modified in farming, is intrinsic to the place. The plant community is relatively immobile, sunlight falls upon the site as does water, nutrients are cycled through the system in place. Animals in ecosystems have circumscribed territories, and the conjunction of plants and animals involves a utilization and cycling of energy and materials in quite limited areas. The modern city is, in this respect, profoundly different in that major natural processes which sustain the city, provide food, raw materials for industry, commerce, and construction, resources of water, and pure air are drawn not from the city or even its metropolitan area but from a national and even international hinterland. The major natural processes are not intrinsic to the locus of the city and cannot be.
# Nature in the Metropolis
In the process of examining the place of nature in the city of man, it might be fruitful to consider the role of nature in the metropolitan area initially, as here, in the more rural fringes, can still be found analogies to the empiricism of the farmer. Here the operative principle might be that natural processes which perform work or offer protection in their natural form without human effort should have a presumption in their favor. Planning should recognize the values of these processes in decision-making for prospective land uses.
A more complete understanding of natural processes and their interactions must await the development of an ecological model of the metropolis. Such a model would identify the regional inventory of material in atmosphere, hydrosphere, lithosphere, and biosphere, identify inputs and outputs, and both describe and quantify the cycling and recycling of materials in the system. Such a model would facilitate recognition of the vital natural processes and their interdependence which is denied today. Lacking such a model, it is necessary to proceed with available knowledge. On a simpler basis, we can say that the major inputs in biological systems are sunlight, oxygen–carbon dioxide, food (including nutrients), and water. The first three are not limiting in the metropolis; water may well be limiting both as to quantity and quality. In addition, there are many other reasons for isolating and examining water in process. Water is the single most specific determinant of a large number of physical processes and is indispensable to all biological processes. Water, as the agent of erosion and sedimentation, is causal to geological evolution, the realities of physiography. Mountains, hills, valleys, and plains experience a variety of climate and microclimate consequent upon their physiography; the twin combination of physiography and climate determines the incidence and distribution of plants and animals, their niches, and habitats. Thus, using water as the point of departure, we can recognize its impact on the making of mountains and lakes, ridges and plains, forests and deserts, rivers, streams and marshes, the distribution of plants and animals. Lacking an ecological model, we may well select water as the best indicator of natural process. In any watershed, the uplands represent the majority of the watershed area. Assuming equal distribution of precipitation and ground conditions over the watershed, the maximum area will produce the maximum runoff. The profile of watersheds tends to produce the steeper slopes in the uplands with the slope diminishing toward the outlet. The steeper the slope, the greater is the water velocity. This combination of maximum runoff links maximum volume to maximum velocity—the two primary conditions of flood and drought. These two factors in turn exacerbate erosion, with the consequence of depositing silt in stream beds, raising flood plains, and increasing intensity and incidence of floods in piedmont and estuary.
The natural restraints to flooding and drought are mainly the presence and distribution of vegetation, particularly, on the uplands and their steep slopes. Vegetation absorbs and utilizes considerable quantities of water; the surface roots, trunks of trees, stems of shrubs and plants, the litter of forest floor mechanically retard the movement of water, facilitating percolation, increasing evaporation opportunity. A certain amount of water is removed temporarily from the system by absorption into plants, and mechanical retardation facilitates percolation, reduces velocity, and thus diminishes erosion. In fact, vegetation and their soils act as a sponge restraining extreme runoff, releasing water slowly over longer periods, diminishing erosion and sedimentation, in short, diminishing the frequency and intensity of oscillation between flood and drought.
Below the uplands of the watershed are characteristically the more shallow slopes and broad plains of the piedmont. Here is the land most often developed for agriculture. These lands, too, tend to be favored locations for villages, towns, and cities. Here, forests are residues or the products of regeneration on abandoned farms. Steep slopes in the piedmont are associated with streams and rivers. The agricultural piedmont does not control its own defenses. It is defended from flood and drought by the vegetation of the uplands. The vegetation cover and conservation practices in the agricultural piedmont can either exacerbate or diminish flood and drought potential; the piedmont is particularly vulnerable to both.
The incidence of flood and drought is not alone consequent upon the upland sponge but also upon estuarine marshes, particularly where these are tidal. Here at the mouth of the watershed at the confluence of important rivers or of river and sea, the flood component of confluent streams or the tidal component of floods assumes great importance. In the Philadelphia metropolitan area, the ocean and the estuary are of prime importance as factors in flood. A condition of intense precipitation over the region combined with high tides, full estuary, and strong onshore winds combines the elements of potential flood. The relation of environmental factors of the upland component and the agricultural piedmont to flood and drought has been discussed. The estuarine marshes and their vegetation constitute the major defense against the tidal components of floods. These areas act as enormous storage reservoirs absorbing mile-feet of potentially destructive waters, reducing flood potential.
This gross description of water-related processes offers determinism for the place of nature in the metropolis. From this description can be isolated several discrete and critical phases in the process. Surface water as rivers, streams, creeks, lakes, reservoirs, and ponds would be primary; the particular form of surface water in marshes would be another phase; the flood plain as the area temporarily occupied by water would be yet another. Two critical aspects of groundwater, the aquifer and its recharge areas, could be identified. Agricultural land has been seen to be a product of alluvial deposition, while steep slopes and forests play important roles in the process of runoff. If we could identify the proscriptions and permissiveness of these parameters to other land use, we would have an effective device for discriminating the relative importance of different roles of metropolitan lands. Moreover, if the major divisions of upland, piedmont, and estuary and the processes enumerated could be afforded planning recognition and legislative protection, the metropolitan area would derive its form from a recognition of natural process. The place of nature in the metropolis would be reflected in the distribution of water and flood plain, marshes, ridges, forests, and farmland, a matrix of natural lands performing work or offering protection and recreational opportunity distributed throughout the metropolis.
This conception is still too bald; it should be elaborated to include areas of important scenic value, recreational potential, areas of ecological, botanical, geological, or historic interest. Yet, clearly, the conception, analogous to the empiricism of the farmer, offers opportunity for determining the place of nature in the metropolis.
# Nature in the City
The conception advocated for the metropolitan area has considerable relevance to the problem of the place of nature in the city of man. Indeed, in several cities, the fairest image of nature exists in these rare occasions where river, flood plain, steep slopes, and woodlands have been retained in their natural condition—the Hudson and Palisades in New York, the Schuylkill and Wissahickon in Philadelphia, the Charles River in Boston and Cambridge. If rivers, flood plains, marshes, steep slopes, and woodlands in the city were accorded protection to remain in their natural condition or were retrieved and returned to such a condition where possible, this single device, as an aspect of water quality, quantity, flood and drought control, would ensure for many cities an immeasurable improvement in the aspect of nature in the city, in addition to the specific benefits of a planned watershed. No other device has such an ameliorative power. Quite obviously, in addition to benefits of flood control and water supply, the benefits of amenity and recreational opportunity would be considerable. As evidence of this, the city of Philadelphia has a twenty-two-mile waterfront on the Delaware River. The most grandiose requirements for port facilities and water-related industries require only eight miles of waterfront. This entire waterfront lies in a flood plain. Levees and other flood protection devices have been dismissed as exorbitant. Should this land be transformed into park, it would represent an amelioration in Philadelphia of incomparable scale.
Should this conception of planning for water and water-related parameters be effectuated, it would provide the major framework for the role of nature in the city of man. The smaller elements of the face of nature are more difficult to justify. The garden and park, unlike house, shop, or factory, have little "functional" content. They are, indeed, more metaphysical symbol than utilitarian function. As such, they are not amenable to quantification or the attribution of value. Yet it is frequently the aggregation of these gardens and spaces which determines the humanity of a city. Values they do have. This is apparent in the flight to the suburbs for more natural environments—a self-defeating process of which the motives are clear. Equally, the selection of salubrious housing location in cities is closely linked to major open spaces which reflects the same impulse. The image of nature at this level is most important, the cell of the home, the street, and neighborhood. In the city slum, nature exists in the backyard ailanthus, sumac, in lice, cockroach, rat, cat, and mouse; in luxury highrise, there are potted trees over parking garages, poodles, and tropical fish. In the first case, nature reflects "disturbance" to the ecologist; it is somewhat analogous to the scab on a wound, the first step of regeneration towards equilibrium, a sere arrested at the most primitive level. In the case of the luxury highrise, nature is a canary in a cage, surrogate, an artifice, forbidden even the prospect of an arrested sere.
Three considerations seem operative at this level of concern. The first is that the response which nature induces, tranquillity, calm, introspection, openness to order, meaning and purpose, the place of values in the world of facts, is similar to the evocation from works of art. Yet nature is, or was, abundant; art and genius are rare.
The second consideration of some importance is that nature in the city is very tender. Woodlands, plants, and animals are very vulnerable to human erosion. Only expansive dimensions will support self-perpetuating and self-cleansing nature. There is a profound change between such a natural scene and a created and maintained landscape.
The final point is related to the preceding. If the dimensions are appropriate, a landscape will perpetuate itself. Yet, where a site has been sterilized, built upon, buildings demolished, the problem of creating a landscape, quite apart from creating a self-perpetuating one, is very considerable and the costs are high. The problems of sustaining a landscape, once made, are also considerable; the pressure of human erosion on open space in urban housing and the inevitable vandalism ensure that only a small vocabulary of primitive and hardy plants can survive. These factors, with abnormal conditions of groundwater, soil, air, atmospheric pollution, stripping, and girdling, limit nature to a very constricted image.
# The Future
Perhaps, in the future, analysis of those factors which contribute to stress disease will induce inquiry into the values of privacy, shade, silence, the positive stimulus of natural materials, and the presence of comprehensible order, indeed natural beauty. When young babies lack fondling and a mother's love, they sometimes succumb to moronity and death. The dramatic reversal of this pattern has followed simple maternal solicitude. Is the absence of nature—its trees, water, rocks and herbs, sun, moon, stars, and changing seasons—a similar type of deprivation? The solicitude of nature, its essence if not its image, may be seen to be vital.
Some day, in the future, we may be able to quantify plant photosynthesis in the city and the oxygen in the atmosphere, the insulation by plants of lead from automobile exhausts, the role of diatoms in water purification, the amelioration of climate and microclimate by city trees and parks, the insurance of negative ionization by fountains, the reservoirs of air which, free of combustion, are necessary to relieve inversion pollution, the nature-space which a biological inheritance still requires, the stages in land regeneration and the plant and animal indicators of such regeneration, indeed, perhaps, even the plant and animal indicators of a healthy environment. We will then be able to quantify the necessities of a minimum environment to support physiological man. Perhaps we may also learn what forms of nature are necessary to satisfy the psychological memory of a biological ancestry.
Today, that place where man and nature are in closest harmony in the city is the cemetery. Can we hope for a city of man, an ecosystem in dynamic equilibrium, stable and complex? Can we hope for a city of man, an ecosystem with man dominant, reflecting natural processes, human and non-human, in which artifice and nature conjoin as art and nature, in a natural urban environment speaking to man as a natural being and nature as the environment of man? When we find the place of nature in the city of man, we may return to that enduring and ancient inquiry—place of man in nature.
# References
Fosberg, F. R. 1958. "The Preservation of Man's Environment." In Proceedings of the Ninth Pacific Science Congress, 1957, 20.
Goddard, David F. 1960. The House We Live In. Transcript of the program broadcast on WCAU-TV (CBS), Channel 10, Philadelphia, Sunday, October 23, hosted and edited by Ian McHarg.
Leopold, Aldo. 1949. A Sand County Almanac. Oxford: Oxford University Press.
Mumford, Lewis. 1956. "Prospect." In William L. THomas, Jr., ed., Man's Role in Changing the Face of the Earth. Chicago: The University of Chicago Press, pp. 1132-52.
Sears, Paul B. 1956. "The Processes of Environmental Change by Man." In William L. Thomas, Jr., ed., Man's Role in Changing the Face of the Earth. Chicago: The University of Chicago Press, pp. 471-84.
# 3
# Ecological Determinism (1966)
During the 1960s, McHarg directed his own efforts as well as those of his graduate students toward ways ecological principles could be applied to landscape architecture and environmental planning. This paper was presented at a conference convened by The Conservation Foundation in April, 1965, at Airlie House, Warrenton, Virginia. Subsequently it appeared in the book Future Environments of North America edited by F. Fraser Darling and John P. Milton and published by The Natural History Press (the eventual publisher of Design with Nature). Darling, a distinguished British ecologist, pioneered the study of human ecology in the Scottish highlands. He showed how small technological innovations changed human settlements and landscapes. This paper is an early discussion of McHarg's theory for an ecological planning method.
# Introduction
In the Western world during the past century transformation of natural environments into human habitats has commonly caused a deterioration of the physical environment. However, much improvement to the social environment has been accomplished in these transformations; city slurb and slum are less attractive physical environments than forest, field, and farm that preceded them. In earlier times, because of the slow rate of change, unity of materials, structural method and expression, this was not so. Few among us regret the loss of ancient marshes on which Venice and Amsterdam sit, the loss of even more ancient hills which now seat Athens and Rome. History testifies to human adaptations, accomplished with wisdom and art, which were and are felicitous. Yet the principles which ensured these successes are inadequate for the speed, scale, and nature of change today. In the seventeenth century it required a third of the treasury of France, the mature life of Louis XIV, and the major effort of André Le Nôtre to realize Versailles. Three centuries later greater New York will urbanize at the rate of 50,000 acres and absorb 600,000 people into its perimeter each year without any plan. In the interim the classical city has been superseded by the industrial city, by metropolis, megalopolis, and now, in the opinion of Lewis Mumford, is en route to Necropolis. Paradoxically in this period of change the city plan has remained the Renaissance archetype which motivated Versailles, a poor symbol of man–nature in the seventeenth century, an inexcusable prototype for the twentieth century.
It is clear that the principles which contributed to historic successes in urban form have failed dismally since the industrial revolution. The success of the subsequent city as provider of employment and social services is its best testimony, but as a physical environment it has continually retrogressed. New principles must be developed for human adaptations, for city, metropolis, and megalopolis.
The problem is an enormous one both in extent and speed of change. Three hundred million Americans are expected to populate the United States in the year 2000. If indeed 80 percent of these will live in urban places, then this involves the urbanization of 55 million acres of non-urbanized land. If one extrapolates from the present megalopolis to this future population, then 10 percent of the land area of the United States, 200 million acres, will fall within urban influence, comparable to megalopolis, in a mere 35 years.
Today, the prescriptions for urban location, form, and growth derive almost exclusively from the social sciences. Both analytic and predictive models are based upon economics. The natural sciences have remained aloof from the problem, yet the understanding of physical and biological processes, which reposes in these sciences, is indispensable for good judgment on the problems of human adaptations of environment.
Many central questions can best be answered by natural scientists but at the onset one alone can suffice. What are the implications of natural process upon the location and form of development? The answer to this is vital to administrators, regional and city planners, architects and landscape architects. For the last it is the indispensable basis for their professional role. As the representative of a profession with a historic concern for the relation of man to nature and the single bridge between the natural sciences and the artificers of the urban environment, it is not inappropriate that the spokesperson for this group ask for the formulation of an ecological determinism.
# Landscape Architecture
In the Western tradition, with the single exception of the English eighteenth century and its extension, landscape architecture has been identified with garden making, be it Alhambra, Saint Gall, the Villa d'Este, or Versailles. In this tradition decorative and tractable plants are arranged in a simple geometry as a comprehensible metaphysical symbol of a benign and orderly world.
Here the ornamental qualities of plants are paramount; no concepts of community or association becloud the objective. Plants are analogous to domestic pets, dogs, cats, ponies, canaries, and goldfish, tolerant to man and dependent upon him; lawn grasses, hedges, flowering shrubs and trees, tractable and benign, man's cohorts, sharing his domestication.
This is the walled garden, separated from nature, a symbol of beneficence, island of delight, tranquillity, introspection. It is quite consistent that the final symbol of this garden is the flower.
Not only is this a selected nature, decorative and benign, but the order of its array is, unlike the complexity of nature, reduced to a simple and comprehensible geometry. This is then a selected nature, simply ordered to create a symbolic reassurance of a benign and orderly world, an island within the world and separate from it. Yet the knowledge prevails that nature reveals a different form and aspect beyond the wall. Loren Eiseley has said that "the unknown within the self is linked to the wild." The garden symbolizes domesticated nature, the wild is beyond.
The role of garden making remains important. Man seeks a personal paradise on earth, a unity of man and nature. The garden is such a quest for a personal oasis, a paradise garden. In these, man can find peace and in tranquillity discover, in Kenneth Rexroth's words, "the place of value in a world of facts." He can respond to natural materials, water, stone, herbs, trees, sunlight and shadow, rain, ice and snow, the changing seasons, birth, life and death. This is a special microhabitat, a healthy cell in the organism of the city, a most humane expression, yet clearly its relevance, depending upon ornamental horticulture and simple geometry, is inadequate for the leap over the garden wall.
In the eighteenth century in England landscape architects "leap't the wall and discovered all nature to be a garden." The leap did not occur until a new view of nature dispelled the old and a new esthetic was developed consonant with the enlarged arena.
Starting with a denuded landscape, a backward agriculture, and a medieval pattern of attenuated land holdings, this landscape tradition rehabilitated an entire countryside, making that fair image persisting today. It is a testimony to the prescience of William Kent, Lancelot "Capability" Brown, Humphry Repton, and their followers that, lacking a science of ecology, they used native plant materials to create communities which so well reflected natural processes that their creations endured and are self-perpetuating.
The functional objective was a productive, working landscape. Hilltops and hillsides were planted to forest, great meadows occupied the valley bottoms in which lakes were constructed and streams meandered. The product of this new landscape was the extensive meadow supporting cattle, horses, and sheep. The forests provided valuable timber, the lack of which Evelyn had earlier deplored, and supported game, while free-standing copses in the meadows provided shade and shelter for animals.
The planting reflected the necessities of shipbuilding but the preferred trees, oak and beech, were climax species and they were planted de novo. On sites where these were inappropriate, northern slopes, thin soils, elevations, pine and birch were planted. Watercourses were graced with willows, alders, osiers, while the meadows supported grasses and meadow flowers. As long as the meadow was grazed, a productive sere was maintained and meanwhile the forests evolved.
The objective, however, was more complex than function alone. Paintings of the campagna by Claude Lorrain, Nicolas Poussin, and Salvator Rosa, a eulogy of nature which obsessed poets and writers, had developed the concept of an ideal nature. Yet it clearly did not exist in the raddled landscape of eighteenth-century England. It had to be created. The ruling principle was that "nature is the gardener's best designer," applied ecology of yesteryear. Ornamental horticulture, which had been obtained within garden walls, was disdained and a precursory ecology replaced it. The meadow was the single artifice, the remaining components were natural expressions, their dramatic and experiential qualities exploited, it is true, but deriving in the first place from that observed in nature.
Nature itself produced the esthetic; the simple geometry, not simplicity but simple-mindedness, of the Renaissance was banished. "Nature abhors a straight line" was declaimed. The discovery of an established esthetic in the Orient based upon occult balance, asymmetry, confirmed this view. In the eighteenth century, landscape began the revolution which banished the giant classicism and the imposition of its geometry as a symbol of man–nature.
This tradition is important in many respects. It founded applied ecology as the basis for function and esthetics in the landscape. Indeed before the manifesto of modern architecture had been propounded, "Form follows function," it had been superseded by the eighteenth-century concept wherein form and process were seen to be indivisible facets of a single phenomenon. It is important because of the scale of operation. One recalls that Capability Brown, when asked to undertake a project in Ireland, retorted, "I have not finished England yet:" Another reason for its importance lies in the fact that it was a creation. Here the landscape architect, like the empiricist doctor, found a land in ill-health and brought it into good heart and to beauty. Man the artist, understanding nature's laws and forms, accelerated the process of regeneration so well indeed that who today can discern the artifice from the untouched?
It is hard to find fault with this tradition but one must observe that while the principles of ecology and its esthetic are general, the realization of this movement was particular. It reflects in agricultural economy, principally based upon cattle, horses, and sheep. It never confronted the city, which in the eighteenth century remained the Renaissance prototype. Only in the urban square, parks and circuses, in natural plantings, was the eighteenth-century city distinguishable from its antecedents.
The successes of this tradition are manifest. No other movement has accomplished such a physical regeneration and amelioration. Its basis lies in applied ecology. It is necessary that modern ecology become the basis for modern interventions particularly at the scale of city, metropolis, and megalopolis, if we are to avert Necropolis.
# Ecological Determinism
Processes are expressive; morphology is a superficial expression of the process examined. The creation of a twentieth-century tradition requires an understanding of natural process and the morphology of the artifacts of man as process. Thus, natural processes are deterministic, they respond to laws; they then give form to human adaptations which themselves contain symbolic content.
Beehive huts, igloos, stilt homes on marshes are morphologically determined. We need today an understanding of natural process and its expression and, even more, an understanding of the morphology of man–nature, which, less deterministic, still has its own morphology, the expression of man-nature as process. The eighteenth century developed a morphology for a pastoral landscape in England. What are the prerequisites for discerning the appropriate morphologies for our time?
I believe that there are six elements which are required:
1. Ecosystem inventory
2. Description of natural processes
3. Identification of limiting factors
4. Attribution of value
5. Determination of prohibitions and permissiveness to change
6. Identification of indicators of stability or instability
The final for the artificers is then the perception of the revealed morphology and its realization.
# Ecosystem Inventory
The eighteenth-century landscape architects were only fair taxonomists, but, by using collected material, transferring from site to like site and planting in communities, they avoided the errors of caprice, ornamental horticulture, and much traditional forestry. In the intervening years descriptive ecology has developed the concept of community. In the Philadelphia region the identification of open water, reed swamp, sedge meadow, secondary succession, mixed mesophytic forest, and pine barrens has great utility. Recent work which refines these descriptions by identifying gradients adds value to this technique. The conception of range from hydrosere to zerosere is of great value but it is the conception of succession, sere, and climax which adds dynamics to the eighteenth-century view. The first prerequisite for the application of ecology to the planning process is the preparation of ecosystem inventories. This involves the creation of ecological maps at various scales in which communities are identified. The inventory should also include the city. The ailanthus-pigeon-starling "community" is quite as important as the oak-beech-hickory forest. The ecosystem inventory is the basis for planning related to natural processes.
# Description of Natural Processes
Inventories and ecological maps have to be supplemented by explanation of natural processes. In particular the stability or instability stage in succession of ecosystems must be described. While this is important for all communities it is particularly necessary for major physiographic regions—coastal plains, piedmont, uplands, etc—and for certain discrete environments—pine barrens, estuarine environment, mixed mesophytic forest, sand dunes, etc. In the city the relation of atmospheric pollution to isolation photosynthesis CO2 consumption is typical of a human process which affects ecosystems. Transformation of farmland to subdivision and erosion-turbidity-reduced photosynthesis and natural water purification are other examples. Descriptions of natural processes and the degree to which they are affected by man are a vital component of this body of information.
# Identification of Limiting Factors
It is important to establish what factors are necessary to ensure the perpetuation of any ecosystem; apart from factors in abundance, which elements are critical—water—plane—table elevation, alkalinity, acidity, fire, first and last frost, etc. This category must be extensive enough to include limiting factors external to the ecosystem under study such as, for example, transformation of a fresh-water into a salt-water marsh through channel deepening and river widening.
# Attribution of Value
In eighteenth-century England the land was thought to be the arena for the creation of a metaphysical symbol—all nature is a garden. Nature was attributed a value which transcended any concept of productivity, the landscape was also productive. In the twentieth century, when nature desperately needs a defense in the face of disdain, disinterest, and remorseless despoliation, the first defense is certainly non-economic, the insistence that man is a co-tenant of the universe, sharing the phenomenal world of living and inert processes, an organism in an interacting biosphere. From this some people will conclude that nature is a manifestation of God, others that the cosmos is unitary, all processes subject to physical law yet having infinite potential, that man is an inhabitant free to develop his potential. Each of these views contains an inherent deference, a posture of serious inquiry, and the instinct to exercise care, intelligence, and art in accomplishing human interventions. Such a view characterized eighteenth-century landscape architects, nineteenth-century naturalists, and today is associated with conservationists.
The search for a theology of man–nature–God does not exclude exchanges which involve the coinage of the time and place. This requires that the proponents of nature also attribute values to nature processes so that these may be recognized as parameters in the planning process. Indeed, given ecological inventories, explanation of natural processes, and identification of limiting factors, the next increment of knowledge essential for an applied ecology is precisely the attribution of value to these natural processes.
Four major divisions of value can be discerned: intrinsic value, value as productivity, value as work performed, and, finally, negative value as discouragement to use.
Intrinsic value is thought to exist wherein the landscape neither is "productive" nor "performs work" but simply is. Areas of wilderness, scenic beauty, scientific value, and educational value might fall into this category.
Productivity would include agriculture, forestry, fisheries, extractive minerals, recreation, gamelands, a concept in common usage.
The attribution of value based upon work performed might include water catchment, water purification and storage, flood, drought and erosion control, hydroelectric power generation, "airshed;" climate, and microclimate amelioration.
Negative value would include those areas wherein there is a hazard and whence occupancy should be discouraged. No occupancy would avert costs and damages. Thus areas subject to earthquakes, volcanism, hurricanes, avalanches, floods, drought, subsidence, and forest fires should fall into this category.
All of these subdivisions can be subject to the concept of replacement value, a most useful concept which can apply at several scales. For example, in the case of a city park planned for an expressway intersection, the value is not "land value" alone but rather the entire cost of replicating the park including the cost of equally mature trees, shrubs, etc. Where it is intended to fill a marsh, the replacement value would include the cost of equal flood protection, water equalization, and wildlife habitats on another site. In the case of transforming prime agricultural land to housing, replacement value would include the cost of upgrading poorer soils to prime capability. Given attribution of value to natural processes, the concept of replacement value provides an important measuring device for alternative choices. No other device offers a comparable restraint to thoughtless despoliation.
Clearly the concept of value poses many difficulties, the change of value over time is one. Low-grade ores, presently marginal farmland, undistinguished rural areas can increase in value with increased demand and shrunken supply. In addition, value is relative. If, as in the Netherlands, survival is linked to the stability of the dunes, then marram and sedge are valuable. If no such a role exists, then dune grasses are merely decorative. If diatoms are needed for water treatment, then they have value. If water is treated with coagulants, rapid sand filter, and chlorine, then diatoms have, in this local case, no value. Marshes can be seen either as costly obstructions to development or as invaluable defenses against flood; in one case they represent costs, in another they represent values. Another problem arises from the geographic scale of natural process and interdependence. The Mississippi watershed unites suburban Chicago with New Orleans; effects upon water quality will affect values in the entire downstream area. The requirements of clean air unite western and eastern United States. There is no method of accounting which relates snowfall in the Rockies to the value of water in California, no accounting which attributes value to forests of upstream watersheds and flood control in the lower Mississippi. The final difficulty in attribution of value lies in unmeasurable qualities. Who will attribute value to the whooping crane, grizzly bear, or pasque flower? Yet the inability to attribute value to serenity, happiness, and health has not deterred economic determinism. In spite of difficulties, the attribution of value to natural process is a necessary precondition for applied ecology as a basis for determining non-intervention, intervention, and the nature, scale, and location of such intervention.
# Determination of Prohibitions and Permissiveness to Change
Given descriptive ecological inventories, supplemented by descriptions of natural process and limiting factors, with a scale of values, the necessary information is available to establish the constraints inherent in natural process which should affect the location and nature of development.
This finally produces the program. No longer is nature an undifferentiated scene, lacking values, defenseless against transformation; it is seen to be a complex interrelated system, in process, having discernible limiting factors, containing values, present or prospective, and, finally, containing both constraints and opportunities. For example, the Arctic and Antarctic; bare mountains, oceans, and perhaps beaches are highly tolerant to human use. Other systems are highly intolerant, wild animals retreat or succumb, natural plant communities are erased and superseded by man's cohorts in the processes of agriculture, industry, and urbanism. Can we select environments, more suitable than these extremely tolerant examples, which, satisfactory for man, are unusually tolerant to him? Can one set limits on transformation of natural habitats implicit in the processes themselves? Thus, how much housing can a forest absorb and still perform an attributed role in the water regimen? How much development can occur on a marsh without destroying its role of water equalization, emergency flood storage, or wildlife habitat? What proportion of an aquifer can be waterproofed by urbanism before its percolation performance is diminished below a required level? The answer to such questions, and many others, is prerequisite to the application of ecology to the planning process.
For the regional planner, landscape architect, city planner, and architect, the development of concepts of prohibition and permissiveness inherent in natural process is the beginning of a modern applied ecology, the gift of natural form, the program for intervention which has relevance to the house and its site, the subdivision, hamlet, village, town, city, metropolis, megalopolis, and the nation.
# Identification of Indicators of Stability or Instability
The concept of ecological determinism requires criteria of performance. For pond ecosystems it may be possible to determine stability or instability from the entropy in the system. It seems unlikely that this concept will be capable of dealing with extensive geographic areas or with the problem of the city. Clearly for the moment some simpler devices are necessary by which stability or instability, succession, sere or climax can be discerned; indeed, there is a desperate need for a concept of a "healthy" and "healthful" environment. We need analogies to litmus paper, the canary in the cage, indicators of health and change. Ruth Patrick has developed histograms from which the "state of health" of Pennsylvania streams may be discerned. Luna Leopold has propounded the measure of a "duration curve" against which stability or instability of stream processes can be examined. This writer has advocated the use of the "coefficient of runoff" as a planning determinant for land use relative to the water regimen. "Sky blue" is a measure of the relative presence of atmospheric pollution. The presence of trace metals in the environment is being investigated as an indicator of health. The technique of indicators is established, but it must be extended, not only to "natural" environments, but also to include those dominated by man.
Can we proceed from broad presumptions which have utility while evidence is collected and analyzed? Can one say that where trees cannot live, then man should not try? Can one say that when the most abundant inhabitants, with man, are pigeons, starlings, and rats, a threshold has been crossed and either evacuation or redevelopment is necessary?
It is important that stable and healthy forests, marshes, deserts, rivers, and streams can be defined, that succession and retrogression can be identified, but it is even more necessary to find comparable definitions and indicators for the city. This moves the subject from the orthodoxy of ecology yet those concerned with the environment are the cohorts of medicine. Pathology is the concern of the medical sciences, the environment of health must be the concern of the artificers, but they require from ecologists an identification of healthy and healthful environments.
The role of the ecologists would include the identification of "healthy environments"—that is, for example, a forest wherein trees, shrubs, animals, and fish were conspicuously healthy and a determination of those factors which were contributory to this condition. Healthy natural environments could then be used as criteria against which adjacent urban environments could be compared.
In the city, examination of health or pathology devolves upon the social and medical sciences. It is necessary to determine stressors, the pathology of stress, and the environment of stress. Among stressors the poisons of food, atmosphere and water, density, noise, sensory overload, and sensory deprivation would be included. The pathology of stress—mortality, cancer, heart disease, suicide, alcoholism, crime, neuroses, and psychoses—might be mapped as isobars, and from the incidence and distribution of both stressors and stress disease the environments of stress may be located.
Given identification of healthy "natural" environments and the urban environments of stress pathology, diagnosis of environments becomes possible and therapy can be prescribed. There are environmental variables linked to most urban stressors. In this search for health and pathology in environments, indicators could serve a valuable role in diagnosis. In Philadelphia does the presence of hemlock indicate a tolerable level of particulate matter in the atmosphere? What plant pathology indicates a dangerous level of lead? Are there indicators of sulphur dioxide or nitrogen dioxide at "safe" levels? What creatures can coexist with levels of noise preferred by man? What indicators can be found for the optimum distribution of trace metals? Which reveal optimal concentrations? What can be inferred of human environments when trees degenerate?
This line of inquiry might well be unproductive yet there are now no operative standards of environmental quality, no limits are placed upon density, poisons, noise, no criteria are available as measures of existing environments or as programs for new environments.a It is clear that criteria are need-ed, and for the empiricist planner, indicators could be a vital tool. The conception of succession or retrogression, stability and instability must be utilized in the examination not only of wild environment, but of those environments dominated by man.
Where does the canary expire, the litmus change color? Where is the dis-genic zone, the area of apathetic survival, the environment of full health—and what are the indicators?
# The Morphology of Ecological Determinism
Each year a million acres of rural land succumb to hot-dog stands, sagging wires, billboards, diners, gas stations, split-levels, ranches, asphalt, and concrete. Most of this is accomplished without benefit of planning, landscape architecture, or architecture. Where these professionals intervene they utilize some part of available knowledge, the best of them are pressing for better information, yet action must occur even when information is inadequate.
This apologia precedes a description of three experiences in which the writer has been involved, each crude in terms of the available information and the interpretations made, yet, for all of their crudity, so effective in giving form as to justify description. These experiences permit extrapolation on the form of ecological determinism.
# New Jersey Shore
From the fifth to the eighth of March 1962 a violent storm lashed the northeast coast from Georgia to Long Island. For three days sixty-mile-per-hour winds whipped high spring tides across 1,000 miles of ocean. Forty-foot waves pounded the shore and vast stretches of barrier islands and bay shore were flooded. In New Jersey alone, 2,400 houses were wrecked, 8,300 partially damaged, and eighty million dollars' worth of direct damages incurred over a three-day period. Almost all of this damage was caused by ignorance of natural process and the resultant destruction of the natural defenses against flood. In this case an ecological inventory existed, natural processes in the beach-bay-shore communities had been described and limiting factors identified, the values involved could be inferred from damages and costs. The requirements requested by this chapter for all ecosystems were satisfied in this one situation.
The theory of dune formation is well understood, as is stabilization by vegetation. The ecological communities from beach dune to bay shore have been identified, as have been their limiting factors. In the Netherlands the value of dunes and their stabilizing grasses, and the important role of groundwater are known and attributed value, but not, however, in New Jersey. It is common knowledge that beaches are highly tolerant to human use but that dunes and their grasses are not. Development of the Jersey shore included breaching of dunes for many purposes—home building, beach access, etc. No constraints were placed upon use of dunes so that vegetation died and the dunes became unstable; no effective restraints were placed upon withdrawals of groundwater which inhibited vegetation growth. Considerable areas were waterproofed by buildings, roads, parking areas, which diminished recharge of the aquifers. The consequences were inevitable: With its natural defenses destroyed, the shore was vulnerable and was extensively damaged.
Had development responded to an understanding of natural process, ecological determinism, much if not all of the damage could have been averted. The form of development, however, would have been quite different. The beach, highly tolerant, would have been intensively utilized; the dunes, generally two in number between sea and bay, would have been inviolate, protected, stabilized by marram; access to the beach would have been made possible by bridges over the dunes; woody vegetation would have been protected in the trough; development would have been excluded from dunes and located only in wide areas of trough and on the bay side. Roads and parking would have been constructed of flexible, permeable materials; drainage would have been used for aquifer recharge; buildings would have been of columnar construction; withdrawals of groundwater would have been regulated. The application of this determinism, responsive to natural process, would have located buildings, roads, and drainage systems and determined their form. It would have been the principal determinant of planning, landscape architecture, and architecture. It would indeed produce the morphology of man-dune-bay for the New Jersey shore.
# The Green Springs and Worthington Valleys
Seventy square miles of beautiful farmland adjacent to Baltimore were made accessible by the construction of a new expressway. The present population [in 1966] of 17,000 was predicted to rise to 75,000 by 1980, to 110,000 by the year 2000. It became necessary to determine where best this development would be located to ensure the conjunction of optimum development and optimum preservation. The conception of ecological determinism was invoked.
The area is characterized by three major valleys contained by wooded slopes with a large intervening plateau. It transpired that the valley bottoms were coterminous with the major aquifers in the region, that in the valleys were the major surface water systems, flood plains, and extensive soils unsuitable for development using septic tanks. The major source of erosion lay in the slopes defining the valleys. The plateau in contrast contained no major aquifers or minor streams; flood plains were absent, as were soils unsuitable for development using septic tanks.
Ecological determinism suggested prohibition of development in the valleys, prohibitions of development on bare slopes, development limited to one house per three acres on wooded slopes. In contrast, development was concentrated upon the plateau in a hierarchy of country town, villages, and hamlets, related in response to physiography. Development in wooded plateau sites was limited to one house per acre; this standard was waived on promontory sites, where high-rise apartments were recommended.
In this example, ecological analysis revealed the morphology of development; it selected the plateau as most tolerant, the valley bottoms and side slopes as least tolerant. It suggested densities of use appropriate to natural process. When this analysis was carried to a more detailed examination of the plateau, this, too, demonstrated variability in both opportunity and intolerance which suggested a hierarchy of communities, hamlets, villages, and a country town instead of pervasive suburbanization. Here the utilization of a very few determinants—surface water, groundwater, flood plains, alluvial silts, steep slopes, forest cover, and an interpretation of their permissiveness and prohibitions, revealed the morphology of man and the Maryland piedmont.
# Metropolitan Open Space from Natural Processes
The Urban Renewal Agency and the states of Pennsylvania and New Jersey supported a research project to establish criteria for the selection of metropolitan open space. The hypothesis advanced was that such criteria can best be discerned by extending the inquiry to examine the operation of the major physical and biological processes in the region. It was suggested that when this is understood, and land-use policies reflect this understanding, not only will the values of natural process be ensured, the place for nature in the metropolis of man, but also there will be discerned a structure of natural process in the metropolitan area, form-giving for metropolitan growth, and the identification of areas for metropolitan open space, recreation, and amenity.
In the country at large and even in metropolitan areas, open space is seen to be absolutely abundant. In the Philadelphia Standard Metropolitan Statistical Area (PSMSA) only 19.1 percent of the area is urbanized; twenty years hence this may reach 30 percent, leaving 2,300 square miles of open space.
The problem of metropolitan open space lies, then, not in absolute area, but in distribution. The commodity concept of open space for amenity or recreation would suggest an interfusion of open space and population. The low attributed value of open space ensures that it is transformed into urban use within the urban area and at the perimeter. Normal process excludes interfusion and consumes peripheral open space.
Yet as the area of a circle follows the square of the radius, large open-space increments can exist within the urban perimeter without major increase in the radius or in the time distance from city center to urban fringe.
The major recommendation of this study was that the aggregate value of land, water, and air resources does justify a land-use policy which reflects both the value and operation of natural processes. Further, that the identification of natural processes, the permissiveness and prohibitions which they exhibit, revealed a system of open space which directs metropolitan growth and offers sites for metropolitan open space.
The characteristics of natural processes were examined; an attempt was made to identify their values, intrinsic value, work performed, and protection afforded. Large-scale functions were identified with the major physiographic divisions of uplands, coastal plain, and piedmont; smaller-scale functions of air-water corridors were identified; and finally, eight discrete parameters were selected for examination. These were: surface water, marshes, flood plains, aquifers, aquifer recharge areas, steep slopes, forests and woodlands, and prime agricultural lands.
For each of the discrete phenomena, and for each successive generalization, approximate permissiveness to other land uses and specific prohibitions were suggested. While all were permissive to a greater or lesser degree, all performed their natural process best in an unspoiled condition. Clearly, if land is abundant and land-use planning can reflect natural process, a fabric of substantially natural land can remain either in low-intensity use or undeveloped, interfused throughout the metropolitan region. It is from this land that metropolitan open space can best be selected.
When this concept was applied to the PSMSA, the uplands were discerned as performing an important role in natural process and offering a specific range of recreational opportunity. The coastal plain was observed to perform an equally important but very different role and offered another range of recreational potential. Uniting uplands and coastal plain to the central city are major air-water corridors, while at the lowest level exist the distribution of the eight parameters selected.
In general, planning for natural process would select uplands, coastal plains, and air-water corridors to remain relatively undeveloped; it would confirm urbanization in the piedmont. It would protect surface water, and riparian lands, exempt flood plains from development by land uses other than those unimpaired by flooding or inseparable from waterfront locations, exclude development from marshlands and ensure their role as major water storage areas, limit development on aquifer resources and their recharge areas, protect prime farmland as present and prospective resources of agricultural productivity and scenic beauty, ensure the erosion-control function of forested steep slopes and ridges, and ensure their role, with forests and woodlands, in the water economy, and as a scenic-beauty and recreational potential.
Land Type | Recommended Uses
---|---
Surface water and riparian lands | Ports, harbors, marinas, water-treatment plants, water-related industry, certain water-using industry, open space for institutions and housing, agriculture, forestry, recreation.
Marshes | Recreation.
Fifty-year flood plain | Ports, harbors, marinas, water-treatment plants, water-related and water-using industry, agriculture, forestry, recreation, institutional open space, open space of housing.
Aquifers | Agriculture, forestry, recreation, industries that do not produce toxic or offensive effluents. All land uses within limits set by percolation.
Aquifer recharge areas | As aquifers.
Prime agricultural lands | Agriculture, forestry, recreation, open space of institutions, housing at one house per twenty-five acres.
Steep lands | Forestry, recreation, housing at maximum density of one house per three acres.
Forests and woodlands | Forestry, recreation, housing at densities not higher than one house per acre, other factors permitting.
Certain environmentally sensitive lands can be used for some purposes with restrictions. For example, recommended uses for such lands in the Philadelphia metropolitan area are as follows:
This concept, if realized, would ensure a structured system of open space within the metropolitan area within which only limited development would occur. It would produce the "natural" morphology of the metropolis. Due to the small number of parameters examined, this is a coarse-grain study. By increasing the number of phenomena and replacing the descriptive account of natural process by accurate quantitative analysis, the value of this information could be compounded.
The importance of these three examples lies simply in the fact that the commonplaces of natural science, where applied to planning, can be intensely illuminating and provide form in a most dramatic way from the level of the house on the shore to the form of metropolis.
# Conclusion
Today, in the face of momentous change in which urbanization is one of the most dramatic phenomena, nature is seen to be defenseless against the positive acts of transformation. The proponents of nature emphasize preservation: a negative position, one of defense, which excludes positive participation in the real and difficult tasks of creating noble and ennobling cities in fair landscapes. Meanwhile, the positive acts of urbanism proceed without perception of natural process, blind to its operation, values, and form-giving rules, scarcely modified by appeals for preservation and protection, remorseless in destruction, and impotent in constructive creation.
The negative view of conservation and the disinterest of natural science in problems of planning and urbanism are disservices to society. The redirection of concern to include not only wild environments, but also those dominated by people is a challenge which the natural sciences and their public arm—conservation—must confront and accept. Understanding of natural process is of central importance to all environmental problems and must be introduced into all considerations of land utilization.
The burden of this chapter is a request to natural scientists, particularly ecologists, to provide the indispensable information which the artificers require—ecological inventories, explanation of natural processes and identification of their limiting factors, the attribution of value, the indicators of healthy and unhealthy environments, and, finally, the degree of permissiveness or prohibition to transformation, implicit in natural processes.
Given this information, those who bring goodwill to the problems of resource planning and urbanism may add knowledge. This information can provide the basis for persuasion in both private and public domains; it can indeed provide the basis for a federal land-use policy based upon natural processes.
Such information can identify roles with geographic locations and attribute values to them. Thus, catastrophe-prone and danger areas would be prohibited to development—earthquake, volcanic, hurricane, avalanche, flood, drought, fire, and subsidence zones; areas having great intrinsic value—wilderness areas, wildlife habitats, areas of great scenic beauty, areas having important scientific and educational value, would be identified and exempted from development; the concept of "work performed" would permit identification of constituent roles—water—catchment areas, dunes, flood, drought-and erosion-control areas, flood plains, airsheds, etc., on which only limited development, responsive to natural process, could occur. Positively this information could select areas suitable for development, tolerant to man. The final component, the indicator, would permit diagnosis and prescription for existing environments.
This ecological information is thus deterministic and might well be called ecological determinism. It reveals the megaform of natural processes at the national level, the mezoform of the region, the microform of the city. From the city down to the level of its parts, ecological determinants become participant parameters in the planning process, but still influential as form-giving.
Precursory ecology made possible the leap over the garden wall in the eighteenth century, but the landscape created was a pastoral one. Modern ecology can enable us to leap more surely into many different environments, reserving some, intervening with deference in others, accomplishing great and wise transformations for human habitats in certain selected locations. When the program is developed and understood, when pervasive, then the artist will make it manifest. His interventions will become metaphysical symbols revealed in art.
The search is then for the place of nature in the city of man and, even more profoundly, for the place of man in nature.
# References
McHarg, Ian L., Roger D. Clemence, Ayre M. Dvir, Geoffrey A. Collins, Michael Laurie, William J. Oliphant, and Peter Ker Walker. 1963. Sea, Storm and Survival, A Study of the New Jersey Shore (mim.). Philadelphia: Department of Landscape Architecture, University of Pennsylvania.
McHarg, Ian L., David A. Wallace, Ann Louise Strong, William Grigsby, Anthony Tomazinis, Nohad Toulan, William H. Roberts, Donald Phimister, and Frank Shaw. 1963. Metropolitan Open Space from Natural Processes (mim.). Philadelphia: Department of Landscape Architecture, University of Pennsylvania.
Wallace, David A., and Ian L. McHarg. 1964. Plan for the Valleys. Philadelphia: Wallace-McHarg Associates.
# 4
# Values, Process and Form (1968)
This paper was published while Design with Nature was being written. McHarg first tested his theoretical concepts on applied ecology in this article, to an audience at the Institute for Advanced Study at Princeton University. He was "absolutely terrified" because of the stature of the audience. McHarg was so gratified by the positive response that he gave the paper again at the Smithsonian Institution, the version that appears here and that was published in The Fitness of Man's Environment (Harper and Row). Essentially, McHarg was seeking confirmation for his ideas about ecology and was able to present theories, which he considered risky coming from a landscape architect, to audiences of distinguished scientists at Princeton and the Smithsonian.
It is my proposition that, to all practical purposes, Western man remains obdurately pre-Copernican, believing that he bestrides the earth round which the sun, the galaxy, and the very cosmos revolve. This delusion has fueled our ignorance in time past and is directly responsible for the prodigal destruction of nature and for the encapsulating burrows that are the dysgenic city.
We must see nature and man as an evolutionary process which responds to laws, which exhibits direction, and which is subject to the final test of survival. We must learn that nature includes an intrinsic value-system in which the currency is energy and the inventory is matter and its cycles—the oceans and the hydrologic cycle, life forms and their roles, the cooperative mechanisms which life has developed, and, not least, their genetic potential. The measure of success in this process, in terms of the biosphere, is the accumulation of negentropy in physical systems and ecosystems, the evolution of apperception or consciousness, and the extension of symbioses—all of which might well be described as creation.
This can be pictured simply in a comparison between the early earth and the present planet. In the intervening billions of years the earth has been transformed and the major change has been in the increase of order. Think of the turbulence and violence of the early earth, racked by earthquakes and vulcanism, as it evolved toward equilibrium, and of the unrestrained movements of water, the dust storms of unstabilized soils, and the extreme alternations of climate unmodified by a green, meliorative vegetative cover. In this early world physical processes operated toward repose, but in the shallow bays there emerged life and a new kind of ordering was initiated. The atmosphere which could sustain life was not the least of the creations of life. Life elaborated in the seas and then colonized the earth, thus increasing the opportunities for life and for evolution. Plants and decomposers created the soils, anchored the surface of the earth, checked the movements of soil particles, modified water processes, meliorated the climate, and ordered the distribution of nutrients. Species evolved to occupy and create more habitats, more niches, each increase requiring new cooperative relationships between organisms—new roles, all of which were beneficial. In the earth's history can be seen the orderings which life has accomplished: the increase to life forms, habitats and roles, symbiotic relationships, and the dynamic equilibrium in the system—the total an increase in order. This is creation.
In the early earth, the sunlight which fell upon the planet equaled the degraded energy which was radiated from it. Since the beginning of plant life, some of the sun's energy has been entrapped by photosynthesis and employed with matter to constitute the ordered beings of plants; thence, to the animals and decomposers, and all of the orderings which they have accomplished. This energy will surely be degraded, but the entrapped energy, with matter, is manifest in all life forms past and present, and in all of the orderings which they have accomplished. Thus, creation equals the energy which has been temporarily entrapped and used with matter to accomplish all of the ordering of physical, biological, and cultural evolution. This, physicists describe as negentropy, in contrast with the inevitable degradation of energy which is described as entropy.
By this we see the world as a creative process involving all matter and all life forms in all time past and in the present. Thus, creation reveals two forms: first, the physical entrapment and ordering which is accomplished primarily by plants and by the simplest animals; and, second, apperception and the resulting ordering for which an increased capacity is observed as species rise in the phylogenetic scale. In this, man is seen to be especially endowed. This view of the world as a creative process involving all of its denizens, including man, in a cooperative enterprise, is foreign to the Western tradition that insists upon the exclusive divinity of man, his independent superiority, dominion, and license to subjugate the earth. It is this man in whose image was God made. This concept of nature as a creative, interacting process in which man is involved with all other life forms is the ecological view. It is, I submit, the best approximation of the world that has been presented to us, and the indispensable approach to determining the role of man in the biosphere. It is indispensable also for investigation, not only of the adaptations which man accomplishes, but of their forms.
The place, the plants, the animals, and man, and the orderings which they have accomplished over time, are revealed in form. To understand this it is necessary to invoke all physical, biological, and cultural evolution. Form and process are indivisible aspects of a single phenomenon: being. Norbert Weiner described the world as consisting of "To Whom It May Concern" messages, but these are clothed in form. Process and fitness (which is the criterion of process) are revealed in form; form contains meaning. The artifact, tool, room, street, building, town or city, garden or region, can be examined in terms of process, manifest in form, which may be unfit, fit, or most fitting. The last of these, when made by man, is art.
The role of man is to understand nature, which is also to say man, and to intervene to enhance its creative processes. He is the prospective steward of the biosphere. The fruits of the anthropocentric view are in the improvement of the social environment, and great indeed are their values, but an encomium on social evolution is not my competence, and I leave the subject with the observation that, while Madison, Jefferson, Hamilton, and Washington might well take pride in many of our institutions, it is likely that they would recoil in horror from the face of the land of the free.
An indictment of the physical environment is too easy, for post-industrial cities are such squalid testimony to the bondage of toil and to the insensitivity of man, that the most casual examination of history reveals the modern city as a travesty of its antecedents and a denial of its role as the proudest testimony to peoples and their cultures. The city is no longer the preferred residence for the polite, the civilized, and the urbane, all of which say "city." They have fled to the illusion of the suburb, escaping the iridescent shills, neon vulgarity of the merchants, usurious slumlords, cynical polluters (household names for great corporations, not yet housebroken), crime, violence, and corruption. Thus, the city is the home of the poor, who are chained to it, and the repository of dirty industry and the commuter's automobile. Give us your poor and oppressed, and we will give them Harlem and the Lower East Side, Bedford-Stuyvesant, the South Side of Chicago, and the North of Philadelphia—or, if they are very lucky, Levittown. Look at one of these habitats through the Cornell Medical School study of Midtown Manhattan, where 20 percent of a sample population was found to be indistinguishable from the patients in mental institutions, and where a further 60 percent evidenced mental disease. Observe the environments of physical, mental, and social pathology. What of the countryside? Well, you may drive from the city and search for the rural landscape, but to do so you will follow the paths of those who preceded you, and many of them stayed to build. But those who did so first are now deeply embedded in the fabric of the city. So as you go you will transect the annular rings of the thwarted and disillusioned who are encapsulated in the city as nature endlessly eludes pursuit. You can tell when you have reached the edge of the rural scene for there are many emblems: the cadavers of old trees, piled in untidy heaps beside the magnificent machines for land despoliation, at the edge of the razed deserts; forests felled; marshes filled; farms obliterated; streams culverted; and the sweet rural scene transformed into the ticky-tacky vulgarity of the merchants' creed and expression. What of the continent? Well, Lake Erie is on the verge of becoming septic, New York suffers from water shortages as the Hudson flows foully past, and the Delaware is threatened by salt water intrusion. Smog, forest fires, and mud slides have become a way of life for Los Angeles. In San Francisco, the Bay is being filled and men build upon unconsolidated sediments, the most perilous foundations in this earthquake-prone area. DDT is in Arctic ice and ocean deeps, radioactive wastes rest on the Continental Shelf, the Mississippi is engorged with five cubic miles of topsoil each year, the primeval forests are all but gone, flood and drought become increasingly common, the once-deep prairie soils are thinner now and we might as well recognize that itinerant investment farming is just another extractive industry.
This is the face of our Western inheritance—Judaism, Christianity, Humanism, and the Materialism which is better named "Economic Determinism." The countryside, the last great cornucopia of the world's bounty, ravaged; and the city of man (God's Junkyard, or call it Bedlam) a vast demonstration of man's inhumanity to man, where existence, sustained by modern medicine and social legislation, is possible in spite of the physical environment. Yet we are the inheritors of enormous beauty, wealth, and variety. Our world is aching for the glorious cities of civilized and urbane men. Land and resources are abundant. We could build a thousand new cities in the most wonderful locations—on mountains and plains, on rocky ocean promontories, on desert and delta, by rivers and lakes, on islands and plateaus. It is within our compass to select the widest range of the most desirable lands and promulgate policies and regulations to ensure the realization of these cities, each in response to the nature of its site. We can manage the land for its health, productivity, and beauty. All of these things are within the capacity of this people now. It is necessary to resolve to fulfill the American Revolution and to create the fair image that can be the land of the free and the home of the brave. But to resolve is not enough; it is also necessary that society at large understand nature as a process, having values, limiting factors, opportunities, and constraints; that creation and destruction are real; that there are criteria by which we can discern the direction and tests of evolution; and, finally, that there are formal implications revealed in the environment which affect the nature and form of human adaptations.
What inherited values have produced this plight, from which we must be released if the revolution is to be completed? Surely it is the very core of our tradition, the Judeo-Christian-Humanist view which is so unknowing of nature and of man, which has bred and sustained his simple-minded anthropocentrism and anthropomorphism. It is this obsolete view of man and nature which is the greatest impediment to our emancipation as managers of the countryside, city builders, and artists. If it requires little effort to mobilize a sweeping indictment of the physical environment which is man's creation, it takes little more to identify the source of the value system which is the culprit. Whatever the origins, the text is quite clear in Judaism, was absorbed all but unchanged into Christianity, and was inflated in Humanism to become the implicit attitude of Western man to nature and the environment. Western man is exclusively divine, all other creatures and things occupy lower and generally inconsequential status; man is given dominion over all creatures and things; he is enjoined to subdue the earth. Here is the best of all possible texts for him who would contemplate biocide, carelessly extirpate great realms of life, create Panama Canals, or dig Alaskan harbors with atomic demolition. Here is the appropriate injunction for the land rapist, the befouler of air and water, the uglifier, and the gratified bulldozer. Dominion and subjugation, or better call it conquest, are their creeds. It matters little that theologians point to the same source for a different text, and choose rather the image of man the steward who should dress the garden and keep it. It matters little that Buber and Heschel, Teilhard de Chardin, Weigel and Tillich retreat from the literality of the dominion and subjugation text, and insist that this is allegory. It remains the literal injunction which has been so warmly welcomed and enshrined at the core of the Western view. This environment was created by the man who believes that the cosmos is a pyramid erected to support man on its pinnacle, that reality exists only because man can perceive it, that God is made in the image of man, and that the world consists solely of a dialog between men. Surely this is an infantilism which is unendurable. It is a residue from a past of inconsequence when a few puny men cried of their supremacy to an unhearing and uncaring world. One longs for a psychiatrist who can assure man that his deep-seated cultural inferiority is no longer necessary or appropriate. He can now stand erect among the creatures and reveal his emancipation. His ancient vengeance and strident cries are a product of an earlier insignificance and are now obsolete. It is not really necessary to destroy nature in order to obtain God's favor or even his undivided attention. To this ancient view the past two centuries have added only materialism—an economic determinism which has merely sustained earlier views.
The face of the city and the land are the best testimony to the concept of conquest and exploitation—the merchants' creed. The Gross National Product is the proof of its success, money is its measure, convenience is its cohort, the short term is its span, and the devil take the hindmost is its morality. The economists, with some conspicuous exceptions, have become the spokesmen for the merchants' creed and in concert they ask with the most barefaced effrontery that we accommodate our values to theirs. Neither love nor compassion, health nor beauty, dignity nor freedom, grace nor delight are true unless they can be priced. If not, they are described as nonprice benefits and relegated to inconsequence, and the economic model proceeds towards its self-fulfillment—which is to say more despoliation. The major criticism of this model is not that it is partial (which is conceded by its strongest advocates), but more that the features which are excluded are among the most important human values, and also the requirements for survival. If the ethics of society insist that it is man's bounden duty to subdue the earth, then it is likely that he will obtain the tools with which to accomplish this. If there is established a value system based upon exploitation of the earth, then the essential components for survival, health, and evolution are likely to be discounted, as they are. It can then come as no surprise to us that the most scabrous slum is more highly valued than the most beautiful landscape, that the most loathsome roadside stand is more highly valued than the richest farmland, and that this society should more highly prize tomato stakes than the primeval redwoods whence they come.
It is, in part, understandable why our economic value system is completely blind to the realities of the biophysical world—why it excludes from consideration, not only the most important human aspirations, but even those processes which are indispensable for survival. The origins of society and exchange began in an early world where man was a trifling inconsequence in the face of an overwhelming nature. He knew little of its operation. He bartered his surpluses of food and hides, cattle, sheep and goats; and valued such scarcities as gold, silver, myrrh, and frankincense. In the intervening millennia the valuations attributed to commodities have increased in range and precision and the understanding of the operation of this limited sphere has increased dramatically. Yet, we are still unable to identify and evaluate the processes which are indispensable for survival. When you give money to a broker to invest you do so on the understanding that this man understands a process well enough to make the investment a productive one. Who are the men to whom you entrust the responsibility for ensuring a productive return on the world's investment? Surely, those who understand physical and biological processes realize that these are creative. The man who views plants as the basis of negentropy in the world and the base of the food chain, as the source of atmospheric oxygen, fossil fuels and fibers, is a different man from one who values only economic plants, or that man who considers them as decorative but irrelevant aspects of life. The man who sees the sun as the source of life and the hydrologic cycle as its greatest work, is a different man from one who values sunlight in terms of a recreation industry, a portion of agricultural income, or from that man who can obscure sky and sunlight with air pollution, or who carelessly befouls water. The man who knows that the great recycling of matter, the return stroke in the world's cycles, is performed by the decomposer bacteria, views soils and water differently from the man who values a few bacteria in antibiotics, or he who is so unknowing of bacteria that he can blithely sterilize soils or make streams septic. That man who has no sense of the time which it has taken for the elaboration of life and symbiotic arrangements which have evolved, can carelessly extirpate creatures. That man who knows nothing of the value of the genetic pool, the greatest resource which we bring to the future, is not likely to fear radiation hazard or to value life. Clearly, it is illusory to expect the formulation of a precise value system which can include the relative value of sun, moon, stars, the changing seasons, physical processes, life forms, their roles, their symbiotic relationships, or the genetic pool. Yet, without precise evaluation, it is apparent that there will be a profound difference in attitude—indeed, a profoundly different value system—between those who understand the history of evolution and the interacting processes of the biosphere, and those who do not.
The simpler people who were our ancestors (like primitive peoples today) did not subscribe to anthropocentric views, nor did the eighteenth-century English landscape tradition which is the finest accomplishment of Western art in the environment, and which derives from a different hypothesis. The vernacular architecture in the Western tradition and the attitudes of the good farmer come from yet another source, one which has more consonance with the Orient than the West. But the views which ensured successes for the hunter and gatherer, for the vernacular farmer, and for the creation of a rich and beautiful pastoral landscape are inadequate to deal with twentieth-century problems of an inordinate population growth, accelerating technology, and transformation from a rural to an urban world. We need a general theory which encompasses physical, biological, and cultural evolution; which contains an intrinsic value system; which includes criteria of creativity and destruction and, not least, principles by which we can measure adaptations and their form. Surely, the minimum requirement for an attitude to nature and to man is that it approximate reality. Clearly, our traditional view does not. If one would know of these things, where else should one turn but to science. If one wishes to know of the phenomenal world, where better to ask than the natural sciences; if one would know of the interactions between organism and environment, then turn to the ecologist, for this is his competence. From the ecological view, one can conclude that by living one is united physically to the origins of life. If life originated from matter, then by living one is united with the primeval hydrogen. The earth has been the one home for all of its evolving processes and for all of its inhabitants; from hydrogen to man, it is only the bathing sunlight which changes. The planet contains our origins, our history, our milieu—it is our home. It is in this sense that ecology, derived from oikos, is the science of the home. Can we review physical and biological evolution to discern the character of these processes, their direction, the laws which obtain, the criteria for survival and success? If this can be done, there will also be revealed an intrinsic value system and the basis for form. This is the essential ingredient of an adequate view of the world: a value system which corresponds to the creative processes of the world, and both a diagnostic and constructive view of human adaptations and their form.
The evolution of the world reveals movement from more to less random, from less to more structured, from simplicity to diversity, from few to many life forms—in a word, toward greater negentropy. This can be seen in the evolution of the elements, the compounds, and life. It is accomplished by physical processes, as in the early earth when matter liquefied and coalesced, forming the concentric cores of the planet. Vulcanism revealed the turbulence of early adaptations toward equilibrium. So, too, did the creation of the oceans. Evaporation and precipitation initiated the processes of erosion and sedimentation in which matter was physically sorted and ordered. When, from the aluminosilicate clays in the shallow bays, there emerged that novel organization, life, there developed a new agency for accomplishing ordering. The chloroplast of the plant was enabled to transmute sunlight into a higher ordering, sustaining all life. The atmosphere, originally hostile to life, was adapted by life to sustain and protect it, another form of ordering. The emergence of the decomposers, bacteria and fungi, permitted the wastes of life forms—and their substance after death—to be recycled and utilized by the living, the return stroke in the cycle of matter in the biosphere. The increasing number of organisms in the oceans and on land represent negentropy in their beings and in the ordering which they accomplish. We can now see the earth as a process by which the falling sunlight is destined for entropy, but is arrested and entrapped by physical processes and creatures, and reconstituted into higher and higher levels of order as evolution proceeds. Entropy is the law and demands its price, but while all energy is destined to become degraded, physical and biological systems move to higher order—from instability towards steady-state—in sum, to more negentropy. Evolution is thus a creative process in which all physical processes and life forms participate. Creation involves the raising of matter and energy from lower to higher levels of order. Retrogression and destruction consist of reduction from the higher levels of order to entropy.
As life can only be transmitted by life, then the spore, seed, egg, and sperm contain a record of the entire history of life. The journey was shared with the worms, the coelenterates, the sponges, and, later, with the cartilaginous and bony fishes. The reptilian line is ours, the common ancestor that we share with the birds. We left this path to assume mammalian form, live births, the placenta, and suckling of the young; the long period of infantile dependence marks us. From this branching line the monotremes, marsupials, edentates, and pangolins followed their paths, and we proceeded on the primate way. Tree shrew, lemur, tarsier, and anthropoid, are our lineage. We are the line of man—the raised ape, the enlarged brain, the toolmaker—he of speech and symbols, conscious of the world and of himself. It is all written on the sperm and on the egg although the brain knows little of this journey. We have been through these stages in time past and the imprint of the journey is upon us. We can look at the world and see our kin; for we are united, by living, with all life, and are from the same origins. Life has proceeded from simple to complex, although the simplest forms have not been superseded, only augmented. It has proceeded from uniform to diverse, from few to many species. Life has revealed evolution as a progression from greater to lesser entropy. In the beginning was the atom of hydrogen with one electron. Matter evolved in the cosmic cauldrons, adding electron after electron, and terminating in the heaviest and most ephemeral of elements. Simple elements conjoined as compounds, thus reaching the most complex of these as amino acids, which is to say life. Life reached unicellular form and proceeded through tissue and organ to complex organisms. There were few species in the beginning and now they are myriad; there were few roles and now they are legion. There were once large populations of few species; now there is a biosphere consisting of multitudes of communities composed of innumerable interacting species. Evolution has revealed a progression from simple to complex, from uniform to diverse, from unicellular to multicelled, from few to many species, from few to many ecosystems, and the relations between these processes have also evolved toward increased complexity.
What holds the electrons to the nucleus? The molecules in rocks, air, and water may have ten atoms, but the organic molecule may have a thousand. Where is the catalytic enzyme which locks and unlocks the molecules? The single cell is very complex indeed; what orchestrates the cytoplasm and nucleus, nucleolus, mitochondria, chromosomes, centrosomes, Golgi elements, plastids, chromoplasts, leucoplasts, and, not least, chloroplasts? The lichen shows an early symbiosis at the level of the organism as the alga and the fungus unit. The plant and the decomposer enter into symbiosis to utilize energy and matter, to employ the first and recycle the latter. The animal enters the cycle, consuming the plant, to be consumed by the decomposer and thence by the plant. Each creature must adapt to the others in that concession of autonomy toward the end of survival that is symbiosis. Thus parasite and host, predator and prey, and those creatures of mutual benefit develop symbioses to ensure survival. The world works through cooperative mechanisms in which the autonomy of the individual, be it cell, organ, organism, species, or community is qualified toward the survival and evolution of higher levels of order culminating in the biosphere. Now these symbiotic relationships are beneficial to the sum of organisms although clearly many of them are detrimental to individuals and species. While the prey is not pleased with the predator or the host far from enamored of the parasite or the pathogen, these are regulators of populations and the agents of death—that essential return phase in the cycle of matter, which fuels new life and evolution. Only in this sense can the predator, parasite, and pathogen be seen as important symbiotic agents, essential to the creative processes of life and evolution. If evolution has proceeded from simple to complex, this was accomplished through symbiosis. As the number of species increased, so then did the number of roles and the symbiotic arrangements between species. If stability increases as evolution proceeds, then this is the proof of increased symbiosis. If conservation of energy is essential to the diminution of entropy, then symbioses are essential to accomplish this. Perhaps it is symbiosis or, better, altruism that is the arrow of evolution.
This view of the world, creation, and evolution reveals as the principal actors: the sun, elements and compounds, the hydrologic cycle, the plant, decomposers, and the animals. Further, if the measure of creation is negentropy, then it is the smallest marine plants which perform the bulk of the world's work, which produce the oxygen of the atmosphere, the basis of the great food chains. On land it is the smallest herbs. Among the animals the same is true; it is the smallest of marine animals and the terrestrial herbivores which accomplish the greatest creative effort of raising the substance of plants to higher orders. Man has little creative role in this realm although his destructive potential is considerable. However, energy can as well be considered as information. The light which heats the body can inform the perceptive creature. When energy is so considered, then the apperception of information as meaning, and response to it, is also seen as ordering, as antientropic. Noise to the unperceptive organism, through perception becomes information from which is derived meaning. In an appraisal of the world's work of apperception, it is seen that the simpler organisms, which create the maximum negentropy, are low on the scale of apperception which increases as one rises on the evolutionary scale. Man, who had no perceptible role as a creator of negentropy, becomes prominent as a perceptive and conscious being. We have seen that the evolution from the unicellular to the multicellular organism involved symbiotic relationships. Hans Selyé has described intercellular altruism as the cooperative mechanism which makes 30 billion human cells into a single integrated organism. He also has described interpersonal altruism. Surely one must conclude that the entire biosphere exhibits altruism. In this sense, the life forms which now exist on earth, and the symbiotic roles which they have developed, constitute the highest ordering which life forms have yet been able to achieve. The human organism exists as a result of the symbiotic relationships in which cells assume different roles as blood, tissues, and organs, integrated as a single organism. So, too, can the biosphere be considered as a single superorganism in which the oceans and the atmosphere, all creatures, and communities play roles analogous to cells, tissues, and organs. That which integrates either the cell in the organism or the organism in the biosphere is a symbiotic relationship. In sum, these are beneficial. This then is the third measure, the third element, after order and complexity, of the value system: the concession of some part of the autonomy of the individual in a cooperative arrangement with other organisms which have equally qualified their individual freedom toward the end of survival and evolution. We can see this in the alga and fungus composing the lichen, in the complex relationships in the forest, and in the sea. Symbiosis is the indispensable value in the survival of life forms, ecosystems, and the entire biosphere. Man is superbly endowed to be that conscious creature who can perceive the phenomenal world, its operation, its direction, the roles of the creatures, and physical processes. Through his apperception, he is enabled to accomplish adaptations which are the symbioses of man—nature. This is the promise of his apperception and consciousness. This examination of evolution reveals direction in retrospect—that the earth and its denizens are involved in a creative process of which negentropy is the measure. It shows that creation does have attributes which include evolution toward complexity, diversity, stability (steady-state), increase in the number of species, and increase in symbiosis. Survival is the first test, creation is the next; and this may be accomplished by arresting energy, by apperception, or by symbiosis. This reveals an intrinsic value system with a currency: energy; an inventory which includes matter and its cycles, life forms and their roles, and cooperative mechanisms.
All of the processes which have been discussed reveal form; indeed, form and process are indivisible aspects of a single phenomenon. That which can be seen reveals process. Much of this need not be superficially visible; it may lie beneath the skin, below the level of vision, or only in invisible paths which bespeak the interactions of organisms. Yet, the place, the plants, animals, men, and their works, are revealed in form.
All of the criteria used to measure evolutionary success apply to form. Simplicity and uniformity reveal a primitive stage, while complexity and diversity are evidence of higher evolutionary forms: few elements or species as opposed to many, few interactions rather than the multitude of advanced systems. Yet, there is need for a synoptic term which can include the presence or absence of these attributes in form. For this, we can use "fitness" both in the sense that Henderson employs it, and also in Darwinian terms. Thus, the environment is fit, and can be made more fitting; the organism adapts to fit the environment. Lawrence Henderson speaks of the fitness of the environment for life in the preface to his book, The Fitness of the Environment.
Darwinian fitness is compounded of a mutual relationship between the organism and the environment. Of this, fitness of environment is quite as essential a component of the fitness which arises in the process of organic evolution; and in fundamental characteristics the actual environment is the fittest possible abode for life.
Henderson supports his proposition by elaborating on the characteristics of carbon, hydrogen, oxygen, water, and carbolic acid, saying that "No other environment consisting of primary constituents, made up of other known elements, or lacking water and carbolic acid, could possess a like number of fit characteristics, or in any manner such great fitness to promote complexity, durability, and the active metabolism and the organic mechanism we call life."
The environment is fit for life and all of the manifestations which it has taken, and does take. Conversely, the surviving and successful organism is fitted to the environment. Thus, we can use fitness as a criterion of the environment, organisms and their adaptations, as revealed in form. Form can reveal past processes and help to explain present realities. Mountains show their age and composition in their form; rivers demonstrate their age and reflect the physiography of their passage; the distribution and characteristics of soils are comprehensible in terms of historical geology, and climate and hydrology. The pattern and distribution of plants respond to environmental variables represented in the foregoing considerations, while animals respond to these and to the nature of the plant communities.
Man is as responsive, but he is selective; the pattern and distribution of man is likely to be comprehensible in these same terms. The term "fitness" has a higher utility than art for the simple reason that it encompasses all things—inert and living, nonhuman and those made by man—while art is limited to the last. Moreover, it offers a longer view and more evidence. Nature has been in the business of form since the beginning, and man is only one of its products. The fact that things and creatures exist is proof of their evolutionary fitness at the time, although among them there will be those more or less fit. There will be those which are unfit and will not persist, those are the misfits; then, those which are fit; and finally, the most fitting—all revealed in form. Form is also meaningful form. Through it, process and roles are revealed, but the revelation is limited by the capacity of the observer to perceive. Arctic differs from rain forest, tundra from ocean, forest from desert, plateau from delta; each is itself because. The platypus is different from seaweed, diatom from whale, monkey from man . . . because. Africans differ from Asians, Eskimo from Caucasoid, Mongoloid from Australoid . . . because; and all of these are manifest in form.
When process is understood, differentiation and form become comprehensible. Processes are dynamic, and so coastlines advance and recede as do ice sheets, lakes are in the process of filling while others form, mountains succumb to erosion and others rise. The lake becomes marsh, the estuary a delta, the prairie becomes desert, the scrub turns into forest, a volcano creates an island, while continents sink. The observation of process, through form and the response, represents the evolution of information to meaning. If evolutionary success is revealed by the existence of creatures, then their fitness will be revealed in form; visible in organs, in organisms, and in communities of these. If this is so, then natural communities of plants and animals represent the most fitting adaptation to available environments. They are most fitting and will reveal this in form. Indeed, in the absence of man, these would be the inevitable expression. Thus, there is not only an appropriate ecosystem for any environment, and successional stages toward it, but such communities will reveal their fitness in their expression. This is a conclusion of enormous magnitude to those who are concerned with the land and its aspect: that there is a natural association of creatures for every environment. This could be called the intrinsic identity of the given form.
If this is so, then there will be organs, organisms, and communities of special fitness, and these will, of course, be revealed in form. This might well be described as the ideal. The creation of adaptations which seek to be metaphysical symbols is, in essence, the concern with representing the ideal. Adaptation of the environment is accomplished by all physical processes and by all life. Yet, certain of these transformations are more visible than others, and some are analogous to those accomplished by man. Throughout the natural world, many creatures are engaged in the business of using inert material to create adaptive environments. These reveal the individual, a society, or a population. Can the criteria of fitness be applied then to the artifact? We can accept that the stilt's legs, the flamingo's beak, and the mouth of the baleen whale are all splendid adaptations, and visibly so. It is no great leap to see the tennis serve, the left hook, and the jumping catch, as of the same realm as the impala's bound, the diving cormorant, or the leopard's lunge. Why then should we distinguish between the athletic gesture and the artifacts which are employed with them: the golf club, bat, glove, or tennis racquet? The instrument is only an extension of the limb.
If this is so, then we can equally decide if the hammer and saw are fit, or the knife, fork, and spoon. We can conclude that the tail-powered jet is more fit for the air than the clawing propellers. If we can examine tools, then we can as well examine the environments for activities: the dining room for dining, the bedroom for sleeping or for loving, the house, street, village, town, or city. Are they unfit, misfit, fit, or most fitting? It appears that any natural environment will have an appropriate expression of physical processes, revealed in physiography, hydrology, soils, plants, and animals.
There should then be an appropriate morphology for man-site, and this should vary with environments. There will then be a fitting-for-man environment. One would expect that as the plants and animals vary profoundly from environment to environment, this should also apply to man. One would then expect to find distinct morphologies for man–nature in each of the major physiographic regions. The house, village, town, and city should vary from desert to delta, from mountain to plain. One would expect to find certain generic unity within these regions, but marked differentiation between them.
If fitness is a synoptic measure of evolutionary success, what criteria can we use to measure it? We have seen that it must meet the simplicity-complexity, uniformity-diversity, instability-stability, independence-interdependence tests. Yet, in the view of Ruth Patrick, as demonstrated by her study of aquatic systems, these may all be subsumed under two terms: ill-health and health. A movement towards simplicity, uniformity, instability, and a low number of species characterizes disease. The opposites are evidence of health. This corresponds to common usage: ill-health is unfit; fitness and health are synonymous. Thus, if we would examine the works of man and his adaptations to the countryside, perhaps the most synoptic criteria are disease and health. We can conclude that which sustains health represents a fitting between man and the environment. We would expect that this fitness be revealed in form. This criterion might well be the most useful to examine the city of man: wherein does pathology reside? What are its corollaries in the physical and social environment? What characterizes an environment of health? What are its institutions? What is its form? Know this, and we may be able to diagnose and prescribe with an assurance which is absent today.
What conclusions can one reach from this investigation? The first is that the greatest failure of Western society, and of the post-industrial period in particular, is the despoliation of the natural world and the inhibition of life which is represented by modern cities. It is apparent that this is the inevitable consequence of the values that have been our inheritance. It is clear, to me if to no one else, that these values have little correspondence to reality and perpetrate an enormous delusion as to the world, its work, the importance of the roles that are performed, and, not least, the potential role of man. In this delusion the economic model is conspicuously inadequate, excluding as it does the most important human aspirations and the realities of the biophysical world. The remedy requires that the understanding of this world which now reposes in the natural sciences be absorbed into the conscious value system of society, and that we learn of the evolutionary past and the roles played by physical processes and life forms. We must learn of the criteria for creation and destruction, and of the attributes of both. We need to formulate an encompassing value system which corresponds to reality and which uses the absolute values of energy, matter, life forms, cycles, roles, and symbioses.
We can observe that there seem to be three creative roles. The first is the arresting of energy in the form of negentropy, which offers little opportunity to man. Second is apperception and the ordering which can be accomplished through consciousness and understanding. Third is the creation of symbiotic arrangement, for which man is superbly endowed. It can be seen that form is only a particular mode for examining process and the adaptations to the environment accomplished by process. Form can be the test used to determine processes as primitive or advanced, to ascertain if they are evolving or retrogressing. Fitness appears to have a great utility for measuring form: unfit, fit, or most fitting. When one considers the adaptations accomplished by man, they are seen to be amenable to this same criterion but, also, synoptically measurable in terms of health. Identify the environment of pathology; it is unfit, and will reveal this in form. Where is the environment of health—physical, mental, and social? This, too, should reveal its fitness in form. How can this knowledge be used to affect the quality of the environment? The first requirement is an ecological inventory in which physical processes and life forms are identified and located within ecosystems, which consist of discrete but interacting natural processes. These data should then be interpreted as a value system with intrinsic values, offering both opportunities and constraints to human use, and implications for management and the forms of human adaptations.
The city should be subject to the same inventory and its physical, biological, and cultural processes should be measured in terms of fitness and unfitness, health and pathology. This should become the basis for the morphology of man-nature and man-city. We must abandon the self-mutilation which has been our way, reject the title of planetary disease which is so richly deserved, and abandon the value system of our inheritance which has so grossly misled us. We must see nature as a process within which man exists, splendidly equipped to become the manager of the biosphere; and give form to that symbiosis which is his greatest role, man the world's steward.
# 5
# Natural Factors in Planning (1997)
The idea for this essay came from Lloyd Wright, then director of conservation and ecosystem assistance for the U.S. Natural Resources Conservation Service (NRCS). A dedicated member of the federal bureaucracy, Lloyd Wright believed other government employees needed inspiration and challenge in light of the harsh criticism that had been leveled against them beginning with the Ronald Reagan administration and perhaps culminating in the bombing of the Alfred P. Murrah Federal Office Building in Oklahoma City.
Wright identified McHarg as the best person for such inspiration and challenge. McHarg's charge was to provide a theoretical framework for using natural resource information in planning. The NRCS provided support through a cooperative agreement with the Arizona State University School of Planning and Landscape Architecture to commission McHarg to write this essay. It was subsequently published in the Journal of Soil and Water Conservation, which is widely read by NRCS staff and other governmental natural resource managers. The paper was also translated into Italian by Danilo Palazzo and published in Urbanistica, an Italian planning journal.
In earlier times, in predominantly agricultural societies, conventional wisdom and folklore included an understanding of the region, its phenomena, processes, and calendar—time to plow, farrow, seed, harvest, first and last frosts, the probability of precipitation. Cultural memories recalled past events, flood and drought, pestilence, earthquakes, hurricanes, tornadoes. Given the crucial importance of this knowledge, essential to survival and success, it seems pointless to restate the obvious, but, sadly, it is increasingly necessary, more so than in earlier, simpler times. The phenomenal increase in urban concentrations, combined with exponential population growth and the reduction of the agricultural component in society and economy, have produced asphalt people who know little of nature and care less. To such people, knowledge of nature is apparently irrelevant to their success or future.
This view is not merely wrong. It is diametrically opposed to need. The greater the population and the greater the urban concentration, the greater the need to understand nature and to act prudently, using the best available knowledge.
The world population is growing exponentially; five billion now, doubling how soon? This enlargement is increasingly concentrated; megacities of 30 million, unknown only decades ago, are commonplace now. Mexico City, Sao Paulo, Calcutta, Tokyo are the most conspicuous examples and more will follow. The presence of such mammoth populations, of unprecedented size and concentration, is a recipe for disaster.
Surely the combination of massive populations in such urban concentrations should be warning enough but the situation is exacerbated by location—the annual flooding in the Gangetic flood plain takes an enormous toll now; the view of the circle of volcanoes, surrounding Mexico City, is not reassuring.
But this is not enough; it appears that the future includes increased frequency and violence in climatic events. Warm a pot of water, see activity increase with temperature until boiling occurs. So too with the earth. Whether or not from the accumulation of greenhouse gases, world temperatures rise and with them climatic violence. This is thought to include not only flood and drought, hurricanes, tsunamis, and tornadoes but earthquakes and volcanoes too. So we have actively increased the threat of death, disaster, pain, and loss by those concurrent processes, population increase, concentration, and now climatic violence.
The consequences of national disasters are awesome now. What of the future? A flood in Bangladesh and an earthquake in the Crimea accounted for half a million deaths. Two hundred thousand lives were lost from twenty events in the last forty years. There are also damages. In 1989, Hurricane Hugo incurred over $6 billion in damages and 80 lives were lost. In the same year, the Loma Prieta earthquake caused $10 billion in damages. Hurricane Andrew caused another $25 billion in 1992, approximately seven percent of Florida's Gross State Budget. The 1994 Midwest floods in the Mississippi River basin and the Northbridge earthquake the same year together resulted in over $50 billion in destruction. Insurance claims for the first three years of this decade exceeded claims for the entire earlier one. These are significant costs, only exceeded by the U.S. Department of Defense budget and the national debt. These costs can be reduced significantly.
In the face of this massive indictment, it would seem impossible to disregard natural calamities in the planning process. The burden of discussion paper is that natural factors have for too long been either excluded or inadequately incorporated in planning. Surely, you say, this is a restatement of the obvious. How can effective planning proceed without the inclusion of this crucial realm? It has, can, and does, and the absence of such data causes planning to be either irrelevant, exclusionary, or inconsequential. How did this paradoxical situation come to be? How can it be corrected?
There are many reasons. The first reposes in the historic disinterest of leading scientists and institutions in the environment. Fashions tend to vary, but for half a century particle physics has dominated the physical sciences, molecular biology has preempted attention in the biological sciences. As recently as the 1950s, the environment was barely an accepted term. I recall when in the 1950s the Audubon Society was preoccupied with birds, the Sierra Club concentrated on preserving the scenic American West. Then the only organization concerned with the national environment was the Conservation Foundation, housed in a one-room office in New York City. In the 1960s, there were few writers or speakers prepared to discuss the environment: Rachel Carson, Ralph Nader, Paul Ehrlich, Barry Commoner, René Dubos, and me.
Rachel Carson was the first spokesperson for the environment: Her message was that we were poisoning natural systems and ourselves; Ralph Nader brought the subject of the environment to national attention and illustrated the impact of the issue on the consumer; Paul Ehrlich gave vivid attention to the population problem; Barry Commoner emphasized chemical pollutants; René Dubos, pathologist, the most humane of the group, insisted that the way the French peasant lived with the environment provided the appropriate answer; my theme was design with nature.
# The Fragmentation of Knowledge
Science itself was fragmented. Reductionism held sway. Integration requires bridging between separate sciences, an attitude resisted by universities and governmental institutions. Meteorologists study climate, geologists rocks, hydrologists address water, pedologists focus on soils. Vegetation is addressed by ecologists, limnologists, and marine biologists; animals and wildlife have their appropriate investigators: animal ecologists, ethnologists, fish and wildlife scientists. For almost every discipline there will be an associated institution. National Oceanic and Atmospheric Adminstration (NOAA) links to climate, the U.S. Geological Survey controls rocks and water, soils repose in USDA Natural Resources Conservation Service, fish and wildlife are located in the National Biological Survey and in the U.S. Fish and Wildlife Service. The U.S. Environmental Protection Agency (EPA) limits itself to regulation while natural vegetation alone has no institutional sponsor. The Smithsonian Institution seems to have no niche.
Each of these elements, either in universities or in government, is like a piece of a jigsaw puzzle which has never been assembled. Is there anyone with the wit, capability, and energy to do this? Unlikely
Humpty Dumpty sat on a wall;
Humpty Dumpty had a great fall;
All the king's horses and all the king's men;
Couldn't put Humpty together again.
The environment is like a pile of eggshell fragments. Not only has the environment been disdained by all but a handful of our leading scientists but inevitably by the most prestigious institutions. There the mandarins scorned the workmen of the environment with their black rimmed fingernails, soiled shirt collars, inhabiting land-grant colleges in declining programs in agriculture and forestry. And moreover, the single integrative science, ecology, absent or little represented in the Ivy League, was housed in broom closets in the land-grant and state colleges.
# The Fragmentation of Government
In addition, of course, there are redundant and often conflicting policies, evidence of cross purposes. The U.S. National Park Service is charged with the preservation and interpretation of national scenic wonders but the U.S. Forest Service is charged with exploiting our national forests, the U.S. Bureau of Land Management (BLM) fills a comparable role for rangeland. While the U.S. Geological Survey (USGS) studies geological and hydrological systems, the U.S. Army Corps of Engineers concentrates on intervention in riverine systems, as does the BLM. Regulation mainly reposes in the EPA, but the Army Corps of Engineers is charged with managing wetlands; the agent which supervised the filling of wetlands is now charged with protecting them! The study of climate by NOAA is very impressive. Satellites, sensors, and computers have advanced meteorology. This advance is not paralleled in hydrology. The massive midwestern floods of 1994 may well have been enhanced by public actions—floods as a public gift at great public cost.
As molecular biology assumed dominance biology became a crucial component of medicine. Organismic and community biology was abandoned. Generalists were cytologists, preoccupied with the cell. Organismic biology declined with the retirement or death of the ninteenth-century naturalists in biology—never to be replaced. In geology this same interest focused physical sciences on carbon dating and planetary physics. It was only the discovery of plate tectonics which brought geology back to spatial studies. But here the disdain of geologists for hydrologists or soil scientists remains an important obstruction to integration.
# Natural Processes Affect Our Health and Safety
If science has been indifferent to the environment, victims of natural calamities certainly have not been. These natural disasters—flood, drought, hurricanes, tsunamis, tornadoes, volcanoes, and earthquakes—have caused enormous loss of life, plus considerable damage, suffering and pain, and promise more. Indeed, it would appear that the sum of global behavior is to increase this damage, pain, suffering, and loss of life.
This is particularly poignant because so much of it could be avoided. Surely it is neither wise nor necessary to locate populations in areas prone to natural disasters. Nor is it necessary to exacerbate them. One example might suffice.
The 1994 Mississippi flooding was exacerbated by failure to understand the role of river ecologies in mitigating flooding and by years of engineering practices that neutralized much natural protection.
The explanation for the lack of interest in natural processes in planning began with the long-term rejection of the environment as a fitting subject of research by both prestigious academic institutions and governmental agencies. The commitment to reductionism and the fragmentation of science provide other reasons. The passion for obtaining data as a commonplace obsession is paralleled by a disinterest in integration. A widespread public belief that natural calamities are acts of God and must be endured is also a culprit. Yet management of natural disaster is one area where profound advances have been made. The understanding of tectonic and meteorological processes, improved monitoring and prediction, and better communication have reduced the consequences of natural calamities.
# The Inadequacies of Planning to Respond to the Environmental Challenge
Yet another explanation lies in the composition of those engaged in city and regional planning. In the postwar years planning was described as an "applied social science," certainly in the dominant planning schools, such as the University of Pennsylvania, the University of North Carolina, the Massachusetts Institute of Technology, the University of California-Berkeley, and Harvard University. Such a description had a profound effect on the planning operation.
Go into a local city or county planning office, anywhere in the United States. Establish that you have been commissioned to write an article on the importance of natural factors in the processes of planning and community design. Speak to the staff. You will find that their training has been predominantly in the applied social sciences. They are likely to have studied economics, sociology, computation, statistics, perhaps (hopefully) land-use and zoning law. Only the planners with an education in geography are likely to have received any training in physical and biological science. So your investigation will not assure you of the central importance of natural science in planning and community design. Indeed, the absence of planners trained in the natural sciences is an explanation for the exclusion of environmental science.
# The Emergence of the Environment in Public Policy
Since the 1970s, the environment has emerged as a significant subject. However, the science departments of most prestigious universities still focus on molecular biology and subatomic physics. While the environment may have emerged from oblivion in the public consciousness it is still not prominent. Agencies continue in their ad hoc, reductionist ways, although there are timid efforts of integration—the National Science Foundation-Environmental Protection Agency combined focus on watersheds is a welcome and belated innovation ; the creation of the National Biological Survey (NBS) in the U.S. Department of Interior is an attempt at integration. The U.S. Fish and Wildlife Service Gap Analysis Program and the Clinton administration's Interagency Ecosystem Management Task Force are other hopeful efforts.
But in sum, it is a lugubrious review. What is the remedy? Well, clearly there must be widespread recognition of the necessity for all of the sciences to address the problems of the environment. In this fragmented and reductionist world it is necessary to recognize and to acknowledge that the greatest progress can be accomplished, not by providing additional data (we already have quantities beyond our ability to use them). No, the greatest advance will occur when it is recognized that integration and synthesis constitute the greatest challenge and provide the greatest promise of success.
On the positive side the increase in scientific knowledge, the availability of sensors, not the least global positioning systems (GPS), offer great opportunities for ecological planning. As important are computers, their ability to digitize massive data sets, retrieve data, analyze them, and undertake automatic analytic procedures and finally perform complex planning syntheses.
# Human Ecological Planning
I have called such a process human ecological planning. It includes data and precepts from the relevant physical, biological, and social sciences. In the 1970s, the Ford Foundation made grants to support ecology at Princeton University, the University of Georgia, and the University of Pennsylvania. At Penn, we designed a new curriculum in ecologically based regional planning, recruiting persons with undergraduate qualifications in the natural sciences. An integrated, ecology-based curriculum was developed, a natural scientist faculty was hired, including such luminaries as Ruth Patrick, a 1996 recipient of the National Medal of Science. This program prospered for twenty years. It produced 15 deans, 38 chairmen and directors, and 150 professors worldwide. From this cadre came 15 new programs emphasizing ecological planning and design. The Penn program ultimately shrank in parallel with the opposition to planning and the environment initiated by former Presidents Ronald Reagan and George Bush and continued by the current U.S. Congress.
Ecological planning, developed at Penn and employed in instruction and research, produced over a thousand graduates who employ ecological planning in many academic institutions and government agencies in North America and around the world. This initiative should receive massive support. It represents an unmatched degree of integration, the incorporation of all of the environmental sciences—physical, biological, and social—combined with advanced computational capability. Descriptions of human ecological planning have been published elsewhere but can be presented in synopsis here. First, it requires that all of the environmental sciences be included and unified by chronology and depicted as a "layer cake:" These would include meteorology, geology, physical oceanography, surficial geology, geomorphology, groundwater and surficial hydrology, soils, vegetation, wildlife (including marine biology), and limnology. The social sciences should be included in ecological inventories too with ethology, ethnography, cultural anthropology, economics, sociology, and geography, particularly computational science.
Our faculty teams undertook numerous studies with the landscape architecture and regional planning students at Penn. From that work, we identified the baseline natural resource data necessary for a layer-cake inventory of a place (table 1).
In addition to being comprehensive and inclusive, the integrating device recommended is chronology. That is, all studies should involve a historic recapitulation ordered by time. Information from the sciences should be organized in a "layer cake" fashion, with older components including rocks at the bottom of the cake and younger layers, such as soil and vegetation above. The completion of such a human ecological study presents an understanding of how the region came to be, what it is, how it works, and where it tends to go. A human perspective is essential—how people relate to their environments both historically and today. The ethnographic history should describe the populations, structured in constituencies, having characteristic values, settlement patterns, resource utilization, and specific issues and attitudes to them. When the inventory is complete, and the data accumulated chronologically, and thus meaningfully, the history of the region under study, its constituent processes, its past, its current form, and its tendencies for the future will become apparent. The product should be an interacting biophysical model on which a human population is causally located.
Table 1. Baseline Natural Resource Data Necessary for Ecological Planning
The following natural resource factors are likely to be of significance in planning. Clearly the region under study will determine the relevant factors but many are likely to occur in all studies.
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Climate. Temperature, humidity, precipitation, wind velocity, direction, duration, first and last frosts, snow, frost, fog, inversions, hurricanes, tornadoes, tsunamis, typhoons, Chinook winds
Geology. Rocks, ages, formations, plans, sections, properties, seismic activity, earth-quakes, rock slides, mud slides, subsistence
Surficial geology. Kames, kettles, eskers, moraines, drift and till
Groundwater hydrology. Geological formations interpreted as aquifer with well locations, well logs, water quantity and quality, water table, flood plains
Physiography. Physiographic regions, subregions and features, contours, sections, slopes, aspect, insulation, digital terrain model(s)
Surficial hydrology. Oceans, lakes, deltas, rivers, streams, creeks, marshes, swamps, wetlands, stream orders, density, discharges, gauges, water quality
Soils. Soil associations, soils series, properties, depth to seasonal high water table, depth to bedrock, shrink-swell, compressive strength, cation and anion exchange, acidity-alkalinity
Vegetation. Associations, communities, species, composition, distribution, age and conditions, visual quality, species number, rare and endangered species, fire history, successional history
Wildlife. Habitats, animal populations, census data, rare and endangered species, scientific and educational value
Human. Ethnographic history, settlement patterns, existing land use, existing infrastructure, population characteristics
# Determining Suitability
This database is now available for queries. One realm of such queries will involve correlation. Where are hurricane-prone zones? Where is landslide susceptibility? Tornado zones? What aspects of elevation, geology, climate, soil, slope are associated with vegetation types? What environmental aspects combine to proffer a habitat for, let us say, an endangered species? One can say that the richer the data set, the better the answers.
There is a process which I have called suitability analysis. This requires that prospective land uses be identified. Normally this would include forestry, agriculture, extractive minerals, recreation, and urbanization. Each of these can be further subdivided, urbanization might include housing of various densities, commerce, and industry. When the consumers have been identified, the inventory is reviewed to identify all factors on the legends of all maps and to determine those which are propitious, neutral, or detrimental for each prospective land use. Maps, either manually or computer produced, will depict all propitious factors for each land use. They will also depict all detrimental attributes. Finally, the computer will solve the command "show me those locations where all or most propitious factors are located, and where all or most detrimental factors are absent" by superimposing the propitious and detrimental factors. A reasonable convention is to colour all propitious factors in shades of green, all detrimental ones in oranges and reds. Dominant green shows the intrinsically suitable locations.
Of course, the method can be enhanced by weighting and scaling each of the factors. I have generally produced a map entitled "protection" which depicts all factors and regions wherein exist hazards to life and health. These environmentally sensitive areas will vary from location to location but are likely to include the following: flood plains, ocean surges, hurricane zones, tornadoes, tsunamis, earthquakes, vulcanism, wildfires, and subsidence. This protection map and associated text should identify the degrees of hazard represented by these processes and recommend a prudent response. As a result, suitability analysis suggests both constraints and opportunities for future land uses.
# Reasons for Optimism
There are reasons for optimism. The subject of the environment has been embraced by the media, surely more for vicarious thrill than for science and understanding. Children's books and television carry much superior information about the environment than they did in your or my childhood. Indeed, I have an excellent sample: my older two sons, in their thirties and forties now, are sympathetic to the environment and lived in a household where ecology was a common word, but their stepbrothers, fourteen and eight, have experienced richer insights into the environment from Nova, National Geographic specials, The Planet Earth, and more. There is an emerging generation, well informed on the environment, indignant at pollution and destruction. They may follow the lead of the 1970s which gave us the flood of environmental legislation. The environmental groups—the World Wildlife Fund, Audubon, Sierra Club, Friends of the Earth, The Nature Conservancy, the National Resources Defense Fund, the Soil and Water Conservation Society—should all constitute youth clubs to embrace these young potential soldiers for the environment.
Scientific knowledge of the environment has grown. The study of plate tectonics alone have advanced our knowledge of the environment; the world warming hypothesis and accompanying research have expanded our knowledge of the atmosphere and meteorology. The development of sensors has added to our ability to monitor the environment, not least GPS which has so simplified surveys and thus landscape inventories. The continuing advances in computation may be the greatest reason for optimism. More data can be ingested, evaluated, synthesized faster, more accurately than ever before.
There are also some sinister reasons for optimism. I would assert that the greatest impetus to improved ecological planning is the last disaster. A disaster amplifies awareness. The succession of alphabetic hurricanes, Edward, Fran, added to Hugo, Agnes, and Andrew have sensitized coastal communities to incipient disaster. Increased population, increased concentration combined with greater climatic violence guarantees more and bigger disasters. If the assertion is true, then each increment of pain and suffering should lead to the greater application of knowledge and wisdom to the planning process.
Can we use our massive brain, augmented by the great prosthesis of sensors and computers, to diminish such pain? Let us hope so.
The last and most important steps begin with resolve. Let us applaud the benefits of analysis and reductionism. They have contributed much to modern science which because of its fragmented nature clearly is not enough. It need not be superseded, but certainly it must be augmented to include synthesis and holism. The environment is now subdivided by science and language. We must direct our energies toward synthesis. There is a vital role here for all scientific institutions, all government departments and agencies. There should be a serious effort to reorganize contradictory views, programs at cross purposes, above all exclusionary myopia, and initiate cooperative procedures. Can we study watersheds, physiographic regions, oceans, tectonic plates, ecosystems? There is a new instrument which does not insist upon scientific disciplines or boundaries, which can accomplish synthesis. It is GIS, geographic information systems, capable of handling immense quantities of data, describing processes and undertaking planning. Not only can GIS undertake the most complex of planning studies undertaken manually it is possible to do studies which cannot be manually done.
Moreover, the computer and the GIS fraternity are already constituencies with magazines, conferences, vendors of hardware and software, oriented to the computer world, irrespective of university, agency, or country. Can this new constituency accomplish the integration which orthodox science has so resolutely rejected? Will GIS provide the belated integration required for planning? Will it spur an integration of inter- and intradisciplinary research and application? Will it lead to integration and synthesis?
The availability of GIS makes inventories a necessity. These should be extensive. A few years ago, I wrote a report with John Radke, Jonathan Berger, and Kathleen Wallace for the EPA entitled "A Prototype Database for a National Ecological Inventory." Our proposal includes a demonstration 40-km 2 sample, the Washington Crossing Hexagon. We chose this to examine two states, with different counties each with distinct nomenclature of geology, soils, vegetation, land use, and zoning. We mapped geology, produced sections, a digital terrain model, digitized aquifers, wells, well logs, discharge gauges, rivers, streams, marshes, wetlands, flood plains, and soils. We digitized 1929 and 1992 vegetation. We mapped all properties, land uses, and zones. It should be noted that making inventories is also an integrative process, particularly if extensive, employing the layer-cake method and using chronology as the underlying principle for organization.
In sum the opportunities for integration, organization of rich data, its analysis and planning using GIS are true, they are available. Will they be used to fulfill their promise, meet society's needs, and give government a larger role in understanding, planning, and regulating the environment and by so doing diminish death, damage, pain, and enhance human health and well-being? Let us hope so—even more, resolve to make it so.
# Conclusion
Newt Gingrich, speaker of the house, archcritic of bureaucracy, insists upon across-the-board reductions of 33 percent for USGS, NBS, EPA, and many more agencies. This approach is unable to distinguish excellence from indolence; it mutilates, rather than accomplishes remedial surgery. Yet there are bums, fools, incompetents, failed programs galore, many in competition. Can we not dispose of surplus, failures, incompetence and, above all, eliminate cross-purposes and ensure integration? But can programs be seen as complementary, contributing to our national purpose?
A White House Task Force should be convened to investigate the integration of scientific perceptions of the environment by accomplishing the following:
* Propose a uniform ecological planning method to be employed by all agencies
* Propose a process to develop and monitor the sets of environmental data that must be employed by all agencies
* Create a master agency devoted to assembling, updating, and interpreting all digital environmental data; it should develop a national GIS
* Produce a uniform policy to handle "greenhouse gases" and plan for carbon fixing
* Develop government-wide strategies to improve biodiversity
* Develop plans to minimize the losses from environmental disasters
* Produce a plan to construct a national ecological inventory
The White House Task Force should design a training process to be offered by selected institutions to instruct senior officials from all agencies on the ecological planning process. A further training program should be designed as a prerequisite for existing junior staff and for all new employees on all environmental agencies.
So, the natural sciences have either been excluded or lightly incorporated in the planning process. As we have seen, natural calamities cost thousands of lives annually, billions of costs, and enormous insurance claims. There is a mood for economy present in Congress but economies in environmental protection are not popular so, why not achieve economies by diminishing the pain and cost of hurricanes, floods, tornadoes, fires, and other catastrophes? To accomplish such real economies, we should undertake a national ecological inventory, monitor the environment and improve both understanding and prediction of natural phenomena, employ these data in ecological planning processes. Announce the intention to convene a White House Task Force to address the subjects of the environment, reorganization of government, a commitment to integration, address the problems of world warming, biodiversity, greater climatic violence, the increased threats from megacities and the necessity of bringing the environmental sciences into an improved planning process.
Let us plan to save lives, to protect the environment, to achieve savings from appropriate ecological planning, to improve prediction and placement, and to improve the human condition.
# Part II
# Planning the Ecological Region
Ian McHarg is part of the organic tradition of American planning that has two clear, although somewhat overlapping, lineages. The first is that of landscape architecture from Frederick Law Olmsted, Sr., to Charles Eliot, Frederick Law Olmsted, Jr., John C. Olmsted, and today's John Lyle, Julius Fabos, Anne Spirn, Forester Ndubisi, Donna Erickson, Glenn Eugster, Catherine Brown, and Rob Thayer. The second line is that of planning, from Patrick Geddes, Lewis Mumford, Benton MacKaye, Artur Glikson to today's Bob Yaro, Randall Arendt, Deborah Popper, Frank Popper, and Tim Beatley. McHarg is the heir to and propagator of both lines. He also cross-pollinates both strains with that of the naturalist-scientist-conservationist tradition, that of George Perkins Marsh, John Wesley Powell, and Gifford Pinchot, the Leopolds and the Odums, Loren Eiseley and Paul Sears, and Rachel Carson and Ruth Patrick.
The connection with Mumford is especially strong. Mark Luccarelli provides a thoughtful analysis of Lewis Mumford's contributions to planning in Lewis Mumford and the Ecological Region (1995). Luccarelli's description of Mumford's efforts "to reorient urban life, to address the crisis of over urbanization, and to recontextualize cities in relation to nature" nicely captures McHarg's work as well:
Mumford cultivated the possibility of a third way: not rural or urban—at least not as these have been defined as polar opposites in the modern era—and certainly not the "middle landscape" of the pastoral suburb; rather, a city as polis in relation to the organic complexities of the regional ecosystem (p. 3).
Mumford used regionalism as a unifying principle to combine ideas about neotechnics, organicism, and community. He drew much from the work of the Scottish botanist-sociologist-town planner Patrick Geddes as well and the French sociologist and biogeographer Frédéric Le Play. According to Luccarelli (1995, see also Wojtowicz 1996), "Mumford was concerned with planning an ecological and regional restructuring of the city and its environs, he insisted that regional planning be informed cultural criticism" (p. 35, emphasis added).
To this end, Mumford joined with several other intellectuals to form the Regional Planning Association of America (RPAA) in 1923. This small group included forester Benton MacKaye, architect Clarence Stein, landscape architect Henry Wright, housing advocate Catherine Bauer, architect Frederick Ackerman, and a few others. The RPAA in general and Mumford especially sought to expand Ebenezer Howard's garden city idea. According to Luccarelli, "Mumford believed the garden city could be a 'regional city': a new kind of modern city in creative relationship with the surrounding countryside" (1995, p. 77). Mumford's colleague MacKaye contributed ideas about the processes needed to achieve such a "creative relationship": planning as exploration and as applied science for charting the future from an understanding of natural processes and patterns.
Benton MacKaye (1940) explicitly linked regional planning to ecology. He defined regional planning as "a comprehensive ordering or visualization of the possible or potential movement, activity or flow (from sources onward) of water, commodities or population, within a defined area or sphere, for the purpose of laying therein the physical basis for the 'good life' or optimum human living" (p. 351).
According to MacKaye, a comprehensive ordering referred to "a visualization of nature's permanent comprehensive 'ordering' as distinguished from the interim makeshift orderings of man" (1940, p. 350). MacKaye quoted Plato to emphasize this thought, "To command nature we must first obey her." MacKaye felt that what he called the "good life" or "optimum human living" was expressed by what the U.S. Congress has called the "general welfare" and what Thomas Jefferson called the "pursuit of happiness." He concluded that "regional planning is ecology" and gave the following definition:
Human ecology: its concern is the relation of the human organism to its environment. The region is the unit of environment. Planning is the charting of activity therein affecting the good of the human organism, its object is the application or putting into practice of the optimum relation between the human and the region. Regional planning in short is applied human ecology (MacKaye, 1940, p. 351).
In his introduction to McHarg's Design with Nature (1969) Mumford clearly passed the regional ecological planning torch to McHarg, who would link the new knowledge derived from ecology to action. Mumford wrote: "he demonstrates, by taking difficult concrete examples, how this new knowledge may and must be applied to actual environments, to caring for natural areas, like swamps, lakes and rivers, to choosing sites for further urban settlements, to reestablishing human norms and life-furthering objectives in metropolitan conurbations."
To accomplish this purpose, McHarg realized that he had to change planning education. He addresses this topic in "Regional Landscape Planning," a 1963 paper based on a talk given at the University of Massachusetts during a conference on natural resources. The talk addressed open space concerns and how to create systems of open spaces. McHarg's approach differs from putting "a stencil over the land and saying that a Green Belt should surround the city." McHarg illustrated his approach through actual projects, a reflection of his design training. He created a model for applied, or action, research where hypotheses were developed and methods tested. In essence, he laid the foundation for a case study approach to planning education and research. In his writings and speeches, he would then narrate the case studies or projects as "stories."
His systems approach was interdisciplinary and had teams of talented individuals. One of the "stories" related in his 1963 paper notes the involvement of Ann Strong, who later emerged as one of the most significant environmental attorneys of our time. The study included early explorations of important concepts such as development rights, wetlands zoning, conservation easements, and preferential taxation for open space lands.
In his quest to redefine the planning profession, McHarg recognized that "the normal professional training of the landscape architect is totally inadequate" for the elevated mission he was proposing. He felt that planning education in the 1960s was equally inadequate. Regional landscape planning is what today would be called ecological planning or environmental planning.
A revealing aspect of that day's talk can be seen in the comments by the two discussants from two different disciplines. Professor Lawrence Hamilton, from the Cornell University Department of Conservation, offered the perspective of a natural scientist engaged in professional resource management education. Professor John Friedmann, then of the Harvard-MIT Joint Center for Urban Studies and later of UCLA, commented from the point of view of a social scientist concerned with graduate education and research in the planning discipline.
Professor Hamilton provides McHarg with a hearty endorsement. John Friedmann, who became one of the leading planning theorists from an applied social scientist perspective, is more skeptical: "my colleagues [Hamilton and McHarg] and I are taking utterly different approaches to pursuing the subject of resource development on a regional scale." Professor Friedmann attributes this difference to their disparate disciplinary origins (i.e., forestry, landscape architecture, and economics), overlooking (or unaware of) McHarg's academic background in city planning and the social sciences at Harvard or that the forester was quick to grasp McHarg's arguments for a new approach to regional planning. Furthermore, Friedmann notes that one "of the great difficulties in planning is that the social scientist finds it nearly impossible to talk with the physical planners." As the planning movement followed Friedmann this schism was exacerbated, and problems of communication developed between social scientists and architects and landscape architects.
Still, Friedmann did advocate "a common orientation, a common outlook, a common vocabulary, and a common concern" for planners both in education and practice. To establish such a base, he argued that three major subject areas should form the core of planning curricula: (1) resource analysis and policies; (2) the politics, process, and theory of planning decisions; and (3) social values and goals. I believe that McHarg followed Friedmann's ideas for both a common orientation and the three major subject areas. In the Master of Regional Planning curriculum McHarg developed at Penn, the common outlook and vocabulary revolved around ecology. The curriculum was structured to address (1) environmental inventories and analysis; (2) the planning process and environmental law; and (3) social values. A basic understanding of ecological science and geology was considered a prerequisite.
The planning of open space is also the topic of his 1970 "Open Space from Natural Processes," which appeared in a book on the topic edited by McHarg's then-partner, the innovative planner David Wallace. McHarg promotes a positive role for open space, one that complements rather than avoids the built environment. The optimum result would be a system of two intertwining webs, one composed of developed land and the second consisting of open space in a natural, or near natural, state.
In that article McHarg evokes Mumford and MacKaye again. He also begins to identify his work with that of Charles Eliot, the brilliant son of the Harvard president and the protégé of Frederick Law Olmsted. McHarg discovered Eliot in the early 1960s and mentioned him in 1964 in "The Place of Nature in the City of Man" (see chapter 2). As the 1970 article illustrates, McHarg remained a champion of Eliot throughout his career.
It also shows that McHarg was well aware of the work of his contemporaries, such as the pioneering river corridor plans of Philip Lewis in Wisconsin and Illinois (Lewis 1996). McHarg also provides an insightful discussion about the merits of using either hydrological or physiographic structure for regional open space planning. (McHarg clearly prefers the use of physiographic structure.) The open space work of Lewis and McHarg during the 1960s provided the foundation for subsequent greenways planning from the late 1970s to the present.
McHarg's article also introduces the intriguing concept of using open space and "airsheds" to ameliorate air pollution problems. Pollution-free areas, "tributary to the city," such as parks and other open spaces, help to create airsheds. McHarg suggests that planning such airsheds at the regional scale would reduce air pollution. For cities with "high pollution days," such as Denver, Los Angeles, and Phoenix, this concept is even more timely today than in 1970. Airsheds, in a more narrow sense, are used by the U.S. Environmental Protection Agency to implement federal clean air laws. Air quality could be improved by implementing open-space systems.
Like the 1963 "Regional Landscape Planning" article, "Must We Sacrifice the West?" is based on a speech. The paper provides insight into Ian McHarg's provocative speaking style as well as his self-depreciating wit. A distinguishing aspect is his ability to adapt his rather consistent main themes to various specific times and places. "Must We Sacrifice the West?" was delivered at the height of the "energy crisis" of the 1970s. His talk was given in Colorado, where at the time an energy development boom threatened the quality of the Rocky Mountain environment. As a result, Ian McHarg provides cogent advice for energy planning.
Because it argues that calculations of energy expenses should include social costs, the paper demonstrates the social concerns that underlie McHarg's ecological planning approach. Planning is advocated as a means to address social issues, "the device" by which people "confront the future." Such confrontation requires that human values be explicit. McHarg holds that such values, when clarified and linked to the environment, have considerably more influence on planning than any amount of data.
Throughout his academic career, McHarg continued to rub up against the "orthodox" city planning tradition, frequently irritating planning theorists but also influencing and changing their ideas about planning. His 1978 "Ecological Planning" was included in an orthodox, but comprehensive, planning theory text. He calls the planner who pursued the traditional approach attempting to come from out of town to help people solve problems "a menace." The role envisioned by McHarg is that of catalyst. He views such a role as a creative activity. Planning, then, is an art, but one based on the predictability that science provides. Science helps the planner understand "the consequences of different courses of action."
McHarg argues that planners should not impose their values on a people or a place. Rather, values should be derived from those who inhabit the landscapes. Developing values from a local perspective based on regional biophysical processes differs from importing values from outside the region. Determining, identifying, and making explicit values are central to McHarg's planning theory. The value system determines the planning process.
His 1981 "Human Ecological Planning at Pennsylvania" returns to the topic of planning education. Regional Landscape Planning has been replaced by Human Ecological Planning in this Landscape Planning essay. However, the actual degree that was implemented at Penn in 1965 was a Master of Regional Planning. Between the early 1960s and the early 1980s interest in regional planning waned in the United States. Meanwhile, interest in environmental and even ecological planning increased, partially as a result of McHarg's influence. His 1981 article offers a reflection on the goals for planning (and for McHarg ecological was increasingly the correct modifier) that he established in the 1960s.
In spite of McHarg's influence, the term environmental planning has gained wider acceptance in the United States than ecological planning. Why? There are at least three reasons. First, environmental design and environmental planning are terms that have enjoyed much popularity within the influential California academic circles since the 1960s. For example, the University of California-Berkeley, the University of California-Davis, the California State Polytechnic University-Pomona, and the California Polytechnic State University-San Luis Obispo have colleges or schools of environmental design, and California-Berkeley offers a Ph.D. in environmental planning. Second, the National Environmental Policy Act of 1970 mandated the use of "the environmental design arts" in federal decision making. As a result, environmental design became institutionalized in the federal bureaucracy.
Third, the concepts of environmental design and environmental planning evolved from architecture and planning. The design disciplines have a long history of intervening in and manipulating our surroundings and in creating places. Architects give form to built urban environments. Planners suggest policy options for human settlements. Environments have a strong visual connotation. Architects are comfortable with visual aesthetics. Ecology, the understanding of interactions, is more unsettling, even subversive.
In contrast to environmental design and planning, ecological design and planning developed in the United States within academic programs of landscape architecture. Landscape architecture education began in the mid-nineteenth century in agricultural and horticulture colleges that were established as a result of land-grant legislation signed by Abraham Lincoln in 1862. This law provided land grants to the states to establish public agricultural and technical colleges. A strong supporter of this system was the pioneer landscape architect Frederick Law Olmsted, Sr., who was involved in planning several land-grant college campuses. The Olmsted brothers' firm (Frederick, Jr., and John C.) subsequently carried on this tradition of campus planning, especially at growing land-grant schools.
A second tradition of landscape architecture education was established when Harvard University founded a landscape architecture program in 1900 with close ties to architecture. In contrast to the land-grant system, Harvard's program emphasized design instead of agriculture and horticulture. Under Frederick Law Olmsted, Jr., and John Nolen's leadership, Harvard initiated a city planning program in the early twentieth century. Many American universities followed the Harvard example of landscape architecture closely aligned with architecture and planning. As a result, two traditions in landscape architecture were established: one emphasizing rural concerns and natural resources; the other more focused on design and urban planning.
Ecological design and planning are a fusion of those traditions. For McHarg ecological planning is a method "to promote human health and well being." He distinguishes ecological planning from economics, transportation, and orthodox city planning. The key, noted in the 1981 human ecological planning article, is McHarg's advocacy to understand the locality, that is, the natural and social processes of a specific region, a specific place. A role of the planner is to synthesize information from the various sciences about a locality to develop hypotheses about its possible futures.
In several other publications, McHarg explains how to organize biophysical information chronologically and arrange it in a "layer cake" model to display causality and to reveal landscape patterns. He also discusses, in many places, how to incorporate the human values of such information to reveal opportunities and constraints for future land uses. The 1981 Landscape Planning (now called Landscape and Urban Planning) article is his most complete work on the relevance of human ecological information in the planning process.
Human ecology dominated the Penn Department of Landscape Architecture and Regional Planning research agenda throughout the 1970s. With the publication of Design with Nature McHarg received some criticism for ignoring social factors in the planning process. He openly admitted that the focus of Design with Nature was on natural processes, on ecological determinism, because such information had been ignored by most traditional planning, design, and decision-making processes. To balance that one-sided approach, he began in the 1970s to incorporate social factors into ecological planning.
At the time anthropology was a strong discipline at Penn, and McHarg was a friend and admirer of Loren Eiseley of the anthropology department. Anthropologists also had strong interest in ecology and, at Penn, they had discovered the city. Because of criticism for sending ethnographers to exotic Third World nations ("cultural imperialism") while ignoring the contemporary urban settlements in developed nations, the Center for Urban Ethnography was founded at Penn. Led by John Szwed, it attracted anthropologists interested in urban America (see Dan Rose's wonderful Black American Street Life [1987], for example).
As a result, McHarg imported anthropologists into his department during the 1970s as he had a decade earlier with ecologists, geologists, hydrologists, and soil scientists (as well as architects, landscape architects, the economist Nick Muhlenberg, and the attorney-city planners Ann Strong and John Keene). Anthropologists Yehudi Cohen and Setha Low taught courses that introduced planners and landscape architects to the basic concepts of ecological and medical anthropology. Even more significantly, ethnographer Dan Rose collaborated with regional planner Jon Berger on a series of human ecology projects that were patterned after McHarg's work.
The essential contribution of the discipline of anthropology to environmental planning is the idea that an ethnographic history of a place should be included in the inventory phase of the planning process. Like a layer-cake diagram of the biophysical processes, the chronology provided by an ethnographic history can help to explain causality. The result is a combination of Patrick Geddes' folk-work-place with ideas about randomness and order reminiscent of chaos theory (see Prigogine and Stengers 1984, for example).
John Friedmann includes a table—a family tree—of the intellectual influences on American planning theory in his Planning in the Public Domain (1987). He commented on McHarg's 1963 paper at the University of Massachusetts, so obviously he is aware of him, but McHarg, Patrick Geddes, Charles Eliot are not included in the family tree. The organicists are generally excluded, except for Mumford, who is listed on the far right branch under the category "Utopians, Social Anarchists and Radicals." From an orthodox perspective this is probably an accurate placement. "Sociology" and "Scientif ic Management" (i.e., Emile Durkheim, Max Weber, and Frederick Taylor) are placed in the center of Friedmann's table. Ian McHarg, like his mentor Mumford, indeed offered a radical view of planning, one based on ecology.
One younger theorist, Manuel Castells, whom Friedmann identifies as a neo-Marxist (the neo-Marxists also fall on the far right of Friedmann's table) has begun to espouse an ecological view. Castell's observations of 1992 are similar to those expressed by McHarg two decades earlier, especially as they relate to the destructiveness of planning systems based on economic determinism as well as to the importance, even prominence, of environmentally based planning. According to Castells, "The danger of total environmental destruction under the new worldwide economic dynamism imposes a new, more comprehensive conception of land use. Environmental planning will be the most rapidly expanding planning frontier in the United States in the next decade" (1992, p. 77). If the neo-Marxists have adopted such ideas, perhaps the organic, ecological tradition of Eliot–Geddes–Mumford–McHarg belongs at the center of planning's intellectual influences, not the margins.
# References
Castells, Manuel. 1992. "The World Has Changed: Can Planning Change?" Landscape and Urban Planning 22:73-78.
Friedmann, John. 1987. Planning in the Public Domain: From Knowledge to Action. Princeton: Princeton University Press.
Lewis, Philip H., Jr. 1996. Tomorrow by Design: A Regional Design Process for Sustainability. New York: Wiley.
Luccarelli, Mark. 1995. Lewis Mumford and the Ecological Region: The Politics of Planning. New York: The Guilford Press.
MacKaye, Benton. 1940. "Regional Planning and Ecology." Ecological Monographs 10:349-53.
McHarg, Ian. 1969. Design with Nature. Garden City, N.Y: Doubleday, Natural History Press. 1992. Second edition, New York: Wiley.
Prigogine, Ilya and Isabelle Stengers. 1984. Order Out of Chaos. Toronto, Ontario: Bantam.
Rose, Dan. 1987. Black American Street Life: South Philadelphia, 1969–1971. Philadelphia: University of Pennsylvania Press.
Wojtowicz, Robert. 1996. Lewis Mumford and American Modernism. New York: Cambridge University Press.
# 6
# Regional Landscape Planning (1963)
In 1997 the comment by John Friedmann at the end of this chapter was confirmed for accuracy. In the book Resources, the Metropolis and the Land-Grant University, where the essay appeared, his name was misspelled, so it may have been another John Friedmann, one with a single "n" in his family name, who commented. Now retired in Australia, Professor Friedmann confirmed by e-mail that, yes, indeed, he was the author of the remarks in question. University of Massachusetts conference organizer and book editor Andrew Scheffey was a friend of Friedmann. From Australia, Professor Friedmann made the following observations:
Thank you for this piece of forgotten history in my academic life. Andy Scheffey was an old buddy of mine from Korea days, and it was he who put the conference together. Rereading my commentary and your [Steiner's] historical introduction, I confess that I was quite intolerant of the "organic" school of regional planning at the time. Intolerant and somewhat ignorant of its contributions. The two probably go together. Today, I have become more tolerant, but I have not become an ecologically informed regionalist. Although I admire the work of (some) landscape architects, I have not found a way to integrate their work with the approach to regional planning (or spatial planning) that comes out of the socio-economic tradition with which you are familiar. The discourse on "sustainable development" has helped to highlight the importance of environmental/ecological dimensions of economic growth, but the idealism implicit in Ian McHarg's and Mumford's formulations is altogether absent from the discourse.
So: two streams that touched in 1963 but had little to share and ultimately went their separate ways.
No, I do not wish to add anything to my commentary at the time. I would put things differently today, but it wouldn't put me any closer to Ian than I was thirty-four years ago.
Professor Friedmann's candid observations reveal the gap between "orthodox" planning theory and the Mumford-McHarg "organic" tradition. As Friedmann reveals, this gap persists, although interest in "sustainability" and "sustainable development" may indeed create a bridge. McHarg's quest then was, with the advice and encouragement of Mumford, to re-create the field of planning. This chapter represents a first step.
I differentiate the professions of regional planning and regional science from that of regional landscape planning which we hope to develop at the University of Pennsylvania. Although there are as yet no specific accomplishments to refer to, I want to talk with you about the direction of this prospective curriculum. I have been asked, wisely I believe, to approach this subject indirectly by describing some of the research work being carried out in the Department of Landscape Architecture at the University of Pennsylvania, as well as some of the research projects which I am involved in as a private practitioner. From a discussion of these projects one can discern some of the skills which are required for Regional Landscape Planning. This discussion hopefully will have some value to you for its generality, although quite obviously it is based upon several particular experiences within a particular context.
The first project concerns open space. When the open space provisions of the 1961 Housing Act were made law, the Urban Renewal Administration confronted the necessity of producing criteria for guiding the purchase of sites within metropolitan areas. As you may recall, the law contains provisions under which 20 percent of the cost of open space land acquisition would be made available from a $50 million fund established for this purpose, or 30 percent in the event that the open space sought was part of a comprehensive metropolitan plan. But the act contained no regulations or criteria concerning the type of land areas to be acquired for such open space purposes.
While my department knew nothing of this particular quandary, we had just finished a year-long study of the Delaware River Basin carried out by twelve graduate landscape architecture students. We had conducted this study in cooperation with the U.S. Army Engineers, the Soil Conservation Service, and various state departments of conservation, economic development, and so forth. In a memorandum describing the preliminary results of this study, sent to the Director of the Institute for Urban Studies at the University of Pennsylvania, I had stated several tentative conclusions of the study. These were not revolutionary findings or concepts in any sense of the term, and they will probably appear relatively simple to those of you who are natural scientists by training and profession, but within our own academic planning departments they had the aspect of original innovations.
The purpose of the study was to develop criteria for land-use planning in the Delaware River Basin. It was obvious that the criteria should be based upon an understanding of the natural processes in the region. Criteria used within this penumbra selected discreet natural processes which performed work, which had intrinsic value, or which offered protection in their natural state. These land classifications were mapped and measured. In addition a panoply of legal tools was related to the selected classifications as prospective instruments of land-use planning.
The Institute for Urban Studies was interested in the ideas presented, and took the matter up with the Urban Renewal Administration, which ultimately decided to support a research project designed to test these various concepts, and to come up with qualitative measurements of these functional values. The unit of study was determined to be the eight-county Philadelphia-Camden metropolitan area.
The criteria advanced were, again, fundamentally quite simple: surface water, marshes, flood plains, wetlands, aquifers-recharge areas, forests, agricultural land, ridges, and steep land. We advanced the concept that the upland sponge has a preponderant role in matters of flood control and water supply, particularly low-flow problems. The disproportionate role of estuary marshes in the diminution of the tidal component of floods was discussed, and beyond that the concept that the coefficient of runoff within a flood control plan might be an appropriate determinant of land use. A final concept was presented, that of an urban airshed. It assumes that atmospheric pollution is going to stay with us, and that inversion conditions will continue at the normal incidence, but that the increasing growth, increasing combustion, and increasing pollution, will make the quality of the air-relieving inversion more important. Thus, in the future we may want to know more about the area of clear air, without combustive processes, that will be required in any metropolitan area to ensure that the air which replaces the foul air is significantly cleaner than that which it replaces.
It is not difficult to discern the wide range of professional competencies necessary to provide the basic information required to establish such criteria—information on water-based classifications, surface waters in various forms, the aquifers and their aquifer recharge areas, the flood plains, the marshes, the wetlands. The basic information on these topics was provided by the U.S. Geological Survey and the U.S. Army Corps of Engineers by their hydrologists. The information on alluvium and flood plains was secured from the Soil Conservation Service county soil surveys. A correlation of alluvium to cyclical floods will be done in collaboration with hydrologists of the Army Corps of Engineers relating these direct observations to the deposition of alluvium as recorded in the various soil surveys.
Having mapped these factors in the study areas, the next problem was the measurement of each of the parameters of which there were eight. With eight counties and four hundred municipalities involved, overlapping two states, this could have become a fantasy of planimetry, and perhaps the only real innovation of this research project was the discovery that the photo electric cell is adaptable, and can be used to measure reflected and incident light. By making a negative of each of these parameters, of which the parameter itself is black against a white background, and knowing the total area of the metropolitan region, it is possible to determine the area covered by the parameter. This proved to be a most inexpensive and accurate method.
The next problem involved the calculation of land values, and this required an economist. It proved to be an extremely difficult physical problem indeed, because of the disparate quality and quantity of available information. In addition, properties are seldom assessed on the basis of constituent parts, for example, in one farm the constituent value of flood plain, steep lands, woodlands, and so forth. Clearly this investigation falls to the economist.
The final step in this first project involved an analysis of existing and proposed legislation. This called for legal skills. We secured the services of a lawyer, Ann Louise Strong, and her role was to consider a sequence of means including tax exemptions, tax deferrals, preferential assessments, development rights, scenic easements, recreational easements, various types of zoning—wetland zoning, aquifer zoning, agricultural zoning, steep land zoning, forestry zoning, and so on, with the final possibility of outright purchase.
This process, then, included the definition of a range of criteria, mapping of these criteria, their measurement, assembly of legislative tools, and analysis of costs. Many disciplines were involved in this process. In this case the landscape architect was the coordinator. Equally clear, the normal professional training of the landscape architect is totally inadequate in preparation for such a role. Hydrology and geology, agriculture and forestry, zoology, botany and ecology, economics and law were all integral to this process.
This project was done for a total cost of $15,000. The second project I want to mention is being done in cooperation with the states of Pennsylvania and New Jersey, each providing $10,000, and the federal government which is providing $42,000. The purpose of this project is to go beyond the steps already described, to determine the future recreational demand, and to produce a primer on the selection of metropolitan open space land areas. A number of elements are involved.
The first is to interpret and modify and adopt the estimates of recreation demand made by the Outdoor Recreation Resources Review Commission. We believe that these figures are good, but must be refined and elaborated. The calculations on demand will be the responsibility of economists with the involvement of persons qualified in transportation. In this region we have data from the Penn-Jersey Transportation Study which permit us to construct a transportation model related to recreational demand.
The next thing we shall have to do, after determining the location of all available lands now used for open space purposes, and assessing the utilization in these facilities, will be to calculate the degree to which these areas might be further developed to permit more intensive use. It is quite apparent that these existing areas will not meet future demand. Therefore, we shall have to examine all the natural areas which have been defined, many of which are permissive to various types of recreational uses, and indicate the extent to which these areas can support some of this recreational load, consistent with their other functions. Again, it is reasonable to presume that these two resources collectively will not fully satisfy future demands, and so the final step in this project will be to determine that increment of unsatisfied demand which will have to be met through the acquisition of additional open space. These will be determined by the types of demand anticipated, physical location of the sites, and economic criteria—the values of the land. The latter will be the task of the economist and transportation expert. The determination of the appropriateness of the land in terms of scenery and its suitability for various recreation functions will be the role of the landscape architect.
At that point we shall be confronted with the problem of reconciling a number of different criteria: existing open space which has been acquired for a number of different reasons, few of them rational; a number of natural areas, the selection of which has been based upon various different criteria, none of which were really recreational; and finally, those areas of open space which will have to be acquired to meet the residual recreational demand. We shall be confronted with the necessity of deciding whether or not these should be systematic. It must be realized that the prototype of most regional planning of this sort, certainly in Great Britain, is to put a stencil over the land and say that a Green Belt should surround the city. We do not believe that this idea of simply proscribing a Green Belt is very valuable or appropriate for the needs of today. We are concerned with producing metropolitan open space for the Philadelphia metropolitan area, but we are also concerned with the criteria which should produce systems of open space.
From this brief discussion it should be clear that the skills and professional qualifications of many persons other than the landscape architect are called for in research projects of this nature. Yet this is the type of regional planning that will become increasingly needed in the future. I suspect that there is no single person who could be trained to do it alone.
Let me now tell you about another project which is quite different, but which in a sense is focused on the same objective. This project happens to be in the private domain rather than the public.
Some time ago a colleague of mine and I were asked to be consultant planners by a group of inhabitants in two valleys in Baltimore County, Maryland. The problem these people face is a very common one. They are located on the outskirts of Baltimore, and they are soon going to be overwhelmed by the excremental type of development. They are reluctant to have this beautiful land ruined. And it is beautiful land, typical hunt country, covered with beech and hickory, a lovely landscape which they do not want despoiled. The problem facing these people is to avoid the cataclysm of development, while at the same time reaping some of the fruits of this development.
The project which we have proposed, in bare outline, involves the following elements. First, we shall make an economic and physical projection based upon the status quo. That is, if nothing else intervenes, if the only future intervention is the existing zoning power, which is to say no intervention whatsoever, then what will be the result? We propose to make an economic projection of the changes in land value, changes of building values, the type and location of development which can be anticipated under prevailing circumstances. We know already, of course, that such development will be catastrophic, and that it will cause everybody who has been there and made the area what it is to leave.
The next step will be to determine the best possible solution to this. What of this land is actually indispensable to the image which the present inhabitants have of it? What parts of it are in effect amenable to development and should be developed?
Answering these questions will require a number of steps. First will be a particular analysis of the physiography of the region, of the flood plains, of the agricultural land and the forests, the marshes and streams, and an analysis of the degree to which parts of the valley are indivisible to other parts. Following upon an analysis of the genius loci there is the task of developing alternative plans which locate development without despoilation. This is the orthodox task of the landscape architect-planner. By far the more difficult task is to develop a method through which a plan may be realized. This area contains some 500 landowners. They must act in concert to ensure realization of the plan. We can heighten their willingness by demonstrating the specter of despoilation resulting from unplanned development. Our proffered solution is the creation of a Development Corporation. We suggest that the landowners concede the development rights of their land to the corporation in exchange for stock. This method has a number of merits. It will tend to halt prodigal development creating low values and it will thus induce intensive development and high values. This device will also produce a higher increment of development profits, which permits giving payments to members of the corporation whose lands should not, in the plan, be developed. This process calls for the services of the tax lawyer, the corporation lawyer, the land-use lawyer, and the economist. The earlier portions of this project will be the province of geologists, hydrologists, agriculturalists, soil conservationists, and landscape architects. And in the very first stage the economist will predominate. So here again, dealing with quite a different sort of problem, we have the same wide spectrum of professionals.
One purpose in telling you about this is to show what is involved, not only in doing landscape planning jobs, but also in finding these jobs and persuading people that they have to be done! In this sense the architect is lucky, because people really have to keep the rain out, and so there will always have to be buildings. But the landscape architect is really involved in purveying a metaphysical symbol, and purveying it to a society which really doesn't give a damn about metaphysical symbols. And so, he not only has to be propagandist for metaphysical symbols, but he has to do this in a materialistic society; he has to show them that they can have metaphysical symbols and Cadillacs, too!
We have in Philadelphia a comprehensive plan which envisions spending $12.5 billion in 20 years to acquire open space, at which time there will be less open space then, than there is now. This struck me as rather expensive retrogression and I was somewhat distressed by this finding. After thinking about it I hit upon the idea of creating new usable open space. We have in Philadelphia some 300 piers, all but a few of which are derelict. I thought that perhaps the area between the bulkhead and the pierhead might be filled in and converted into riverside parks. The Philadelphia Planning Commission retained me to prepare a preliminary investigation of this hypothesis.
This again involved a great number of scientific investigations. First came the hydrologic work. Would the diminution of the water perimeter affect the flood capacity of the Delaware River? Here we obviously turned to the hydrologists of the U.S. Army Corps of Engineers. The next consideration concerned the problems of building foundations, and for this we went again to the Army Corps of Engineers, and also to the Netherlands to discuss the problem with the people at the Soil Mechanics Institute at Delft. Then there was the question of aquifers and the aquifer recharge areas on both sides of the Delaware. The question of sediment yield came up, and it was learned that sediment deposition would in fact be diminished by reducing the friction by bulkheading instead of having a whole set of finger piers.
Another important consideration was the potential source of fill material. The Army Corps of Engineers were at the time widening and deepening the Delaware channel, and were producing literally hundreds of millions of tons of gravel, sand, and organic silt. At the same time, it transpired that the city sanitation department had the problem of disposing of one million cubic yards of incinerator waste a year. They have a $10 million capital investment program to buy barges, build water transfer stations, and to transfer this material to the marshes of New Jersey, which of course they would then despoil.
A cost-benefit analysis showed that the City of Philadelphia could bulkhead 6 ½ miles of waterfront, create 436 acres of new park land, and spend $10 million on the development of parks, at costs identical to that required to barge incinerator wastes to destroy the marshes of New Jersey. Here again, quite a varied spectrum of professionals was involved, and hardly a landscape problem, but certainly a metropolitan resource situation.
I can conclude very quickly by turning to the question of regional landscape planning at Pennsylvania. We now have a Department of Regional Science, under Walter Isard. We have a Department of City Planning with a faculty of about fifty, primarily social scientists. There is active discussion about the development of a Department of Regional Planning, based largely upon the proclivities and attitudes of the present Department of City Planning, but in conjunction with the Department of Regional Science. This is all very well, and such a development would fit very smoothly into our situation. However, both of these existing departments are staffed almost exclusively by social scientists. Both would like very much to have natural scientists present.
Strangely enough, in my university, where there is no forestry, no agriculture, and where the natural scientists are primarily the biological scientists, concerned with cytology and microbiology, landscape architecture constitutes the only bridge to the natural sciences. This is fortuitous, and we have, therefore, received the assent of the Departments of Regional Science, City Planning, the Dean, and the Provost to investigate creating a Department of Regional Landscape Planning which will be the third department of this triad—regional science, regional planning, regional landscape planning. The last would be the bridge to the natural sciences. The first two departments would include people from statistics, economics, planning, sociology, law, and the latter would attract natural scientists interested in planning problems. We would hope to receive people with at least a master's degree proficiency in their own field, and to this we would add one and perhaps two years of work in planning itself, done in conjunction with courses in regional science and regional planning. Of course, we would hope that the regional scientists and planners would take at least several courses with us.
I think that this new direction is a great adventure. I am very glad that the natural scientists are beginning to be concerned with the problems confronting the physical planner. I believe that the absence of the natural scientists in planning up to this point is very, very sad indeed. I do not think this is due to any exclusion by the social scientists, but rather to the indifference of the natural scientists themselves.
I am delighted that the day has come when the natural scientists are beginning to become interested in this field. I think that their contributions are fundamental and that this interaction will be beneficial in complementing the existing body of planning knowledge, and leading us to fruitful innovation.
# Comments—Lawrence S. Hamilton
I want to take this opportunity to re-emphasize, from the standpoint of the so-called natural sciences, the imperative for action which I see in this open space question generally. Secondly, I wish to stress the need for greater participation by the natural scientists in regional planning activities. I then want to make a few remarks about what we have been doing at Cornell in our graduate training program in trying to produce resource professionals who might meet this need.
With regard to the second of these—the need for greater participation by natural sciences—if I heard Professor McHarg correctly, all that I can do is to say Amen to his plea. He has demonstrated this point so well that I wish I had said it that way.
And now to the first point. In New York State we find the problems of open space planning mounting at a phenomenal rate. The changes in land use are occurring at a bewildering pace. We have the paradox, which I am sure you in Massachusetts have also, of a state that is becoming both increasingly urbanized and increasingly forested at one and the same time. We see something like 200,000 acres passing out of agriculture each year into this wild domain called open space. Some land is going into urban uses, but much is starting down the successional path toward forest. In this forest, along with the floral and fauna constituents, there will, of course, be weekend hideaways, summer residences, and even year-round homes. These present land-use planning problems of new dimensions.
In no place is the change more rapid than on the cutting edge of the urban or metropolitan fringe—this tension zone of rural-urban interpenetration. A group such as this meeting, or any educational institution concerned with natural resources, might well focus its attention on the problems being generated in this fringe area. The problems are legion, and they are on a new scale, making obsolete our traditional tools, techniques, and approaches. This is where the landscape of America is being most drastically altered. We need research directed towards these problems, and we need to have persons with an ecological comprehension of the problems involved in our resource planning and development activities. Again, I strongly concur with the sentiments expressed by Professor McHarg, and with his plea for greater participation by the natural scientists.
It is important, as we decide what we are going to do with this open space, that we recognize that the various components—trees, grasses, sedges, gullies, water, and so forth—are not just there, but that they are part of a process going on, part of a number of processes—life and death, weathering and erosion migration and invasion, deposition—all these and others, contributing to some kind of energy system. I am talking about the ecosystem. These processes tend to become exceedingly complex, and what we do to one element has repercussions which echo around the entire system.
Resource professionals trained to understand the biological and physical environment (some of them do not at the present time), geologists, foresters, wildlife managers, soil scientists or hydrologists, are in a position to identify the natural resource development opportunities which exist, as well as the natural resource or environmental constraints which are operative. Portions of the environment having unique natural characteristics often fall before unplanned or ill-planned urban and suburban uses because they are beyond the ken of subdividers, politicians, real estate developers, lot purchasers, and perhaps even some urban and regional planners who have come primarily from the social sciences.
We are all familiar with the everyday examples of these ill-planned uses of land. There are the gravel deposits of high value that may be lost to commercial exploitation because we put residences on them, or near enough to them that the owners feel gravel exploitation to be a nuisance, and pass ordinances against them. Or, the unique recreation site, important waterfowl areas, key areas for future water storage, that become lost to use, or become too expensive to reclaim, because of ill-advised development. Resources such as these must be identified in advance so that certain types of non-compatible uses generated by urban forces may be guided, wherever possible, to more appropriate and equally satisfactory areas. Similarly, areas of highly productive soils, uniquely adapted to low-cost agricultural production, should be identified and perhaps reserved from development until absolutely needed for building sites, airports, residences, and industrial zones. The evaluation and proper interpretation of soil resources can result in better planning for transportation, industrial location, subdivision development, landscaping, and other functions. A soil survey can identify various soils appropriate for particular uses, thus avoiding such costly ecological consequences as soil slippage, flooded basements, waste disposal problems.
I believe that this aspect of the regional planning process can best be met by persons trained in disciplines dealing with the use and management of natural resources. In addition to professional specialized resource training of undergraduate programs, such persons should receive additional training in ecology and economics. This is what we are trying to do at Cornell. Those resource professionals who come to us with an interest in land-use planning are given an ecological basis by appropriate work in soils, geology, air photo interpretation (which is really understanding the environment from a picture of it), plant and animal ecology. In addition to this they are given work in planning theory. The methods provided by economics for weighing and analyzing alternative possibilities are equally important ingredients. Since this is frequently lacking in the training of many resource specialists, we encourage either a major or minor concentration in economics.
This sort of broad-brush graduate training admittedly has some real dangers. We have tried to hold these to a minimum by insisting that persons coming into the program have depth training in one of the natural sciences, and that they have had professional work experience. By observing these criteria we have been successful in securing mature graduate students, who know why they have come back, and who feel a commitment to making this kind of academic endeavor. They realize that an understanding of the natural environment is necessary if they are to work effectively in their particular field. They are concerned about making the environment a more pleasant place for human occupancy.
An institution such as the University of Massachusetts would be well advised to capitalize on the strengths that currently exist at the institution. Where a strong college of agriculture with competency in natural sciences is within the university, it would seem to me that land-use planning orientation might well have an ecological foundation. Our program of resource training originated within such a framework, although we have established good working relationships with the social sciences and the urban-oriented planning of the college of architecture.
These remarks refer to the training provided for natural resource specialists interested in working in the planning phases of this field. It seems to me that as these people move away from the hinterland and become more involved in metropolitan fringe problems they will become more closely associated with people now trained in regional planning. This is not to say that they will become one and the same, but merely to observe that they will more closely resemble their social science counterparts.
# Comments—John Friedmann
It is of great interest to me to find that my colleagues and I are taking utterly different approaches in pursuing the subject of resource development on a regional scale, reflecting no doubt the three disciplines of our separate origins—Professor McHarg from landscape architecture, Professor Hamilton from forestry and conservation, and I from economics and geography. I think there is a lesson in this to be learned about educating young people who will go out into the world to engage in the practical tasks of regional planning and resource development.
The kinds of programs that Professor McHarg described for us are typical examples, although they are clearly not the only ones that might be cited. They are especially typical in that they all make use of a large number of dif ferent specialists. And so, when we speak of training for resource development and consider the problem of devising appropriate courses of study for those who wish to gain expertise in this area, we are confounded, because we know that, in actual development situations, the resource specialists will have to work, shoulder to shoulder, with professionals representing many different backgrounds and skills and exhibiting altogether different ways of thinking.
I have been closely associated with two major regional development programs, one with the TVA, and another in Venezuela. I have found all sorts of people working on these programs—engineers, agronomists, sociologists, lawyers, and so on down the list. But generalists in resource development and management were not among them. This experience, which I believe is not unique, suggests that we cannot really expect to train generalists who will be competent in the total spectrum of specializations required for resource development. We cannot expect to combine, within a single person, the skills of a hydrologist, ecologist, soil specialist, city designer, urban designer, sociologist, and economist. It is a sheer impossibility for any one person in a single lifetime to acquire the knowledge necessary to absorb the very specialized and diversified knowledge in each of these professions.
What we can expect, however, is that students who have an interest in resource development acquire, partly from their formal education and partly from their working experience, a common orientation, a common outlook, a common vocabularly, and a common concern. One of the great practical dif ficulties in planning is that the social scientist finds it nearly impossible to talk with the physical planners. This is especially true in the newly developing regions of the world, where the economic and physical planners clearly go their own and separate ways. Although they should be working in close coordination with one another, they don't, simply because they do not command the same scientific language.
It seems, therefore, that one of the great tasks confronting a university is to create this universal language for people who are concerned with problems of resource development. The spectrum of specializations which might be involved is so broad that there is no need to put any limit upon it. You have had one roster of specializations brought before you this afternoon.
How then, specifically, can we go about creating this common feeling and understanding, this common vocabulary and knowledge, within a university? It seems to me, in a field so fluid as this one, which is in a continuing state of flux, and in which there is no great body of literature to refer to, that we cannot begin to teach resource development except by associating such teaching with research. Perhaps this is true of all fields which today are characterized less by a received stock of knowledge than by a process of study. But it applies particularly to the field of resource development. I believe that very little progress can be made by teaching that is not closely related to research.
Therefore, I would put first emphasis on that aspect, though I realize that much of the teaching will be done at the undergraduate level, where research has not traditionally been part of the curriculum. However, there is a need for change toward this kind of orientation, and we must begin, even at the undergraduate level, to introduce students to research, and to provide more time for their professors to engage in research.
Another aspect that I would stress in this connection is the need to ensure that research in resource development be undertaken across the broad spectrum of disciplinary specializations. This in itself will help to build up a common vocabulary, to create some common understanding, and perhaps new theoretical insights. It is not going to be done by one specialist thinking "... how can I be cross-disciplinary. . . ." It can be done only by keeping up a constant stream of communication, information, and conversation among a group of concerned specialists, scientists, and researchers working on a common problem. This is the way in which a common stock of experience, knowledge, and language can be built up. In this sense I look upon the teaching of resource development not merely as a job of communicating knowledge, but as participation in a flow of constantly evolving knowledge.
Secondly, I want to suggest the desirability for establishing, in addition to this research phase, a common core curriculum for students interested in resource development. We can, of course, have lengthy arguments as to what this core should consist of. This is not the place to enter into such a discussion. Since I am of the conviction that we must, for the time at least, continue to train specialists rather than generalists in the field of resource development, the core curriculum I envision would be a rather restricted one. It should neither be all-encompassing nor take too much out of the student's total time spent at the university.
I would suggest that, as a minimum, three major subject areas should be included in a core curriculum. The first should focus on resource analysis and policies. The second should deal with the politics, process, and theory of planning decisions. The third should be in the area of social values and goals as they relate to resource development and planning. Since development is always for a purpose, we must become sensitive to the social values and objectives involved, must have some familiarity with them, and be able to talk and think about them in a reasonably sophisticated way.
There are, of course, many ways in which a core curriculum might be established. Probably the most feasible, in most situations, is to incorporate it into existing schools and departments, perhaps at the initiative of an interdisciplinary Committee on Education and Research in Resource Policy and Development. Such a committee might not only introduce new courses bearing on resources problems, but guide students in developing course sequences appropriate to their specialization and interest. Another possibility is the establishment of a separate facility or organization within the university—such as a center or an institute—which would begin to pull together related activities, and to focus on interdisciplinary resource problems and situations.
In conclusion, I wish to stress the importance, in this whole problem complex, of educating the public, that is, the consumers of resource development planning. Here your university has accumulated a great deal of relevant experience, stemming from many decades of work in agricultural extension. In considering new activities and new subject fields for adult education, emphasis might well be given to the question of resource development on a community-wide and regional scale.
There is a clear need for the public to be alerted to the sorts of issues that are being discussed here. It is important that we have an intelligent market for our planning activities, a market that knows pretty much what it wants and how it wants to get there. Ultimately, the binding decisions will have to be made by the consumers of planning, not by the specialists. It is a case of where the specialist proposes and the community disposes. I would therefore put great stress upon public education as an integral part of the total education program.
# 7
# Open Space from Natural Processes (1970)
This essay resulted from a study of metropolitan open space sponsored by the U.S. Department of Housing and Urban Development (HUD) and the states of New Jersey and Pennsylvania. McHarg's partner and colleague, David Wallace, edited the book Metropolitan Open Space and Natural Process (published by the University of Pennsylvania Press), which was based on the HUD study. Contributors to the study and the book included McHarg and Wallace's Penn colleagues: Ann Strong, William Grigsby, William Roberts, and Nohad Toulan.
McHarg's very specific study for open space in the Philadelphia metropolitan region became the model for other open space studies. Essentially, the study was based on the "simple working method for open space" that he had been exploring for a decade (see, for example, chapter 2 of this book, "The Place of Nature in the City of Man"). McHarg's concept of protecting environmentally sensitive areas (marshes, swamps, flood plains, steep slopes, and prime habitats) was followed in many subsequent regional open space plans. For example, the 1995 Desert Open Spaces plan for the Phoenix region undertaken by Design Workshop for the Maricopa Association of Governments has many similarities to McHarg's efforts three decades earlier. The concept also influenced environmental planning rule making, as for example, the elevated position wetlands (in 1970 called marches and swamps) eventually received in environmental policy. Although the open space plan for Philadelphia was not implemented, McHarg later collaborated on a similar effort for Tulsa, Oklahoma, which was followed.
There is need for an objective and systematic method of identifying and evaluating land most suitable for metropolitan open space based on the natural roles that it performs. These roles can best be understood by examining the degree to which natural processes perform work for man without his intervention, and by studying the protection which is afforded and values which are derived when certain of these lands are undisturbed.
# Introduction
A million acres of land each year are lost from prime farmland and forest to less sustainable and uglier land uses. There is little effective metropolitan planning and still less implementation of what is planned. Development occurs without reference to natural phenomena. Flood plains, marshes, steep slopes, woods, forests, and farmland are destroyed with little if any remorse; streams are culverted, groundwater, surface water, and atmosphere polluted, floods and droughts exacerbated, and beauty superseded by vulgarity and ugliness. Yet the instinct for suburbia which has resulted in this enormous despoliation of nature is based upon a pervasive and profoundly felt need for a more natural environment.
The paradox and tragedy of metropolitan growth and suburbanization is that it destroys many of its own objectives. The open countryside is subject to uncontrolled, sporadic, uncoordinated, unplanned development, representing the sum of isolated short-term private decisions of little taste or skill. Nature recedes under this careless assault, to be replaced usually by growing islands of developments. These quickly coalesce into a mass of low-grade urban tissue, which eliminate all natural beauty and diminish excellence, both historic and modern. The opportunity for realizing an important part of the "American dream" continually recedes to a more distant area and a future generation. For this is the characteristic pattern of metropolitan growth. Those who escape from the city to the country are often encased with their disillusions in the enveloping suburb.
# The Hypothesis
This pattern of indiscriminate metropolitan urbanization dramatizes the need for an objective and systematic way of identifying and preserving land most suitable for open space, diverting growth from it, and directing development to land more suitable for urbanization. The assumption is that not all land in an urban area needs to be, or even ever is, all developed. Therefore choice is possible. The discrimination which is sought would select lands for open space which perform important work in their natural condition, are relatively unsuitable for development, are self-maintaining in the ecological sense, and occur in a desirable pattern of interfusion with the urban fabric. The optimum result would be a system of two intertwining webs, one composed of developed land and the second consisting of open space in a natural or near natural state.
Heretofore, urbanization has been a positive act of transformation. Open space has played a passive role. Little if any value has been attributed to the natural processes often because of the failure to understand their roles and values. This is all the more remarkable when we consider the high land values associated with urban open space—Central Park in New York, Rittenhouse Square in Philadelphia being obvious examples. This lack of understanding has militated against the preservation or creation of metropolitan open space systems complementary to metropolitan growth. In this situation, governmental restraints are necessary to protect the public from the damaging consequences of private acts which incur both costs and losses to the public, when these acts violate and interrupt natural processes and diminish social values. There is an urgent need for land-use regulations related to natural processes, based upon their intrinsic value and their permissiveness and limitations to development. This in turn requires general agreement as to the social values of natural process.
Planning that understands and properly values natural processes must start with the identification of the processes at work in nature. It must then determine the value of subprocesses to man, both in the parts and in the aggregate, and finally establish principles of development and nondevelopment based on the tolerance and intolerance of the natural processes to various aspects of urbanization. It is presumed that when the operation of these processes is understood, and land-use policies reflect this understanding, it will be evidence that the processes can often be perpetuated at little cost.
The arguments for providing open space in the metropolitan region, usually dependent on amenity alone, can be substantially reinforced if policymakers understand and appreciate the operation of the major physical and biological processes at work. A structure for metropolitan growth can be combined with a network of open spaces that not only protects natural processes but also is of inestimable value for amenity and recreation.
In brief, it is hypothesized that the criteria for metropolitan open space should derive from an understanding of natural processes, their value to people, their permissiveness, and their prohibition to development. The method of physiographic analysis outlined here can lead to principles of land development and preservation for any metropolitan area. When applied as a part of the planning process, it can be a defensible basis for an open space system which goes far toward preserving the balance of natural processes and toward making our cities livable and beautiful.
# Normal Metropolitan Growth Does Not Provide Open Space, Although Land Is Abundant
Without the use of such a method as described earlier, open space is infinitely vulnerable. An examination of the growth in this century of the major metropolitan areas of the United States demonstrates that urbanization develops primarily on open land rather than through redevelopment. The open space interspersed in areas of low-density development within the urban fabric is filled by more intensive uses and open space standards are lowered. Urban growth consumes open space both at the perimeter and within the urban fabric. The result is a scarcity of open space where population and demand are greatest. This phenomenon has aroused wide public concern as the growth of the cities, by accretion, has produced unattractive and unrelieved physical environments. Amenity, breathing space, recreational areas, and the opportunity for contact with nature for an expanding population are diminished. As important, it often exacerbates flood, drought, erosion, and humidity and it diminishes recreational opportunity, scenic, historic, and wildlife resources. Further, the absence of understanding of natural processes often leads to development in locations which are not propitious. When natural processes are interrupted, there are often resultant costs to society.
Demand for urban space is not only relatively but absolutely small. The 37 million inhabitants of megalopolis, constituting 24 percent of the U.S. population, occupied 1.8 percent of the land area of the continental United States (Gottman 1996, p. 26). Only 5 percent of the United States is urbanized today; it is projected that less than 10 percent will be so utilized by the year 2000. Space is abundant. Even in metropolitan regions, where large urban populations exist in a semi-rural hinterland, the proportion of urban to rural land does not basically contradict the assertion of open space abundance. For example, in the Philadelphia Standard Metropolitan Statistical Area (PSMSA), with 3,500 square miles or 2,250,000 acres, only 19.1 percent was urbanized in 1960. Here an increase in population from 4 million to 6,000,000 is anticipated by 1980. Should this growth continue and occur at an average gross density of three families per acre, only 30 percent of the land area would be in urban land use at that time. Some 2,300 square miles or 1,500,000 acres would still remain in open space.
The difficulty in planning lies in the relationship of this open space to urban uses. The market mechanism which raises the unit price of land for urban use tends to inhibit interfusion of open space in the urban fabric. Open space becomes normally a marginal or transitional use, remaining open only while land is awaiting development. The key question then is, if land is absolutely abundant, how can growth be guided in such a way as to exploit this abundance for the maximum good?
# Exceptions to the General Experience
While generally metropolitan growth has been unsympathetic to natural processes, there are exceptions. In the late nineteenth- and early twentieth-century park planning, water courses were an important basis for site selection. The Capper Cromptin Act selected river corridors in Washington, D.C. The Cook County Park System around Chicago consists of corridors of forests preponderantly based upon river valleys. The first metropolitan open space plan, developed for Boston by Charles Eliot in 1893, emphasized not only rivers, but also coastal islands, beaches, and forested hills as site selection criteria. In 1928 Benton MacKaye, the originator of the Appalachian Trail, proposed using open space to control metropolitan growth in America but did not base his open space on natural process.
Patrick Abercrombie's Greater London Plan pays implicit attention to natural process in the location for the satellite towns, in the insistence on open space breaks between nucleated growth, in the recommendation that prime agricultural land should not be urbanized, and in specifying that river courses should constitute a basis for the open space system.
In recent studies conducted by Philip Lewis on a state-wide basis for Illinois and Wisconsin (e.g., State of Wisconsin 1962) physiographic determinants of land utilization have been carried beyond these earlier examples. Corridors have been identified which contain watercourses and their flood plains, steep slopes, forests, wildlife habitats, and historic areas. These characteristics are of value to a wide range of potential supporters—conservationists, historians, and the like—and the studies demonstrate the coincidence of their interests in the corridors. The expectation is that these groups will coordinate their efforts and combine their influence to retain the corridors as open space. Resource development and preservation is advocated for them. In another recent study, ecological principles were developed and tested as part of a planning process for the Green Spring and Worthington valleys, northwest of Baltimore Maryland. Here the design process later described was evolved. Two more elaborate ecological studies, the first for Staten Island (McHarg 1969, pp. 103–15) and the second for the Twin Cities Metropolitan Region in Minnesota (Wallace, McHarg, Roberts, and Todd 1969), have undertaken to analyze natural processes to reveal intrinsic suitabilities for all prospective land uses. These are shown as unitary, complementary, or in competition.
The present study of metropolitan Philadelphia open space is more general in its objective. It seeks to find the major structure of open space in the PSMSA based upon the intrinsic values of certain selected natural processes to set the stage for further investigations.
# Need for the Ecological Approach
There are, of course, several possible approaches. The first of these, beloved of the economist, views land as a commodity and allocates acres of land per thousand persons. In this view nature is seen as a generally uniform commodity, appraised in terms of time-distance from consumers and the costs of acquisition and development. A second approach also falls within the orthodoxy of planning, and may be described as the geometrical method. Made popular by Patrick Abercrombie, the distinguished British planner, this consists of circumscribing a city with a green ring wherein green activities, agriculture, recreation, and the like, are preserved or introduced.
The ecological approach, however, would suggest quite a different method. Beginning from the proposition that nature is process and represents values, relative values would be ascribed to certain discernible processes. Then, operating upon the presumption that nature performs services for man without his intervention or effort, certain service-processes would be identified as social values. Yet further, recognizing that some natural processes are inhospitable to human use—floods, earthquakes, hurricanes—we would seem to discover intrinsic constraints or even prohibitions to man's use or to certain kinds of use.
Objective discussion between the ecologist and the economist would quickly reveal the fallacy of the commodity approach. Nature is by definition not a uniform commodity. In contrast, each and every area varies as a function of historical geology, climate, physiography, the water regimen, the pattern and distribution of soils, plants, and animals. Each area will vary as process, as value and in the opportunities and constraints which it proffers or withholds from human use.
In a similiar discussion between ecologist and green belt advocate, the question which most embarrasses the latter is whether nature is uniform within the belt and different beyond it. The next question is unlikely to receive an affirmative answer, "Does nature perform particular roles within the belt to permit its definition?" Clearly the ecologist emerges a victor in these small skirmishes, but now the burden is upon him. What is the ecological approach to the selections of metropolitan open space?
# The Value of Natural Process in the Ecosystem
There is, at present, no existing ecological model for a city or metropolitan region; it is necessary, therefore, to embark upon a theoretical analysis of natural process without the aid of such a model.
Plant and animal communities require solar energy, food, nutrients, water, protection from climate extremes, and shelter. These conditions must be substantially regular in their provision. In order to ensure these optimal conditions, non-human or primitive-human systems have adapted to the natural environment and its constituent organisms to ensure a complex process of energy utilization, based upon photosynthesis, descending through many food chains to final decomposition and nutrient recirculation. In response to the problem of climatic extremes these communities do modify the microclimate. Such natural systems have mechanisms whereby water movement is modified to perform the maximum work. The aggregate of these processes is a stable, complex ecosystem in which entropy is low and energy is conserved (Odum 1959, ch. 3).
The net result is a system with high energy utilization and production, continuous soil formation, natural defenses against epidemic disease, microclimatic extremes diminished, minimal oscillation between flood and drought, minor erosion, and natural water purification. There obviously are many advantages which accrue to civilized man from this condition—a viable agriculture and forestry, abundant fish and wildlife, natural water purification, stability in the water system, defense against flood and drought, diminished erosion, sedimentation and silting, and a self-cleaning environment with high recreational potential and amenity.
The values of the natural processes far exceed the values which usually are attributed to them. Agriculture, forestry, and fisheries are taken into consideration in the evaluation of regional assets, but atmospheric oxygen, amelioration of climate and microclimate, water evaporation, precipitation, drainage, or the creation of soils tend to be disregarded. Yet the composite picture of the region's resources must include all natural processes. Beginning with the values of agriculture, forestry, and fisheries, the value of minerals and the value of the land for education, recreation, and amenity may be added. Agricultural land has an amenity which is not generally attributed, since it is also a landscape which is maintained as a byproduct. Forests equally have an amenity value and are self-cleaning environments, requiring little or no maintenance.
Water has values which transcend those related to certain discrete aspects of the hydrologic cycle. In this latter category are many important processes—water in agriculture, industry, commerce, recreation, education and amenity, consumption, cooling, hydroelectric generation, navigation, water transport and dilution, waste reduction, fisheries, and water recreation.
Value is seldom attributed to the atmosphere; yet the protection from lethal cosmic rays, insulation, the abundance of oxygen for animal metabolism and the carbon dioxide for plant metabolism which it affords, all demonstrate an indispensability equal to land and water. In terms of positive attributed value the atmosphere has been accorded virtually none. Only when atmosphere has become polluted are the cost and necessity of ensuring clean air recognized.
Even in the exceptional condition when natural processes are attributed value as in agriculture and forestry, these are generally held in such low esteem that they cannot endure in the face of competition from urban or industrial uses. It is impossible to construct a value system which includes the vast processes described. It is, however, quite possible to recognize the fundamental value of these processes, their characteristics, and their relationship to industrial and urban processes. This understanding should lead to a presumption in favor of nature rather than the prevailing disdain.
Working toward the goal of developing working principles for land-use planning in general and the selection of metropolitan open space in particular, it is advantageous to examine the degree to which natural processes perform work for man without his intervention and the protection achieved by leaving certain sub-processes in their natural state without development. While this cannot yet be demonstrated quantitatively, it can be described.
Natural processes which perform work for man include water purification, atmospheric pollution dispersal, microclimate amelioration, water storage and equalization, flood control, erosion control, topsoil accumulation, and the ensurance of wildlife populations.
Areas which are subject to volcanic action, earthquakes, tidal waves, tornadoes, hurricanes, floods, drought, forest fires, avalanches, mud slides, and subsidence, should be left undeveloped in order to avoid loss of life and property. In addition, there are other areas which are particularly vulnerable to human intervention; this category includes beach dunes, major animal habitats, breeding grounds, spawning grounds, and water catchment areas. There are also areas of unusual scenic, geological, biological, ecological, and historic importance. In each of these cases, it is apparent that wise land-use planning should recognize natural processes and respond to them. As many of these processes are water related, it would seem that water may be a useful indicator of these major physical and biological processes described as natural processes.
# Water and Natural Processes
Water, as the agent of erosion and sedimentation, is linked with geological evolution to the realities of physiography. The mountains, hills, valleys, and plains which result, experience a variety of climate and microclimate consequent upon their physiography. The combination of physiography and climate determines the incidence and distribution of plants and animals, their niches and habitats.
The use of water as a unifying concept links marsh and forest, rivers and clouds, ground and surface water, all as interdependent aspects of a single process. It permits us to see that the marsh filled in the estuary and the upland forest felled are comparable in their effect, that pollution to surface water may affect distant groundwater, that building of an outlying suburb may affect the flood height in the city. Although we lack an ecological model, a gross perception of natural process may be revealed through the selection of water as a unifying process. This may suggest land-use policies reflecting the permissiveness and prohibitions of the constituent phases of water in process. By this useful method the constituent roles of open space may be seen and the optimal distribution of open space in the metropolitan region may be discerned.
Water is not the best indicator or theoretical tool for ecological planning. The physiographic region is perhaps the best unit for ecological studies since there tends to be a marked consistency within each physiographic region and distinct variations between them.
# Water and the Roles of Major Physiographic Regions
In the Philadelphia metropolitan area study, the roles of the physiographic regions have been simplified to three components: the first, the uplands of the piedmont, second, the remainder of that region with the final category being the coastal plain. The area to be studied is the PSMSA three and one-half thousand square miles which constitute the metropolitan region and straddle coastal plain and piedmont, a situation typical of many cities on the eastern seaboard.
# The Uplands
The uplands are the hills of the watershed, the highest elevations, wherein many streams begin, their watercourses narrow, steep, and rocky. The soils tend to be thin and infertile from long-term erosion. In the Philadelphia metropolis, the uplands consist of a broad band of low hills, 12 to 20 miles wide, bending northeast-southwest obliquely through the area. As a result of the absence of glaciation in this area the rivers and streams lack natural impoundments and are, therefore, particularly susceptible to seasonal fluctuations.
The natural restraints to flooding and drought, contributing to equilibrium, are mainly the presence and distribution of vegetation, principally forests and their soils, particularly on the uplands and their steep slopes. Vegetation absorbs and utilizes considerable quantities of water. In fact, vegetation and their soils act as a sponge restraining extreme runoff, releasing water slowly over longer periods, diminishing erosion and sedimentation—in short, diminishing the frequency and intensity of oscillation between flood and drought and operating toward equilibrium.
In the uplands, land-use policies should protect and enhance forests and other vegetative cover to diminish runoff, oscillation between flood and drought, and the insurance of natural water purification. On steep slopes, land-use management would notably include forestation programs. Land uses should then be related to permissiveness and prohibitions inherent in such a region related to the primary role of the upland sponge in the water economy.
# The Piedmont
The piedmont is also non-glaciated in the study area, and consists of the gentler slope below the uplands. It sustains fertile soils which, in limestone areas, are equal in fertility to the richest in the entire United States. In the three divisions it is in the piedmont alone that fertile soils abound. Unglaciated, the piedmont, like the uplands, lacks natural impoundment and is flood prone.
Here is the land most often developed for agriculture. These lands, too, tend to be favored locations for villages, towns, and cities. Here, forests are often residues or the products of regeneration on abandoned farms. Steep slopes in the piedmont are associated with the dissected banks of streams and rivers.
The agricultural piedmont does not control its own defenses. It is either affected by or defended from flood and drought by the conditions in the uplands and coastal plain. When cropland is ploughed and lacks vegetation, when building sites are bared, they are subject to erosion and contribute to sediment yield. Even when covered with growing crops, the average runoff here exceeds the forest. Nonetheless, the vegetative cover, configuration, and conservation practices in the agricultural piedmont can either increase or diminish flood and drought. The piedmont is particularly vulnerable to both. The presence of forests and woodlands will act as ameliorative agents as they do in the uplands and the presence of vegetation will diminish runoff by absorption, friction, and through increased percolation.
The fine capillary streams of the uplands become the larger streams and rivers of the piedmont, and their volume increases proportionately, as does their flood potential and their oscillation to low flow.
In the piedmont, fertile soils should be perpetuated as an irreplaceable resource. Both agriculture and urbanization are primary contributors to erosion and sedimentation; they are also the major contributors to water pollution from pesticides, fertilizers, domestic and industrial wastes.
Planning should relate water pollution to specific stream capacities; withdrawals of water to specific capacities of surface water and groundwater; conservation practices to erosion control for farmland and construction sites.
# The Coastal Plain
The coastal plain has been, through much of geologic time, a vast depository of sediments eroded from the uplands. The soils are shallow and acid, overlaying sedimentary material. These are naturally infertile although in New Jersey such sandy soils with abundant fertilizer and water, support extensive truck farming.
The major physiographic characteristics of the coastal plain are its flatness, the poverty of its soils, the abundant marshes, bays, estuaries, the unique flora of the pine barrens, and finally, the great resources of groundwater in aquifers.
The incidence of flood and drought in the piedmont and coastal plains is not only consequent upon the upland sponge, but also upon estuarine marshes, particularly where these are tidal. Here, at the mouth of the watershed, at the confluence of important rivers or river and sea, the flood components of confluent streams or the tidal components of flood assume great importance. In the Philadelphia metropolitan area, the estuary and the ocean are of prime importance as factors in flood.
A condition of intense precipitation over the region combined with high tides, full estuary, and strong on-shore winds brings together the elements that portend floods. The estuarine marshes and their vegetation constitute the major defense against this threat. These areas act as enormous storage reservoirs, absorbing mile-feet of potentially destructive waters, thus reducing flood potential. This function may be described as the estuarine sponge, in contrast to the upland sponge. The water resources of these aquifers represent a significant economic value.
In the coastal plain, the susceptibility to flood of these areas and the invaluable role of marshlands as flood storage reservoirs should also be reflected in land-use policies which could ensure the perpetuation of their natural role in the water economy. The value of groundwater in aquifers should be reflected in land-use policy.
Finally, the extensive forests of the pine barrens and their unique flora depend upon fire, which is an inevitable and recurrent threat. Here extensive urbanization makes such a threat actuality: pine barrens are not suitable sites for development.
These three major divisions are clearly different in their permissiveness and prohibition, and in their roles in the water regimen: the uplands should be viewed and planned as the upstream control area for water processes, flood, drought, water quality, erosion control, and an area of high recreational potential. This region is not normally selected for extensive urbanization and can be easily protected from it. But it performs its role best when extensively forested.
The coastal plain performs a primary flood control, water supply, and water-related recreation function, and is not suited to extensive urbanization. The entire region, characterized by rivers, marshes, bays, and estuaries is critical to wildlife. Properly used, it offers a great recreational potential. In contrast, the piedmont, with the exception of prime agricultural areas, is tolerant to urbanization. Although the prime farmland of this region is an irreplaceable resource, a scenic value, and requires defense against urbanization, the piedmont should be the region for major urban growth in the future.
Planning for natural processes at this scale would regulate urban development in uplands and coastal plains, concentrate it in the piedmont, on non-prime agricultural land. Connecting these regions would be an undeveloped fabric of streams, rivers, marshes, bays, and estuaries, the steep slopes of dissected watercourses, as water corridors permeating the entire metropolis.
# Significant Physiographic Phenomena and Their Roles
The major physiographic divisions reveal the principal roles in the water regimen and should constitute an important generalized basis for identification of natural processes and planning. Yet it is necessary to have a greater specificity as to the constituent roles of natural process. Toward this end, eight components have been selected for identification and examination. It is clear that they are to some extent identifiable and perform discrete roles. These eight constituent phenomena are:
1. Surface water
2. Marshes
3. Flood plains
4. Aquifers
5. Aquifer recharge areas
6. Prime agricultural land
7. Steep lands
8. Forests and woodlands
The first five have a direct relationship to water; the remaining three are water-related in that they either have been determined by, or are determinants of water-in-process.
This group of water-related parameters is selected as containing the most useful indicators of natural process for a majority of metropolitan regions in the United States. They appear to be the most productive for the project area under study, but the underlying hypothesis of natural process would take precedence in these areas where other parameters prove to be more illuminating. That is, the selection of criteria for a system of metropolitan open space can best be developed from an understanding of the major physical and biological process of the region itself. The interconnected phenomena should be integrated into a unified system to which the planning process responds.
# Varying Permissiveness to Urban Uses
Having identified certain sub-processes, it is necessary to describe their intrinsic function and value and then to determine the degree to which the performance of natural process is prohibitive or permissive to other land uses.
The terms prohibitive and permissive are used relatively. In no case does natural process absolutely prohibit development. Yet in a condition of land abundance, there is available a choice as to location of development. This being so, areas of importance to natural process should not be selected where alternative locations of lesser importance are available. Further, development of natural process areas should occur only where supervening benefits result from such development, in excess of those provided by natural process. Still further, the tolerances of natural processes to development by type, amount, and location are an important constituent of land-use policy.
# Surface Water (5,671 linear miles)
In a situation of land abundance, restraints should be exercised on the development of surface water and riparian land. In principle, only those land uses which are inseparable from waterfront locations should occupy riparian lands; land uses thereon should be limited to those which do not diminish the present or prospective value of surface water or stream banks for water supply, amenity, or recreation.
In the category of consonant land uses would fall port and harbor facilities, marinas, water treatment plants, sewage treatment plants, water-related industry, and in certain instances, water-using industries.
In the category of land uses which need not diminish existing or prospective value of surface water would fall agriculture, forestry, institutional open space, recreational facilities, and, under certain conditions, open space for housing.
In the category of land uses which would be specifically excluded would be industries producing toxic or noxious liquid effluents, waste dumps, non-water-related industry or commerce.
The presumption is then that surface water as a resource best performs its function in a natural state and that land uses which do not affect this state may occupy riparian land, but other functions be permitted according to the degree which their function is indivisibly water-related.
# Marshes (133,984 acres; 7.44 percent)
In principle, land-use policy for marshes should reflect the roles of flood and water storage, wildlife habitat, and fish spawning grounds. Land uses which do not diminish the operation of the primary roles are compatible. Thus, hunting, fishing, sailing, recreation, in general, would be permissible. Certain types of agriculture, notably cranberry bogs, would also be compatible. Isolated urban development might also be permitted if the water storage role of marshes was not diminished by the filling and accelerated runoff that such development would entail.
# Flood Plains (339,760 acres; 18.86 percent)
The flood-plain parameter must be attributed a particular importance because of its relation to loss of life and property damage. The best records seem to indicate that the incidence and intensity of flooding in metropolitan areas is on the increase (Witala 1961). The presumption is that this results from the reduction of forest and agricultural land, erosion and sedimentation, and the urbanization of watersheds. This being so, there is every reason to formulate a land utilization policy for flood plains related to safeguarding life and property.
The incidence of floods may be described as recorded maxima. For the Delaware River, the maximum recorded floods are those of 1950 and 1955. The alternate method of flood description relates levels of inundation to cyclical storms and describes these as flood levels of anticipated frequency or probability.
There is, then, a conflict of use for this land between water and other types of occupancy. This conflict is most severe in areas of frequent flooding which, however, occupy the smallest flood plain. The conflict diminishes in frequency with the more severe floods, which occupy the largest flood plain. It would seem possible to relate the utilization of the flood plain to the incidence and severity of cyclical flooding.
Increasingly, the 50-year or two percent probability flood plain is being accepted as that area from which all development should be excluded save those functions which are either benefited or unharmed by flooding or those land uses which are inseparable from flood plains.
Thus, in principle, only such land uses which are either improved or unharmed by flooding should be permitted to occupy the 50-year flood plain. In the former category fall agriculture, forestry, recreation, institutional open space, and open space for housing. In the category of land uses inseparable from flood plains are ports and harbors, marinas, water-related industry, and under certain circumstances, water-using industry.
# Aquifers (149,455 acres; 8.3 percent)
The definition of an aquifer as a water-bearing stratum of rock, gravel, or sand is so general that enormous areas of land could be so described. For any region the value of an aquifer will relate to the abundance or poverty of water resources. In the Philadelphia area the great deposits of porous material parallel to the Delaware River are immediately distinguishable from all other aquifers in the region by their extent and capacity.
Aquifers may vary from groundwater resources of small quantity to enormous underground resources. They are normally measured by yields of wells or by the height of the water table. The aquifer in New Jersey parallel to the Delaware River has been estimated by the Soil Conservation Service [currently called the Natural Resources Conservation Service] to have a potential capacity of one billion gallons per day. This valuable resource requires restraints upon development of the surface to ensure the quality and quantity of aquifer resources. Consequently, development using septic tanks and industries disposing toxic or noxious effluents should be regulated. Injection wells should be expressly prohibited. The matter of surface percolation is also important, and as percolation will be greatest from permeable surfaces there are good reasons for maximizing development at the extremes of density—either sewered high density development or very low density free standing houses. Land-use policy for aquifers is less amenable to generalized recommendation than the remaining categories, as aquifers vary with respect to capacity, yield, and susceptibility. Consequently, there will be ranges of permissiveness attributable to specific aquifers as a function of their role and character.
In principle, no land uses should be permitted above an aquifer which inhibit the primary role as water supply and reservoir regulating oscillations between flood and drought.
Agriculture, forestry, and recreation clearly do not imperil aquifers. Industries, commercial activities, and housing, served by sewers, are permissible up to limits set by percolation requirements. Sources of pollutants or toxic material, and extensive land uses which reduce soil permeability, should be restricted or prohibited.
# Aquifer Recharge Areas (83,085 acres; 4.61 percent)
Such areas are defined as points of interchange between surface water and groundwater. In any water system certain points of interchange will be critical; in the Philadelphia metropolitan area the interchange between the Delaware River, its tributaries and the parallel aquifer, represents the recharge area which is most important and which can be clearly isolated. Percolation is likely to be an important aspect of recharge. Thus, two considerations arise: the location of surface-to-groundwater interchange below the ground, and percolation of surface to groundwater.
By careful separation of polluted river water from the aquifer and by the impounding of streams transecting the major aquifer recharge areas, the aquifer can be managed and artificially recharged. Groundwater resources can be impaired by extensive development which waterproofs the surfaces, by development which occupies desirable impoundments valuable for aquifer management and by pollution.
In principle, all proposed development on aquifer recharge areas should be measured against the likely effect upon recharge. Injection wells and disposal of toxic or offensive materials should be forbidden; channel widening, deepening, and dredging should be examined for their effect upon recharge as should deep excavations for trunk sewers. Surface development and sewage disposal on such an area should be limited by considerations of percolation.
# Prime Agricultural Land (248,816 acres; 11.7 percent)
Prime agricultural soils represent the highest level of agricultural productivity; they are uniquely suitable for intensive cultivation with no conservation hazards. It is extremely difficult to defend agricultural lands when their cash value can be multiplied tenfold by employment for relatively cheap housing. Yet, the farm is the basic factory, the farmer is the country's best landscape gardener and maintenance work force, the custodian of much scenic beauty. Utilization of farmland by urbanization is often justifiable as the highest and best use of land at current land values, yet the range of market values of farmlands does not reflect the long-term value or the irreplaceable nature of these living soils. An omnibus protection of all farmland is indefensible; protection of the best soils in a metropolitan area would appear not only defensible, but also clearly desirable.
Jean Gottman has recommended that "the very good soils are not extensive enough in Megalopolis to be wastefully abandoned to nonagricultural uses" (1961, p. 95). The soils so identified are identical to the prime agricultural soils in the metropolitan area.
While farmland is extremely suitable for development and such lands can appreciate in value by utilization for housing, it is seldom considered that there is a cost involved in the development of new farmland. The farmer, displaced from excellent soils by urbanization, often moves to another site on inferior soils. Excellent soils lost to agriculture for building can finally only be replaced by bringing inferior soils into production. This requires capital investment. Land that is not considered cropland today will become cropland tomorrow, but at the price of much investment.
In the PSMSA by 1980 only 30 percent of the land area will be urbanized; 70 percent will remain open. Prime agricultural lands represent only 11.7 percent of the area. Therefore, given a choice, prime soils should not be developed.
In principle, U.S. Department of Agriculture (USDA) Land Capability System Type 1 soils are recommended to be exempted from development (save by those functions which do not diminish their present or prospective productive potential). This would suggest retirement of prime soils into forest, utilization as open space for institutions, for recreation, or in development for housing at densities no higher than one house per 25 acres.
# Steep Lands (262,064 acres; 14.55 percent)
Steep lands and the ridges which they constitute are central to the problems of flood control and erosion. Slopes in excess of 12° are not recommended for cultivation by the Soil Conservation Service. The same source suggests that for reasons of erosion, these lands are unsuitable for development. The recommendations of the Soil Conservation Service are that steep slopes should be in forest and that cultivation of such slopes be abandoned.
In relation to its role in the water regimen, steepness is a matter not only of degree, but also of vegetation and porosity. Two ranges of slopes are identified in our study of the PSMSA: 15 to 25 percent, and greater than 25 percent. The first category of 15 percent is identified as those slopes for which major site-engineering begins to be necessary to accommodate development. Roads should be equally parallel to the slope rather than perpendicular. Coverage by houses and roads should be compensated by measures to slow down runoff. Examination of a number of examples suggests 15 percent as the danger point for any development at all, but further empiric research is necessary. Above 25 percent, however, there appears widespread agreement that no development should occur and land should be treated to decrease runoff as much as possible.
In summary, erosion control and diminution of velocity of runoff are the principal water problems of steep slopes. Land uses compatible with minimizing these problems would be mainly forestry, recreation, and low-density development on the less-steep slopes. Since such slopes also tend to dominate landscapes, their planting in forests would provide great amenity.
# Forests and Woodlands (588,816 acres; 32.7 percent)
The natural vegetative cover for most of this region is forest. Where present, this exerts an ameliorative effect upon microclimate; it exercises a major balancing effect upon the water regimen, diminishing erosion, sedimentation, flood, and drought. The scenic role of woodlands is apparent, as is the provision of a habitat for game. The recreational potential of forests is among the highest of all categories. In addition, the forest is a low maintenance, self-perpetuating landscape, a resource in which accrual of timber inventory is continuous.
Forests can be employed for timber production, water management, wildlife habitats, as airsheds, recreation, or for any combination of these uses. In addition, forests can absorb development in concentrations which will be determined by the demands of natural process which they are required to satisfy. Where scenic considerations alone are operative, mature forests could absorb housing up to a density of one house per acre without loss of their forest aspect.
Land uses for forests should be determined by the natural process roles which the forest is required to play in the water regimen.
As a result of analyzing the eight significant physiographic phenomena and their roles in the region, several land uses can be recommended. These land uses are summarized in table 2.
Table 2. Limited Development Areas and Recommended Land Uses
Limited Development Areas | Recommended Land Uses
---|---
1. Surface water and riparian lands | Ports, harbors, marinas, water treatment plants, water-related industry, open space for institutional and housing use, agriculture, forestry and recreation
2. Marshes | Recreation
3. 50-year flood plains | Ports, harbors, marinas, water treatment plants, water-related and water-using industry, agriculture, forestry, recreation, institutional open space, open space of housing
4. Aquifers | Agriculture, forestry, recreation, industries which do not produce toxic or offensive effluents, all land uses within limits set by percolation
5. Aquifer recharge areas | As aquifers
6. Prime agricultural lands | Agriculture, forestry, recreation, open space for institutions, housing at 1 house per 25 acres
7. Steep lands | Forestry, recreation, housing at maximum density of 1 house per 3 acres, where wooded
8. Forests and woodlands | Forestry, recreation, housing at densities not higher than 1 house per acre
# Open Space and Airsheds
The atmosphere of a metropolitan area has a capacity to diffuse air pollution based upon temperature, air composition, and movement. Concentration of pollution is associated with cities and industries, replacement of polluted air and its diffusion depend upon air movements over pollution free areas. These will be related to wind movements, and must be substantially free of pollution sources if relief is to be afforded to air pollution over cities. One can use the analogy of the watershed and describe the pollution free areas, tributary to the city, as the airsheds.
The central phase of air pollution is linked to temperature inversion during which the air near the ground does not rise to be replaced by in-moving air. Under inversion, characterized by clear nights with little wind, the earth is cooled by long-wave radiation and the air near the ground is cooled by the ground. During such temperature inversions with stable surface air layers, air movement is limited; in cities, pollution becomes increasingly concentrated. In Philadelphia "significant" inversions occur one night in three. Parallel and related to inversion is the incidence of "high" pollution levels, which occurred on twenty-four "episodes" from two to five days in duration between 1957 and 1959. Inversions then are common as are "high" levels of pollution. The danger attends their conjunction and persistence. Relief other than elimination of pollution sources is a function of wind movement to disperse pollution over cities, and secondly, the necessity that in-moving air be cleaner than the air it replaces.
The windrose during inversions can establish the percentage direction of winds which might relieve pollution; the wind speed, combined with wind direction, will indicate these tributary areas over which the wind will pass to relieve pollution. These areas should be substantially, free of pollution sources.
The concentration of pollution sources in Philadelphia covers an area fifteen miles by ten miles with the long axis running approximately northeast. Let us assume sulfur dioxide to be the indicator of pollution (830 tons per day produced), an air height of 500 feet as the effective dimension, an air volume to be replaced of approximately 15 cubic miles, a wind speed of four miles per hour, selected as a critical speed. Then one cubic mile of ventilation is provided per mile of windspeed and it is seen to require three and three-quarter hours for wind movement to ventilate the long axis, two and one-half hours to ventilate the cross axis. Thus, the tributary to ensure clean air on the long axis is 15 miles beyond the pollution area, ten miles beyond for the cross axis. The windrose for Philadelphia during inversions shows that wind movements are preponderantly northwest, west, and southwest, contributing 51.2 percent of wind movements, the other five cardinal points represent the remainder.
This very approximate examination suggests that airsheds should extend from 10 to 15 miles beyond the urban air pollution sources in those wind directions to be anticipated during inversion. The width of these belts should correspond to the dimension of the pollution core and, in very approximate terms, would probably be from three to five miles. Such areas, described as airsheds, should be prohibited to pollution source industries.
Should this concept be realized, broad belts of land, free of industry, would penetrate radially toward the city center.
Under the heading of atmosphere the subject of climate and microclimate was raised. In the study area the major problem is summer heat and humidity. Relief of this condition responds to wind movements. Thus, a hinterland with more equable temperatures, particularly a lower summer temperature, is of importance to climate amelioration for the city. As we have seen, areas which are in vegetative cover, notably forests, are distinctly cooler than cities in summer, a margin of 10°F is not uncommon. Air movements over such areas moving into the city will bring cooler air. Relief from humidity results mainly from air movement. These correspond to the directions important for relief of inversion. We can then say that the areas selected as urban airsheds are likely to be those selected as appropriate for amelioration of the urban microclimate. However, in the case of the former, it is important only that pollution sources be prohibited or limited. In the case of microclimate control, it is essential that the airsheds be substantially in vegetative cover, preferably forested.
The satisfaction of these two requirements, the creation of urban airsheds as responses to atmospheric pollution control and microclimate control, would create fingers of open space penetrating from the rural hinterland, radially into the city. This is perhaps the broadest conception of natural process in metropolitan growth and metropolitan open space distribution. Clearly, this proposal directs growth into the interstices between the airshed corridors and suggests that metropolitan open space exists within them.
# Conclusions
In summary, it is proposed that the form of metropolitan growth and the distribution of metropolitan open space should respond to natural process. The phenomenal world is a process which operates within laws and responds to these laws. Interdependence is characteristic of this process, the seamless web of nature. Man is natural, as is the phenomenal world he inhabits, yet with greater power, mobility, and fewer genetic restraints; his impact upon this world exceeds that of any creature. The transformations he creates are often deleterious to other biological systems, but in this he is no different from many other creatures. However, these transformations are often needlessly destructive to other organisms and systems, and even more important, by conscious choice and inadvertance, also deleterious to man.
A generalized effect of human intervention is the tendency toward simplification of the ecosystems, which is equated with instability. Thus, the increased violence of climate and microclimate, oscillation between flood and drought, erosion and siltation, are all primary evidence of induced instability.
Human adaptations contain both benefits and costs, but natural processes are generally not attributed values, nor is there a generalized accounting system which reflects total costs and benefits. Natural processes are unitary whereas human interventions tend to be fragmentary and incremental. The effect of filling the estuarine marshes or of felling the upland forests is not perceived as related to the water regimen, to flood or drought; nor are both activities seen to be similar in their effect. The construction of outlying suburbs and siltation of river channels are not normally understood to be related as cause and effect; nor is waste disposal into rivers perceived to be connected with the pollution of distant wells.
Several factors can be observed. Normal growth tends to be incremental and unrelated to natural processes on the site. But the aggregate consequences of such development are not calculated nor are they allocated as costs to the individual incremental developments. While benefits do accrue to certain developments, which are deleterious to natural processes at large (for example, clear felling of forests or conversion of farmland into subdivisions), these benefits are particular (related in these examples to that landowner who chooses to fell trees or sterilize soil), while the results and costs are general in effect. Thus, costs and benefits are likely to be attributed to large numbers of different and unrelated persons, corporations, and levels of government. It is unprovable and unlikely that substantial benefits accrue from disdain of natural process; it is quite certain and provable that substantial costs do result from this disdain. Finally, in general, any benefits which do occur—usually economic—tend to accrue to the private sector, while remedies and long-range costs are usually the responsibility of the public domain.
The purpose of this study is to show that natural process, unitary in character, must be so considered in the planning process that changes to parts of the system affect the entire system, that natural processes do represent values, and that these values should be incorporated into a single accounting system. It is unfortunate that there is inadequate information on cost-benefit ratios of specific interventions to natural process. However, certain generalized relationships have been shown and presumptions advanced as the basis for judgment. It seems clear that laws pertaining to land use and development need to be elaborated to reflect the public costs and consequences of private action. Present land-use regulations neither recognize natural processes, the public good in terms of flood, drought, water quality, agriculture, amenity, or recreational potential, nor allocate responsibility to the acts of landowner or developer.
We have seen that land is abundant, even within a metropolitan region confronting accelerated growth. There is, then, at least hypothetically, the opportunity of choice as to the location of development and locations of open space.
The hypothesis, central to this study, is that the distribution of open space must respond to natural process. The conception should hold true for any metropolitan area, irrespective of location. In this particular case study, directed to the Philadelphia metropolitan region, an attempt has been made to select certain fundamental aspects of natural process, which show the greatest relevance to the problem of determining the form of metropolitan growth and open space.
The problem of metropolitan open space lies then, not in absolute area, but in distribution. We seek a concept which can provide an interfusion of open space and population. The low attributed value of open space ensures that it is transformed into urban use within the urban area and at the perimeter. Normal urbanization excludes interfusion and consumes peripheral open space.
Yet as the area of a circle follows the square of the radius, large open space increments can exist within the urban perimeter without major increase to the radius or to the time distance from city center to urban fringe.
The major recommendation of this study is that the aggregate value of land, water, and air resources does justify a land-use policy which reflects both the value and operation of natural processes. Further, that the identification of natural processes, the permissiveness and prohibitions which they exhibit, reveals a system of open space which can direct metropolitan growth and offers sites for metropolitan open space.
The characteristics of natural processes have been examined; an attempt has been made to identify their values, intrinsic value, work performed and protection afforded. Large-scale functions have been identified with the major divisions of upland, coastal plain, and piedmont; smaller scale functions of air and water corridors have been identified; and, finally, eight discrete parameters have been selected for examination.
For each of the discrete phenomena and for each successive generalization, approximate permissiveness to other land uses and specific prohibitions have been suggested. While all are permissive to a greater or lesser degree, all perform their natural process best in an unspoiled condition. Clearly, if land is abundant and land-use planning can reflect natural process, a fabric of substantially natural land will remain either in low intensity use or undeveloped, interfused throughout the metropolitan region. It is from this land that public metropolitan open space may best be selected.
This case study reveals the application of the ecological view to the problem of selecting open space in a metropolitan region. It reflects the assumption that nature performs work for man and that certain natural processes can best perform this work in a natural or mainly natural condition. Clearly, this is a partial problem; one would wish that simultaneously, consideration were also given to those lands which man would select for various purposes, for settlements, recreation, agriculture, and forestry. Such a study would be more complete than the isolation of a single demand. Yet, it is likely that the same proposition would hold although the larger study would better reveal the degree of conflict. For the moment, it is enough to observe that the ecological view does represent a perceptive method and could considerably enhance the present mode of planning which disregards natural processes, all but completely, and which in selecting open space, is motivated more by standards of acres per thousand for organized sweating, than for the place and face of nature in man's world.
# Notes
Wallace-McHarg Associates (1964). William G. Grigsby was economic consultant, Ann Louise Strong, governmental and legal consultant, and William H. Roberts, design consultant in this first practical application of the new approach. [Wallace-McHarg Associates was later Wallace, McHarg, Roberts, and Todd and is now Wallace Roberts and Todd.]
Ecological model is a theoretical construct, either descriptive or mathematical, by which the energy flow of an organic system can be described.
5,671 linear miles in the PSMSA; surface water has been identified as all permanent streams shown on USGS (U.S. Geological Survey) 1:24,000 Series.
Kates, William, and White, suggest classification of flood zones into prohibitive, restrictive, and warning zones, based on physiographic analysis. The prohibitive zone would protect structures and fill to preserve the channel capacity in flood conditions; the restrictive zone would simply alert users they were within the flood plain and the decision to accommodate would be theirs (Kates et al. 1962).
See Kates and William (1962) for thorough discussion of the conditions and value judgments concerning occupancy of flood plains. It is evident from this analysis that flood controls are most easily established where the certainty of flood occurrence is high.
See Burton (1962) for a more detailed consideration of agricultural occupancy of flood plains.
E.g., in Pittsburgh, 25 percent and greater slopes are now subject to an ordinance prohibiting further development.
Study on the Philadelphia airshed conducted under direction of the writer by Hideki Shimizu, Department of Landscape Architecture, University of Pennsylvania, 1963, unpublished.
# References
Burton, Ian. 1962. Types of Agricultural Occupancy of Flood Plains in the U.S.A. Chicago: Department of Geography, University of Chicago.
Gottman, Jean. 1961. Megalopolis. New York: The Twentieth Century Fund.
Kates, C., and Robert William. 1962. Hazard and Choice Perception in Flood Plain Management. Chicago: Department of Geography, University of Chicago.
Kates, C., Robert William, and Gilbert F. White. 1962. "Flood Hazard Evaluation." In Gilbert F. White, ed., Papers on Flood Problems. Chicago: Department of Geography, University of Chicago, pp. 135–147.
McHarg, Ian. 1969. "Processes as Values." In Design with Nature. Garden City, N.Y: Doubleday, Natural History Press.
MacKaye, Benton. 1928. The New Exploration: A Philosophy of Regional Planning. New York: Harcourt Brace.
Odum, Eugene. 1959. The Fundamentals of Ecology. Philadelphia: Saunders.
State of Wisconsin. 1962. Recreation in Wisconsin. Madison: Department of Resource Development.
Wallace-McHarg Associates. 1964. Plan for the Valleys. Towson, Md.: Green Spring and Worthington Valley Planning Council.
Wallace, McHarg, Roberts, and Todd. 1969. An Ecological Study for the Twin Cities Metropolitan Region, Minnesota. Prepared for Metropolitan Council of the Twin Cities Area. Philadelphia: U.S. Department of Commerce, National Technical Information Series.
Witala, S. W. 1961. Some Aspects of the Effect of Urban and Suburban Development on Runoff. Lansing, Mich.: U.S. Department of Interior, Geological Survey.
# 8
# Must We Sacrifice the West?(1975)
In the midst of the "energy crisis" of the 1970s, McHarg was invited to the fifth Vail Symposium in Vail, Colorado, on August 12–17. The theme of the symposium was growth options for the Rocky Mountain West. McHarg's paper was subsequently published in a book produced in conjunction with the University of Denver Research Institute, edited by Terrell Minger and Sherry Oaks, with an introduction by Tom McCall, the former governor of Oregon and champion of land-use planning.
The symposium was held in a relatively small room packed with leading television journalists such as Walter Cronkite and Roger Mudd. President Gerald Ford and many members of his energy and environmental staffs were in attendance. McHarg based his observations on Howard T. Odum's net energy model. He posed the question, "How much energy do you spend for the energy received and what are the environmental consequences?" This type of cost-benefit analysis seeks to understand how much energy must be employed for each unit of extracted energy. McHarg observed that the energy that could be derived from oil shale in Colorado would be less than that expended to extract it. Subsequently, plans to exploit oil shale in Colorado were abandoned.
I know a great deal about the sacrifices that people are required to make for energy. For I was born outside of Glasgow, and energy exploitation, mineral exploitation, and sacrifice have been synonymous for at least two hundred years in that part of the world. It was in Scotland, my native country, that broadbank coal and iron were discovered. We wondered what to do with all those metallic lumps. Then somebody discovered how to process the iron into a useful form—and the industrial revolution began.
This revolution started in Scotland, moved to northeastern England, South Wales, Germany, the Lorraine, and finally to the United States. With the advent of heavy equipment, strip mining was made possible, allowing us to steal more minerals from the earth while simultaneously wreaking greater havoc on the environment. Now, the industrial revolution is poised for the attack on the West. You may be the next victim of "progress."
So the real question is: "How can we solve our energy needs without obliterating the landscape, destroying our civilization, and then, in the long run, finding our energy supplies exhausted?" This is a complex question; not one that merits a simple answer. But I think I have sufficient time to outline what we should be doing, lest we find that we have exhausted our energy resources. I shall talk about policies for ranking our energy supplies, measuring the true costs of different energy sources, and the concept of computing the net energy consumed. Since I am a "planner" by training, I'm obliged to throw in a plug for planning in America. And finally, I'll try to be obnoxious and contentious by proposing a strategy for implementing some of these concepts.
For us to organize our energy sources in terms of high and low priorities, we must look to nature for direction. We must realize that we are actually confronting a biological imperative that nature has been solving for 2 ½ billion years. Evolution has an incredible energy component: A surviving organism has to be able to find a habitable environment, one that produces the support ingredients necessary to sustain the organism and ensure its survival. Energy is an important ingredient.
The requirement of finding and utilizing energy efficiently has been a precondition for evolutionary success. Nature has always provided ample energy resources for the organisms. Their survival has been determined by their ability to effectively use the energy while expending a minimum amount of work to process it.
The problem we are experiencing now—that of using non-renewable resources—was also the situation during the earth's formative years before photosynthesis existed. During that time, all organisms were engaged in consumptive, depletive processes. Then came the miracle of photosynthesis. I see it now in my mind's eye: a chloroplastic plant in the ocean has just discovered photosynthesis. It says to the sun, "Sun, do you mind if I have some of your energy?" And the sun says, "Sure, but you know the second law of thermodynamics: you've got to give it back." The little plant says, "Sure, I just want to keep some of the sunlight for a little while, then I'll give it back in the form of infrared." So, in the presence of carbon dioxide and water, the little old chloroplast holds its hands up to the sun, and the sunlight falls upon it and is transmuted into its being. In this process, heat is lost. Here we have a miracle, based on the capability of chloroplast to entrap energy and use it for living.
Now that's the solution to the energy problem—using the renewable energy supplies that nature has been using for billions of years. Simply picking up coal (which is really ancient, stored sunlight) is not the answer because someday we will exhaust the supplies. I sometimes wonder who gave the franchises to utility companies to burn the stored sunlight, blowing it up their smokestacks, watching it being wasted. That's not the way nature works. That's not what successful evolution depends on. Successful evolution depends upon meeting the thermodynamic tests, of utilizing the least necessary amount of energy by recycling and conserving it to ensure its continued supply. How can we learn from nature's example? What sort of policies can we develop that would use biological analogies for providing energy? The first would be to array all energies just the way all plants and animals have always done: by mutation and natural selection. That is, we would array available energies by the degree to which production or utilization imposes a stress upon the producer or consumer of the environment. Now that sounds so bloody simple. Basically, it is very simple. And when we array these energies, we'd find no question about which is the least stressful or most abundant, it is sunlight.
Only 1 percent of all available sunlight is captured by plants and this sunlight really fuels the world. We have a fantastic situation where not only do we have a mechanism by which very diffused radiation is encapsulated in plants and in animals, but we also have no environmental stresses as a result of the energy production. And we benefit from atmospheric oxygen upon which the metabolism of all animals exists.
But I don't think we can do anything as clever as photosynthesis. That took perhaps about 2 ½ billion years to develop and we're not going to do that in the thirty years before we run out of oil. But we can do something a little simpler—like developing photovoltaic cells.
Every satellite is powered by photovoltaic cells. And every exploding meteor is a little miniature photovoltaic cell. We should be spending our time and money on projects like producing low-cost efficient photovoltaic cells for our homes.
Why don't we spend our money on solar energy? The answer is very simple: because we've got an oil lobby, a gas lobby, a coal lobby, and they're all vociferous and powerful. Where is the lobby for solar energy? There isn't one. But we desperately need one. We need to throw vast quantities of money into making photovoltaic cells available so that everybody would put some on his house. Nothing would please me more than to put some on my house, and run the energy through my meter, and at the end of the month send a bill to the Philadelphia Electric Company.
We already have some systems for using the sun's energy, such as heating and cooling our buildings. Although this modest technology is rather unrefined, it is, none the less, an important opportunity. We must further our efforts in this area.
There are several other energy sources that we haven't taken full advantage of, such as the wind. The whole Midwest was once covered with windmills, using this power to great benefit. And tidal surge and geothermal energy are both alternatives that we've yet to optimize. Even if these don't offer the Rocky Mountain West potentially direct applications, they can relieve the demands for your coal, oil shale, and oil—and that will surely help forestall the "Rape of the West."
As we order our energy sources, we eventually have to look at the non-renewable supplies that presently constitute our primary energy banks. You know as much about these as I do. You know they produce enormous costs to the physical environment and serious costs to the social environment. I won't even attempt to prioritize these, except for mentioning the most controversial and potentially destructive of them all: the atomic fuels.
I've followed the developments of atomic energy for a long time, since 1945. I have observed one thing: when the Atomic Energy Commission begins to be frightened by nuclear reactors, we had better be frightened too. I am frightened because the errors in the use of atomic energy don't simply attack the living, they attack the genetic inheritance of all life. So I say, here is an area where the circle costs are incalculably high and where errors are of such dimension as to be absolutely insupportable.
And this very nicely leads into my second point, one that has been implicit in my comments, but definitely should be explicit. Calculations of our energy needs cannot only be measured in BTU's and dollars, but must include all of the social expenses too. Traditionally, we haven't included the social costs in our formulas. For example, the present economic problems of Britain, in my opinion, are traceable to the brutality and oppression imposed on British workers, particularly in the mines. I want you to think of these long and enduring social costs. In our computation, in which we first consider the gradient of the environmental stress imposed by the utilization of energy types, we should also include all social costs before pricing energy.
Unfortunately, we don't include social costs in the price of our energy. Whenever I get my bill from the Philadelphia Electric Company, I have the illusion I am paying all the costs that are incurred. Of course, that is not true at all. Philadelphia Electric has never paid all of the costs; therefore, I am not being required to pay all of the costs. And no one has been required to even compute, let alone pay, all of the costs. You see, my costs for electricity should include the destruction of Appalachia—the depravity, hopelessness, misery, black lung disease, and injuries. The cost of remedy should be included. Then, the price I pay the Philadelphia Electric Company would be fair.
So, as we confront the "Rape of the West," let us compute all of the social costs before we determine what the price of energy will be. To do this, we need to employ what may become one of our most important concepts—the concept of net energy. Nature doesn't have any problem with this; it only can deal with net energy. But man has avoided the responsibility of dealing with net energy. Let me explain.
There are many "hidden" costs in the oil we buy from the Middle East. These include the energy necessary to drill 2,000 or 3,000 feet into the earth, to produce the steel, iron, and metals used to build the super tankers, the fuel for the ships to cross the Atlantic, storage and refinement of the oil, and energy for transporting it to the consumer. It's very important to calculate the total amount of energy used in the transaction. The fact of the matter is, we use a large amount of energy in the transaction which reduces the net energy. The deeper we have to drill, the further we have to transport, the more we use in the process, the smaller the net production.
Now, although being able to define the energy priorities, compute the social costs, and measure the net energy point us in the right direction, they don't organize the necessary future strategies. What we need to know is what the consequences of certain acts might be. With that knowledge it would be possible to select certain acts on the grounds that they will not only ensure our survival, but help us achieve the appropriate ends. Unfortunately, we have no such process now. The best we can do is called "planning," and in the United States, it is held in low esteem. Actually, planning is the most enduring biological process.
Planning is an absolutely ancient process and it's an indispensable one. It is the device by which man confronts the future. I define planning as the ability to understand the dichotomy between man and nature with sufficient perception to predict the consequences of contemplated acts, and to select those alternatives likely to guarantee survival. We've established that there are two things going on. There is an environment (biological, physical, and cultural) on one hand, and the consumers on the other hand. As discussed earlier, the thermodynamic test requires that the consumers seek the environment requiring the least energy employment. So if that is the formula, it tells us what the program should be.
First, we have to increase our knowledge of the environment. We've got a host of people who can help us with this. We must call upon these scientists of the environment. We must ask them to array the information so as to display chronology and causality, allowing us better to understand reality.
Chronology is a very good unifying device. First, we have the scientist identify bedrock geology, which is the most ancient and enduring of all subsequent events. Next, we add geomorphology, hydrology, and physiography. And then we look at the climatic impingement which explains soils and distribution of plants and animals. Through this process, the scientist has made a layer cake of chronology helping us to understand the place as an interacting biophysical process.
Next, we ask the scientists to describe the region in terms of dynamic and interacting processes, understanding that everything is in the process of interacting with everything else. Every region is in fact one single interacting region. So the scientists describe the region as an interacting biophysical system. At that point you have a descriptive model. If you can go beyond that, you have a fully or partially predictive model. Then you're in the position to make some sort of predictions about the consequences likely to result from certain human acts.
Next, we would like to understand why people are where they are and doing what they are doing. Of course, this is difficult, but it can be done through an ethnographic history. Begin with the ecological layer cake of the region, and very carefully populate it with primeval flora and fauna and some indigenous population. Carefully place man and all of his artifacts and institutions on this primeval landscape. Working up to the present, we will have a greater understanding of why people made some of the decisions they made.
Armed with all of this information, there are two things to be done: Make some predictions about the future, requiring some hypothesis about the larger setting, such as the region and the country; and at the same time, listen to people's needs, desires, and expectations of their environment. These are just as good as geologist's statements about rocks. If a man says to me, "I care more about the future of this land than about the money that can be taken out as coal," it is a piece of important data. This must be computed in any calculation comparing contemplated acts against potential consequences.
Setting up a system under which the people of the region have the best understanding of the physical and biological system of the region through the collection and interpretation of the data is critical. These data should be available through public libraries, public cable television, educational institutions, and elsewhere. You must be able to solicit the needs, desires, perceptions, affections, and expectations of all the constituencies in the region. The solution to any problem is going to involve the identification of constituencies that have set needs, desires, and expectations from the environment.
But even with all of these data, the critical thing that will determine the solution will be the value system of the problem solver. The solution to any planning problem of utilization of resources will vary not with respect to data, but with respect to the value system. I would like to end on a funny note which I think epitomizes this point.
Once upon a time the State Highway Department of Delaware confronted the intractable problem of putting a beltway around Wilmington. They needed a patsy—a patsy with an environmental name tag. And I fit the bill perfectly. I may be stupid, but I'm not that stupid. I realized I was being set up, so I wrote the contract in such a way that I would produce a method, not a solution.
We studied an area of 30,000 acres, digitized in one-acre cells. We not only had all the geophysical data, but all the socio-economic data, cultural data, and so on. So we took all of the information to the State Highway Department's civil engineers. I said, "Gentlemen, this is the most complete description of this region that you could possibly have. All you must do to locate the highway is weigh all the factors according to their positive or negative effect on the highway—and the highway's effect on the community. Then we'll have a computer print out the solution resulting from your value system." And of course, each did it—and each produced a different solution. I had some of my colleagues do it and they produced yet different solutions. And the State Highway Department said, "Which solution should we use?" "Oh," I said, "don't ask me. Who would I be, a Scottish immigrant living in Philadelphia, to represent myself as a surrogate for the value systems of all the people affected? But, if you are arrogant enough to believe that you could represent the value system for all these complex people, from the Chicano mushroom farm workers to the pre-revolutionary Blacks to the DuPonts, and all the rest in between, the best of luck to you, but I certainly wouldn't do it!"
The moral of the story is that the solution varies with respect to the value system of the person who solves the problem. So you have to identify the interactive biophysical culture system under study. You have to identify all the prospective consumer's needs, desires, and expectations. And then you have to allow each constituent to get a resolution to his particular problems. At that point in the democratic process, the resolution is the will of the majority, the rights of minorities are respected, and the conclusion lies in the courts. I would be content with this. At least those who did it would know what they're doing.
The technology exists. We can use technology as a slave, not as a master. In most cases the thing that obstructs us is not the absence of data or technology, but the absence of an integrated device or institution to utilize, to disseminate the information.
My recommendation is simple (and probably not even as obnoxious or contentious as I promised): Establish a national environmental lab for the United States. Divide the United States into eleven physiographic divisions, each of which includes homogenous environments with respect to geology, physiography, hydrology, soil, etc. Locate an environmental lab in each division and require them to be staffed with appropriate scientists, collect and digitize data, and make models. These people then become the surrogates of each region. They produce the data which allows the constituents to identify their value systems. The process is opened to public debate through cable TV hearings and publicity, making it possible for people to negotiate with each other about the value systems to be used.
If we develop this system and are able to predict the consequences of contemplated events, if we're able to choose between these consequences in terms of short-term or long-term so that our choices guarantee survival and fulfillment, then this indeed would be the way to prevent the sacrifice of the West.
You must in fact find such instruments and institutions because you are guardians, custodians, stewards, not only of a beautiful bit of Colorado, but of a wonderful, rich, and cherished part of the earth. So I commend you to this quest, for you, for me, for all Americans, for children yet unborn.
# 9
# Ecological Planning: The Planner as Catalyst (1978)
Although McHarg challenged orthodox planning theory, especially the economicsocial science deterministic approach of theorists such as John Friedmann and Britton Harris, he embraced the notion of the planner as catalyst for positive change. One role model was his partner David Wallace, who had helped transform Baltimore's downtown through his activism. In many ways, McHarg favored an activist place-making approach to environmental planning over more bureaucratic rule making, although he certainly recognized rules that are necessary to prevent environmental pollution, destruction, and degradation.
This paper was written for Robert Burchell and George Sternlieb for a conference at Rutgers University on planning theory. Other contributors to the conference and the later book included the crème of the orthodox planning theory crop, including Friedmann, Harris, Robert Beauregard, David Harvey, William Grigsby, and the most notable change agent-catalyst from the planning community, Paul Davidoff.
By and large the virtues of the planning I was taught were orderliness and convenience, efficiency and economy. The first set contains minor virtues, and the second set contains less than noble ones. These virtues have little to do with survival or success of plants, animals, and men in evolutionary time.
A fallacy is that planners plan for people. Actually this is not an assumption at all; it is a presumption. The planner who comes from out of town and is prepared to solve problems is a menace.
I prefer to think of planners as catalysts. The planner suppresses his own ego and becomes an agent for outlining available options. He offers predictability that science gives him about the consequences of different courses of action. He helps the community make its values explicit. He identifies alternative solutions with attendant costs and benefits. These vary with different constituencies, as do their needs and values.
This sort of planning might be called ecological. It is based on an understanding of both biophysical and social systems. Ecological planners operate within the framework of a biophysical culture.
Ecological planning addresses itself to the selection of environments. Ecological planners help institutions and individuals adapt these and themselves to achieve fitness.
For example, when I prepare a planning study, I insist that scientists of the environment study the region in terms of the processes which produce the phenomena constituting the region. They describe the phenomena of the region as an interacting biophysical model. Such a model can then be seen to have intrinsic opportunities and constraints to all existing and prospective users. Fitness is defined as the presence of all or most propitious attributes with none or few detrimental areas.
This notion of planning stems from two fundamental characteristics of natural processes: creativity and fitness. Creativity provides the dynamics that govern the universe. There is a tendency for all matter to degrade to entropy, but in certain energetic transactions there is a process by which some matter is transformed to a higher level or order. All of biology subscribes to this law: entropy increases but a local syntropy can be achieved. It is seen in both energetic transactions, in the evolution of matter, life, and man. This biological "creativity" enables us to explain the rich and diverse world of life today, as opposed to the sterile world of yesterday.
The second concept—fitness—stems partially from Darwinian notions about how organisms adapt and survive. Equally important is the thought that the surviving organisms are fit for the environment. The world provides an abundance of environmental opportunity. This teaches us that the world is environmentally variable, offering variable fitness. This results from the most basic elements—hydrogen, nitrogen, oxygen, and carbon—and the earth itself provides environmental opportunity.
All systems are required to seek out the environment that is most fit, to adapt these and themselves, continuously. This is a requirement for survival. This is called adaptation. It is an imperative of all life, it has been, and it always will be. Fitness can best be described as finding an environment—physical, biological, or social—in which the largest part of the work of survival is done by the environment itself. There is then an energetic imperative for evolutionary success. Systems which are "fit" are evolutionary successes; they are maximum success solutions to fitness.
Planning, of course, is more than understanding environments and explaining why they are what they are, and where they are going. It is also explaining why people are where they are, doing what they are doing. An ecological planner would look at this over time, through an ethnographic history: Where did the first people who occupied a given place come from? Why did they leave? Why did they choose the environment they did? What adaptive skills determined their location? What adaptive skills did they practice? What modifications did they make to the environment? What institutions did they develop? What plans?
The social value of a given environment is an amalgam of the place, the people, and their technology. People in a given place with opportunities afforded by the environment for practicing a means of production, will develop characteristic perceptions and institutions. These institutions will have perceptions and values that feed back to an understanding of the environment—both national and social—and that have a modification of technology. Thus, I believe, we have a continuous model, which emanates from the physical and biological, and extends to the cultural.
The most critical factor is the value system, for it determines the planning solution. I strongly object to much of the current planning philosophy as it is emerging in both teaching and practice, for it assumes that the planner imposes values and exercises for the "good of the people." I resist this. Given a set of data, the planning solutions will vary, not with respect to the set, but with respect to the value systems of the people who seek to solve the problem. Most of the important values are particular and there is no substitute for eliciting them from the constituents themselves. These values themselves become the data, whether it be for describing rocks, soils, animals, people, or institutions. Planners must elicit these data from their client if they are going to help solve the problems posed by the particular system within which the client functions. This, in fact, is the planner's most important role. After he has done it, he should step aside, and the resolution of the problem of the explicit system will be found through the political process, and ultimately, in some cases, through the courts.
In sum, the planner is a catalyst and a resource. He determines what skills and branches of knowledge are appropriate to solutions, and what institutions. He helps to describe the interactions of systems. He describes probable alternative courses of action and assists his constituents in making their value system explicit. The planner then helps his clients understand what the consequences of applying that value system are in terms of their costs and benefits. He participates with them in negotiations among different constituencies over the relaxation or change of values in order to come to some agreement about the allocation of resources.
If the process is successful the constituencies will select the fittest environments, adapting these and themselves to achieve a creative fitting. As health can be described as the ability of persons, families, or institutions to seek and solve problems, so planning is not only a measure of the health of a group or institution, but, is health-giving to such agents.
It could make planning more fitting, perhaps even healthier.
# 10
# Human Ecological Planning at Pennsylvania (1981)
In 1981 Arthur Johnson, a faculty member at Penn, edited a special issue of Landscape Planning (now called Landscape and Urban Planning, published by Elsevier). The special issue was devoted to the Penn Department of Landscape Architecture and Regional Planning. McHarg's article described the human ecological approach. He incorporates an understanding of the ecologies of people to the natural-resource approach, described in Design with Nature. To that end, he had encouraged colleagues such as Dan Rose, Jon Berger, and Setha Low, as well as graduate students, to pursue such an approach.
# Abstract
A theory of human ecological planning is presented which is based on the premise that all social and natural systems aspire to success. Such a state can be described as "syntropic-fitness-health." Understanding the process of interaction between the landscape and the people who inhabit it provides a basis for assessing the opportunities and constraints afforded by the environment, and the needs and desires of the population which can be combined to present alternative futures. Such a model allows examination of the impact of any plan upon the health of the inhabitants and the well-being of the social and natural systems.
# Introduction
It is first necessary to define terms. The simplest term is the last in the title of this paper. Pennsylvania is a contraction for the Department of Landscape Architecture and Regional Planning at the University of Pennsylvania (also called Penn). Here a faculty described in Science (Holden 1977) as one of the very few multi-disciplinary and interdisciplinary faculties in the United States, is committed to the development and teaching of human ecological planning. The faculty comprises physical, biological, and social scientists, architects, landscape architects, and city and regional planners.
It is next necessary to define human ecological planning. The central word in this compound noun has primacy: ecology has been defined as the study of the interactions of organisms and environment (which includes other organisms). The word human is adequately defined in standard dictionaries but human ecology is not. While ecology has traditionally sought to learn the laws which obtain for ecosystems, it has done so by investigating environments unaffected or little affected man; it has emphasized biophysical systems. Yet clearly no systems are unaffected by man, indeed studies of the interactions of organisms and environment are likely to reveal human dominance. Hence, ecology simply must be extended to include man. Human ecology can then be defined as the study of the interactions of organisms (including man), and the environment (including man among other organisms). However, if man is assumed to be implicit in both definitions of organisms and environment then the standard definition for ecology can apply to human ecology.
The possibilities for creating a human ecology seem to be afforded by a new extension and integration of existing scientific disciplines. Ecology has been used to integrate the sciences of the biophysical environment. If we extend ecology by adding ethology we introduce the subject of behavior as an adaptive strategy. If we extend further into ethnography and anthropology we can include the study of human behavior as adaptation. If, finally, we extend into medical anthropology and epidemiology we can close the cycle by examining the natural and human environment in terms of human health and well-being.
Planning cannot be defined succinctly. Planning consists of the formulation of hypothetical alternative futures. These are constituted into component actions comprising courses of action. These are subsequently measured in terms of costs and benefits (employing the value system of the initiator of the alternative futures). The least cost—maximum benefit solution selects the preferred hypothetical future.
When "planning" is linked to "ecological" the primacy of goals is modified. Goals are derived from the region. Ecological planning is an instrument for revealing regions as interacting and dynamic natural systems having intrinsic opportunities and constraints for all human uses. Preferred hypothetical futures will be proffered by locations where all or most propitious factors exist with none or few detrimental ones for any and all prospective uses. What constitutes detrimental or propitious is derived from the prospective use and the value system of the initiating person or group.
When the term is compounded into "human ecological planning" the region is expanded into a physical, biological, and cultural region wherein opportunities and constraints are represented in every realm. Geophysical and ecological regions are identified as cultural regions in which characteristic people pursue means of production, develop characteristic settlement patterns, have characteristic perceptions, needs and desires and institutions for realizing their objectives. Hypothetical future alternatives are derived from expressed needs and desires of groups. These are matched against the physical, biological, and cultural resources. Preferred hypothetical futures can be derived for each group with its associated value system.
Human ecological planning is a cumbersome and graceless title. Remedy, however, while possible, is distant. When it becomes accepted that no ecosystem can be studied without reference to man then we may abandon the "human" descriptor and revert to "ecological planning." Better still, when planning always considers interacting biophysical and cultural processes, then we can dispense with the distinction of "ecological" and simply employ the word "planning." However, that state is far in the future, as most planning today excludes physical and biological sciences, ecology, ethnography, anthropology, epidemiology, and concentrates upon economics and sociology. It remains necessary, not only to advance planning to become ecological, but even more, to develop a human ecological planning.
# Theory
Having defined terms it is next necessary to describe the theory which impels human ecological planning. Clearly, as it incorporates the physical, biological, and social sciences it can also employ the theory of these sciences. Planning is not rich in theory, and no statement on a theory of human ecological planning has yet been propounded. In order to initiate this quest I offer the following as a tentative beginning.
It would appear that all living systems tend to oscillate between two extreme states. These can be described as: (1) syntropic-fitness-health; and (2), entropic-misfitness-morbidity. Entropy is the tendency for energy and matter to degrade from higher to lower levels of order in any energetic transaction. All energetic transactions result in an increase in entropy. However, while the preceding is true, certain energetic transactions produce a product of matter and energy at a higher level of order than at the onset. This process is described by Buckminster Fuller as syntropic. The explosion of super novae in a primeval universe composed of hydrogen resulted in an increase in entropy but there was a residuum, not only of hydrogen but also helium, lithium, beryllium, boron, on up the periodic table of elements, each more ordered than hydrogen. The evolution of matter in the universe is syntropic. In life forms, all of which constitute energetic transactions, photosynthesis is the most notable example of syntropy. Given continuous energy, in the presence of carbon dioxide and water, the chloroplast creates glucose, a higher level of order than the ingredients. The evolution of life forms itself is syntropic whereby successive forms represent increased capability, a higher level of ordered energy and matter in their beings.
Fitness has two definitions, each complementary. Charles Darwin (1859) stated that "the surviving organism is fit for the environment." Much later Lawrence Henderson (1913) augmented this proposition by showing that the actual world consists of an infinitude of environments, all exhibiting fitness for appropriate organisms. These two propositions can be linked into an evolutionary imperative. Every organism, system, or institution, is required to find the fittest environment, adapt that environment and itself in order to survive. This imperative is linked to the syntrophy-entropy criteria. An environment is defined as fit for an organism by the degree to which the environment, as found, provides the largest amount of needs for that organism. The corollary, then, is that the organism is required to import and employ the least amount of energy and time to modify the environment and itself to make it more fitting. Thus there is a condition of syntropic fitness. Entropic misfitness would be represented by an organism unable to find a fit environment and/or unable to adapt that environment and/or unable to adapt itself. Its fate would be non-survival, death, and extinction.
All creatures as individuals, species, or ecosystems, aspire to survival and success. The mechanism for achieving this state is adaptation. This has three modes: physiological adaptation by mutation and natural selection used by all life forms; innate behavior, shared by animals and man; and, the unique human instrumentality, cultural adaptation. Until recombinant DNA is employed, physiological adaptation will remain beyond voluntary control; innate behavior is similarly resistant to manipulation. Adaptation through modification of culture is the most plastic instrument for voluntary action leading to survival and success.
While the verb, "to fit," applies to the active selection of environments, adapting those and the self, the noun "fitness" has another meaning. It implies health. From another direction, the definition of a healthy person is one who solves and seeks problems. Yet a second definition of health is the ability to recover from insult or assault. Both definitions accord with evolutionary biology; they could be subsumed by adaptation, particularly the latter. It appears that fitness and health are states linked in a more profound way than is suggested by common usage. Fitness is defined as problem solving, i.e., finding fit environments, adapting these and the self. Health is similarly defined as seeking and solving problems. It would appear that not only is fitness syntropic, but so is health. There seems to be a state of syntropic-fitness-health. The antithesis confirms this assumption. Morbidity reveals a process moving from higher to lower levels of order, decomposition after death is its most complete expression. Morbidity is also a failure in adaptation: the environment is not fit for the cells, tissue, organ or organism; or the system is unable to make it or itself fit, and finally, the system is unable to recover from insult or assault, unable to solve problems resulting in entropic-misfit-morbidity. There is one further observation of significance. Health, it would appear, not only reveals evidence of the presence of syntropic fitness, but, if health is defined as problem seeking and solving, then the quest for fitness is also health-giving. This long preamble finally reaches planning. Of all the instrumentalities available through cultural adaptation (language, religion, symbol, art, philosophy, etc.), it would seem that one above all is most directly connected to the evolutionary imperative of finding fit environments and adapting these, of accomplishing syntropic-fitness-health. This instrument is planning, in particular human ecological planning, or planning for human health and well-being.
The theory of human ecological planning can now be summarized: all systems aspire to survival and success. This state can be described as syntropic-fitness-health. Its antithesis is entropic-misfitness-morbidity. To achieve the first state requires systems to find the fittest environment, adapt it and themselves. Fitness of an environment for a system is defined as that requiring the minimum work of adaptation. Fitness and fitting are indication of health and the process of fitting is health giving. The quest for fitness is entitled adaptation. Of all of the instrumentalities available to man for successful adaptation, cultural adaptation in general and planning in particular, appear to be the most direct and efficacious for maintaining and enhancing human health and well-being.
# Method
Theory produces an objective which requires a method to achieve it. We are required to promote human health and well-being by planning, specifically to select fit environments for all users; to participate in the adaptation of that environment and the user. We must be able to describe regions as interacting physio-bio-cultural systems, reconstitute them as resources and hence as social values, array these attributes as either costs or benefits for prospective consumers, and select the maximum benefit—least cost solution. Thus, we must be able to model regions, insofar as science permits, at least descriptively, at best quantitatively; we must employ all of the predictive skill which science provides to forecast the consequences of contemplated actions on the interacting systems.
Before describing the method employed there is one important observation to be made. Human ecological planning incorporates the physical, biological, and social sciences and in so doing it utilizes the universal laws which those sciences employ. However, the distinction of ecological planning, unlike economic, transportation, or orthodox city planning, is that this planning theory and method emphasize locality. As with ecosystems and bioclimatic zones, it is believed that each region or locality is spatially determined. While responding to universal laws, each region is believed to comprise unique attributes of place-folk-work, as first identified by Patrick Geddes, and these will determine the capabilities, opportunities, and constraints of the region and thus the potential hypothetical futures.
The first objective of the method is to create a model of the region under study. As all of the sciences of the environment agree that things, creatures, places, and people are only comprehensible through the operation of laws and time, it is appropriate that the modeling exercise should employ evolution and history. Matter preceded life which preceded man. We should begin the construction of the model with its physical evolution, continue with biophysical evolution, and conclude with the addition of cultural history. This method selects the participants. Physical evolution is the province of meteorology, geology, hydrology, soil science, and where applicable, physical oceanography. Biophysical evolution adds botanists and marine biologists where appropriate. Human evolution requires anthropologists—physical, cultural and medical ethnographers, and such economists, sociologists, and political scientists as are compatible with the ecological view. The final participant is the planner, the person willing to undertake synthesis, oriented towards problem solving.
As science has divided this "one world" into discrete areas of concern, we are required to accept this situation, although the objective is to unite all of the discrete perceptions into a description of a single interacting system. While we must employ all disciplines which describe the environment we can collect and array their data in a way which employs history and causality to assist in the portrayal of reality. Let us build a layer-cake representation of the region with the oldest evidence on the bottom and new, consequential layers superimposed in place and in time.
We can properly begin with the oldest evidence, bedrock geology, as the basement and superimpose meteorology above. We ask that the geologist, mindful of meteorological history, reconstruct geological history to explain the phenomena of the region and its dynamics. Upon bedrock geology surficial geology is added with meteorology remaining on top. We ask a geomorphologist to add his more recent data and explain physiography in terms of bedrock and surficial geology. The next scientist, a groundwater hydrologist, interprets the previous data to explain historic and contemporary phenomena. A surface water hydrologist follows. Together they describe hydrological processes, contemporary phenomena, and tendencies. The next layer is the domain of the soils scientist. He, as with others, is required to invoke the data of all prior sciences to explain the processes and phenomena within his realm. However, he must also invoke the effect of life forms historically. The plant ecologist follows. He describes plants in terms of communities occupying habitats. The descriptions of these employ geology, meteorology, hydrology, and soils. He recapitulates vegetational history to explain the existing flora and its dynamics; the limnologist populates aquatic systems using a similar method, and finally, the animal ecologist, depending most upon plant ecology, constructs a history of wildlife and explains the contemporary populations and their environments. Arrayed as a cross-section, the layer cake is complete—bedrock geology, surficial geology, groundwater hydrology, physiography, surface-water hydrology, soils, plants, wildlife, micro-, meso- and macro-climate. The layer cake has evolved over time and continues to do so. The entire formulation of the ecological model has emphasized interacting processes and time. All phenomena have been, are now, and are in the process of becoming. The layer cake is an expression of historical causality. It is possible to peer from the surface to the bottom and explain process, reality, and form. We have, at once, a discipline and (to the degree science provides it) a predictive model.
The study of the evolutionary history of the region has revealed that all processes are interacting with each other: geological and meteorological history are expressed in surficial geology which, in turn, are expressed in hydrology and soils. The sum of these is reflected in environments populated by appropriate plant communities while these last are occupied and utilized by consonant animals. It becomes clear that physical processes are synthesized in physiography while biophysical processes are synthesized by ecosystems.
The next task is to populate the region with its inhabitants. Here we confront the threshold between ecological and human ecological planning. The physical and biological sciences assume the operation of order. While there is randomness in systems, randomness cannot explain them. In short, we accept that nature is systematic. Can we also assume that man and nature are systematic, or is man random with respect to a systematic nature? It seems more reasonable to believe that man and nature are systematic and to continue to employ the evolutionary method to explain folk—work—place. We wish to know who the people are, why they inhabit the region, why they are where they are, doing what they are doing.
In order to explain people, place, and work, using an evolutionary method, we need to undertake the study of ethnographic history. We can begin with the biophysical representation of the region before the advent of Western man. The primeval environment is then occupied by its aboriginal people. However, it is necessary to explain the anthropological model which underlies the ethnographic history before we proceed. This construct affirms that, while a region may be described as an interacting biophysical system, it is simultaneously, a social value system, i.e., it contains resources. However, what constitutes a resource is determined by the perception of the inhabitants or observers, and the available technology and capital. Given certain perceptions, technology, and capital, certain natural phenomena will be perceived as resources and exploited. These resources, e.g., minerals, lumber, wildlife, will be locationally determined by the natural history of the region. Thus the means of production, utilizing resource or resources, will also be locationally determined. The means of production implies labor selection, generally persons skilled in the appropriate kinds of employment, e.g., fishing, mining, lumbering, farming. In American history the exploitation of resources, in selecting accomplished practitioners, often attracted ethnic groups who had practiced similar skills in Europe. So we see Scots miners, Portuguese fishermen, Italian masons, Basque shepherds, Russian and Swedish farmers in the northern Great Plains, Spanish farmers in the arid Southwest, French Canadian and Swedish lumbermen, and others. People sharing an occupation tend to have similar perceptions of themselves, particularly when this is accompanied by shared ethnicity and religion. Each occupational type is likely to have characteristic land uses and settlement patterns, and, finally, groups associated with a particular means of production are likely to develop characteristic institutions to promote and enhance their success and well-being, both in the private and public domain.
Thus, we can anticipate that changes in land-use and settlement patterns respond to two causes: the first is technological change and the second includes massive social events, e.g., wars, immigration, and the like.
The method can proceed. It begins with the layer cake occupied by its aboriginal inhabitants, plants, animals and people, their settlement patterns, land uses, transportation corridors. The next land-use map should portray colonial settlement responsive to then-current perceptions and technology, and should explain the differences from the original map in terms of resource, technology, and social events. For example, in the age of sail, the colonizers of the eastern seaboard of the United States sought safe harbors and estuarine and river systems permitting penetration of the interior. The Hudson, Delaware, Christiana, Susquehanna, Potomac, and James rivers provided safe harbors, but penetration of the interior posed serious problems—water—falls on the main rivers and their tributaries. All of the waterfalls occurred on the same geological phenomenon, the interface between the crystalline piedmont and the coastal plain. At this point, settlement occurred on each of the rivers Albany, Trenton, Philadelphia, Wilmington, Baltimore, Washington, D.C., and Richmond, Virginia. New York City was a special case of a granite island located in a drowned tidal estuary.
At the time of the American Revolution, iron for cannon and cannon balls was important. Iron was obtained from bogs, which were abundant in the coastal plain of New Jersey, but scarce elsewhere. This is comprehensible from natural history: the bogs located the bog iron which located the foundries which located settlement and which located operators. So with the grist and saw mills, ports, harbors, fisheries, and farming. Each successive map shows changes responsive to new options presented by technological innovation and/or impelled by social events, revolutions, war, immigration being the most notable. Throughout the ethnographic history, attention is also paid to nonphysical instruments of adaptation, e.g., laws, ordinances, and mores, the evolution of institutions, public and private. Finally, the last product is one of current land use, a document gravid with meaning, as ordered as a map of rocks, soils, or vegetation.
It transpires that people are who they are, where they are, doing what they are doing for good and sufficient reasons. Moreover, they are not reticent to describe themselves, in economic, ethnic, occupational, religious, and spatial terms. They have no doubt about the territory they occupy. They can define themselves, their territory, and their neighbors. The process of eliciting this people-place description is called "consensual mapping." When regions defined by inhabitants are compared to "natural" regions and features, a great conformity becomes apparent. In many cases there are physiographic-social regions and subregions. This applies not only to rural and metropolitan areas but also to urban neighborhoods in the City of Philadelphia. The discovery that people, far from being random with respect to the natural environment, are highly ordered, simplifies the next task. We wish to ascertain the needs and desires of the population. The theory presumes that health and well-being attend the task of finding fit environments, adapting these and the self. In order to ascertain what values constitute "fitness" and "misfitness" we must elicit those values because they are not objective; they cannot be attributed, but only elicited. But as it transpires that populations reveal discrete self-defined groups and locations, it is not necessary to interview large numbers to obtain this information. Key informant interviews with a small-sample questionnaire for each group will suffice. We wish to know what perceptions groups have of themselves and their environment. What are their needs and desires? In order to ascertain the values they apply, subjects are asked to reveal their positions on certain contemporary issues. This method proves more efficacious than direct questions on values.
When groups have been identified, it becomes necessary to ascertain their values and the instruments through which they seek to attain their objectives. Obviously, the formal political arena receives investigation, but informal institutions receive even greater attention. Voluntary associations, e.g., conservation groups, fraternal organizations, voluntary fire companies, parent-teacher associations, the League of Women Voters, the Grange, Mushroom Institute, Shell Fisherman's Association, each reveals its values, its position on issues, its membership, its financial capabilities, its strategies in its publications and public statements. This analysis, combined with one on kinship, reveals the social system as a complex network with a mosaic of overlapping constituencies, each having characteristic needs and desires, with varying capabilities of realizing them through the market, private and public institutions. However, there is a danger that the social data can become so complex as to be unusable. The ordering principle employs land use. Using matrices and maps, discrete social groups, e.g., Italian Catholic mushroom farmers, are located. Next, issues are identified and positions on these issues are also represented on matrices and maps. A conformity becomes apparent between land uses, occupants, locations and positions on issues. All of the preceding data are mapped on a single four-way matrix entitled Community Interaction. One ordinate enumerates the inhabitants as self-described constituencies. At right angles are shown the issues. Opposite constituencies are listed all land uses. The final quadrant identifies all agencies, private and public, through which the constituencies operate to achieve their ends. This matrix can also be employed to show which constituencies suffer or benefit from different reactions to any issue. The data are now ordered and comprehensible. As human ecological planning is future-oriented, it cannot be content with the existing population as the client group. It must make either assumptions or predictions about its size and composition at future times. This is the area of growth modeling and market analysis dominated by economists. The economy of the region is reviewed historically and, in conjunction with assumptions made for the national and regional economy, predictions of growth or decline are made for population, employment categories, and public facilities. This analysis includes the levels of private investment which can be anticipated in different sections of private enterprise and fiscal expenditures on the various levels of government.
We have come to the point where the elements of the Darwin-Henderson imperative can be resolved. We have modeled the natural region as an interacting system in which values repose. We have asked the occupants and users to describe their needs and desires, their conception of benefits (which also reveals dislikes, aversions, and costs). We can now search for fit environments for all users, present and prospective. Fitness remains unchanged—a fit environment requires the least work of adaptation. In other words, where all or most propitious factors co-exist with none or few detrimental ones, is, by definition, "most fitting." This requires a two-part process. First, we must interpret the data maps for their opportunities and constraints for all prospective land uses. For example, the geology map might be interpreted for seismic activity, landslide hazard, fault zones, economic minerals, subsidence, compressive strength, and more. The physiography map will be interpreted to show physiographic regions and features, elevation categories, slope categories, and so on, for each map. The sum of all of the legends on all maps constitutes the sum of attributes to be employed in the allocation process. Obviously, any single phenomenon can be an opportunity for one user and a constraint to another. The users must now be defined, either in terms of self-identified social groups, e.g., Italian mushroom farmers, or more broadly, agricultural types. The appropriate user-category will be determined from the anthropological study. When known, all of the relevant attributes in every data set are compiled in terms of opportunities and constraints for that land use. The region is then shown as a gradient of maximum to minimum suitability (fitness) for each and every land use, i.e., agriculture, industry, commerce, recreation, and housing all by type. In addition, one further map is generally prepared, entitled "Protection." This identifies and delimits all hazards to life and health from physical phenomena in flood plains, hurricane zones, seismic areas, and the like. It also identifies areas which can be made hazardous by human use, e.g., by induced subsidence by withdrawal of water, gas, oil, or by mining. It also locates all rare, endangered, scarce, or valuable species of plants and animals, and all buildings, places, and spaces deemed to be of historic, scenic, or scientific value. At this point, it is possible to synthesize all of the optima for all prospective land uses showing the intrinsic suitability (fitness) of the region for all existing and prospective uses. This should show locations where more than one land use can coexist compatibly in a single area. It should also show where co-existing uses would be in conflict. The preceding procedure is appropriate for government planning as an objective statement on intrinsic fitness of the region. However, planning decisions are not made on objective criteria alone. It is also possible to allocate resources using the value system of the discrete constituencies which constitute the region. The major variable in any allocation process in planning is the value system of the problem solver. Hence, there can be as many plans as there are groups. Moreover, for every allocation one can analyze who suffers and who benefits. The availability of such data would allow groups to see the consequences of the employment of their value system in terms of resource allocation. It would permit groups to see where values would have to be modified to achieve objectives which they sought. In short, it provides the opportunity for an overt, explicit and replicatable planning process, and indeed for planning as a truly informed democratic process.
Ecological planning seeks to fit consumer and environment. This problem-solving and problem-seeking quest conforms to the definition of health. Ecological planning should be health giving. Success in such planning or fitting should be revealed in the existence of healthy communities, physical, biological, and social systems in dynamic equilibrium. However, many persons and institutions may satisfy the definition of health by seeking and solving problems but still succumb to disease and death from causes of which they were ignorant or causes beyond their control. As a result a specialization in health planning has been developed within the realm of human ecological planning. This subscribes to the ecological theory and method which has been described but uses as its viewpoint the degree to which actions by persons, groups, or institutions enhance or diminish human health and well-being.
The first category of such actions comprises natural phenomena and processes which constitute a hazard to life and health. This would include areas susceptible to volcanism and earthquakes, floods, tsunamis, avalanches, landslides, fire, subsidence, and the like. It would also include vectors of disease inducing malaria, yellow fever, schistosomiasis, river blindness, sleeping sickness, cholera, amoebic dysentery, etc.
The subsequent category would comprise those situations where certain human actions would transform a benign situation into a hazardous one, e.g., where earthquakes may be induced by injecting wastes into faults, or subsidence induced by underground mining or the withdrawal of water, natural gas, and oil, the exacerbation of flood or drought, disposal of toxins into the environment, or contemplated inhabitation of environments wherever hazards existed.
A third category would include the hazards to life and health related to resources and occupations. This would include uranium, lead, mercury, zinc, asbestos, and beryllium, among others. It would include occupations which extracted these minerals, processed, and used them. This category of occupational diseases associated with resources would include the effect of pesticides and herbicides both upon farm workers and consumers.
A further realm of investigation would focus upon regions impacted by hazardous processes, e.g., a population subtended by a plume of air pollution from one or more sources, a population utilizing contaminated water sources, or one served by an unsanitary food distribution facility, etc.
The final area is social epidemiology, where populations are analyzed to discern their degree of health/morbidity attributable to multiple factors with an emphasis on behavior.
In all of these cases the human ecological presumption holds. Individuals and society can achieve health and well-being by seeking and solving problems. In the special case of health planning, the connection is visible and incontrovertible. Persons are assisted in identifying problems to which they would otherwise be oblivious and are therefore enabled to confront and resolve them.
As one reviews the amount of work to be undertaken to accomplish human ecological planning, the first response is that it is beyond the financial or human resources of most communities. Indeed, given present priorities it probably is. Yet, if the quest for health and well-being necessarily involves planning as the agent for accomplishing syntropic-fitness-health, is this not the single most important activity of persons, families, institutions and, not least, government? All units of government are enjoined by the U.S. Constitution to promote the health and well-being of citizens. Indeed, if this is the most powerful agency available to man, can he avoid employing it? It has multiple benefits. It requires that all activities, public and private, be reviewed in terms of the degree to which they promote and enhance well-being. The acquisition of both the natural inventory and the ethnographic history, if assimilated, provides the basis for an informed citizenry capable of exercising good judgment. In addition, access to the ecological model gives the opportunity for predicting the consequence of contemplated actions, both in the public and private realm. In terms of costs the initial cost of undertaking an ecological inventory and interpretation, an ethnographic history supplemented by interviews, would be high, but they would only have to be done once. The task of keeping the inventory current and enriched would not be demanding.
I have elsewhere (Wallace, McHarg, Roberts, and Todd, 1973) described a process whereby this conception could be employed as national policy. It was recommended that a National Environmental Laboratory be created to model the ecosystem of the United States and provide a depository of integrated and interpreted data available to all, either in published form, mapped form, or through a display terminal facility, located in all public libraries. In addition there would be thirty-four Regional Environmental Laboratories, one for each of the homogenous physiographic regions comprising the United States. These would provide data at scales appropriate to the nation and its regions. Those data should also be available at scales appropriate to states, countries, and finally, cities and towns.
Of course, university departments do not engage in national policy. While this subject is of intense interest to planners, the purpose of the Penn Department of Landscape Architecture and Regional Planning is to engage in research and instruction to train professional landscape architects and regional planners. The faculty have concluded that it is presumptuous to plan without knowledge of place and people and that planning involves fitting them together, mindful of the thermodynamic imperative of fitness and the supervening criterion of enhancing human health and well-being.
The evolution of the department at Penn over the past quarter century has been from a preoccupation with design in the absence of any scientific prerequisites or training to a continuous increase in the content of both physical and biological science, integrated by ecology. The present phase aspires to extend ecological planning and design into an applied human ecology. It is the hope of this author that this issue will help bring together others engaged in this and similar quests in order that there will be mutual benefit.
# References
Darwin, C. 1859. On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life. London: Murray.
Henderson, L.J. 1913. The Fitness of the Environment. New York: The Macmillan Company.
Holden, C. 1977. "Champion for Design with Nature." Science, 195:379-382.
Wallace, McHarg, Roberts, and Todd. 1973. Towards a Comprehensive Plan for Environmental Quality. Washington, D.C.: American Institute of Planners for the U.S. Environmental Protection Agency.
# Part III
# Form and Function Are Indivisible
Ian McHarg pursued design before planning and, in many ways, remained a designer, at least he retained the flair, the attitude, of the designer. His major contributions to the environmental design arts are theoretical. Like many young people, he was attracted to landscape architecture because of overlapping interests in the outdoors and in drawing. Before the second world war, he apprenticed as a landscape architect while studying in colleges of art and agriculture. At the end of the war, he became exposed to planning through a correspondence course. But, it was his quest to be a landscape architect, a designer, that brought him to study in the United States at Harvard University.
Those were heady times at Harvard, which, with the arrival of Walter Gropius at the Graduate School of Design, had become the center of the modern movement. McHarg fell very much under the spell of modernism and was especially attracted to the Bauhaus philosophy of interdisciplinary interaction as well as its optimistic view of the promise of design for shaping a better future. He formed friendships at Harvard with leading designers that would last a lifetime. In fact, McHarg knew well all of the leading architects of the postwar era from Walter Gropius and Lou Kahn to Robert Venturi and Romaldo Giurgola. Denise Scott Brown was a member of McHarg's faculty very early in her career, just one of many distinguished designers associated with the Penn Department of Landscape Architecture and Regional Planning.
Using ecological knowledge to design and to create forms and space, is an especially difficult task. As a student at Penn, I was required to take a geomorphology course. The regular professor, Bob Giegengack, was on sabbatical, and he had been replaced by a young postdoctoral scholar from Cambridge who was puzzled why his course was filled with landscape architecture and regional planning students. After clarifying that we were not landscape gardeners, we explained that we had come to Penn to study with Ian McHarg and to learn how to design with nature.
The instructor was a flexible Englishman, so he assigned us to identify places on the Penn campus where geological materials were misused, where design was not with nature. Several of us identified the Woodland Avenue Gardens, where a grey, glazed brick in exposed aggregate concrete had been used that did not fit the Philadelphia climate. Frosts would spall the faces off the glazed bricks, resulting in ugly, pockmarked walls. The instructor was thrilled with this and other examples. We were shocked to learn that the Woodland Avenue Gardens had been designed by none other than Ian McHarg.
I do not believe any of us mentioned it to him at the time. Years passed. I left Philadelphia, returned again to teach and to pursue a Ph.D., left again, and would return from time to time. I would always walk through the Woodland Avenue Gardens because it was strategically located above a subway stop and at connecting points to West Philadelphia sidewalks. Although the walls remained ugly, I noticed that the tree canopy was quite pleasant, even room-like, and the connection between the subway stop and pedestrian routes functioned well. Eventually the area was redesigned by, among others, Hanna-Olin, and the walls were remade.
I eventually mentioned the geomorphology assignment to Professor McHarg. He explained the program was to convert a street into a park and to reinforce pedestrian connections to the subway. He had given considerable thought to the trees and, yes, it was his goal to create a room. And, yes, he remembered the grey, glazed bricks. He had been inspired to use them because of a Philip Johnson project, a small bank at 30th and Market in Philadelphia. McHarg is a great admirer of Philip Johnson, who had been McHarg's first guest critic at Penn. Johnson came to Penn at his own expense. Every criticism by Johnson was "a history lesson," according to McHarg. McHarg had also offered Johnson some site-planning advice on his famous Glass House in New Canaan. Specifically, he suggested to Johnson that he purchase the neighboring property to protect the setting of the house and suggested some selective pruning of the grounds. The landscape architect for the Glass House was Dan Kiley. Certainly, the landscape contributes to the quality of the Glass House in numerable ways.
McHarg's explanation about the Woodland Avenue design impressed me on several levels. First, I was only familiar with the "after," which, in spite of the walls, worked well. Designers transform spaces which we often take as a given after the fact. Second, McHarg described the details of the project as a designer. His critics underestimate his commitment to design. Third, the image of Ian McHarg copying the materials of Philip Johnson was fascinating. The problem of the grey, glazed bricks was that before McHarg's use at Woodland Avenue, they had only been used on buildings, not retaining walls. On buildings the inside wall was insulated. On the retaining wall both sides were exposed to the cold, which resulted in the glazing being popped off.
Early in his career Ian McHarg was indeed a modernist, and his ideas were influenced by modernism throughout his career. His process for ecological planning and design is an affirmation that we can use knowledge to build a better future, a central tenet of modernism. But McHarg also was instrumental in challenging the style of modernism.
I believe Nan Ellin's (1996) analysis of the fall of modernism is especially clear and accurate. She credits Mumford with laying the basis for the critique, as does McHarg (1996). Furthermore, Ellin points to the landscape critique that argued for more emphasis on both vernacular and regional ecology. The leaders of this view were J. B. Jackson and Ian McHarg. The second major body of criticism of modern design, according to Ellin, came from the social planners, mainly centered at Penn, such as Herbert Gans, Paul Davidoff, and Linda Davidoff.
Denise Scott Brown and Robert Venturi provided the synthesis, which resulted in a misnamed, that is, "postmodernism," movement and an even sillier style than that which preceded it. Both Scott Brown and Venturi have wisely proclaimed themselves not to be postmodernists. Neither is McHarg. He leapfrogged from modernism over postmodernism to something else. No one wants to live in a "post" anything era. If we do indeed live in an Information Age or an Ecological Era or both, an Eco-Information Age, then McHarg has articulated more than anyone else a theory for design for this new age.
What he advocated, as did Scott Brown and Venturi, was anti-style. What emerged in Philadelphia, in no small part because of the populist influences of Mumford, Gans, and the Davidoffs was an approach to design and planning derived from place. Scott Brown and Venturi, borrowing from J. B. Jackson among others, elevated and celebrated the vernacular. McHarg advocated understanding places through interactions, that is, its ecology. This view rejected fashion. Style from modernism to deconstructionism to whatever-is-next-ism became irrelevant.
The first essay in this section, originally published as a 1957 article in Architecture Record, is connected directly to both Philip Johnson and modernism. McHarg exhibits his knowledge of modernism and his concerns about the future of the city and the housing needs of its residents. During the postwar era, housing was a pressing issue for many architects and planners. McHarg had worked with the Scottish new town program in the early 1950s and was certainly familiar with English and other European housing programs. His wife, Pauline, was Dutch. Among the most extensive postwar redevelopment efforts were those in the Netherlands. Many new communities were planned, designed, and built by the Dutch (Van der Wal 1997).
McHarg brought his firsthand European experience to the New World and attempted to advocate a third housing form between Le Corbusier's high rises and Frank Lloyd Wright's prairie houses. McHarg's solution to urban housing was the town house with an interior courtyard. He observed "no modern town house has been produced to replace the traditional seventeenth and eighteenth century town house."
Although McHarg clearly admires traditional town houses, from Amsterdam to Philadelphia, he also finds much inspiration in Greek, Roman, North African, and Hispanic prototypes. In addition, McHarg identified many of his contemporaries as making advancements in court house design, most notably Johnson, whose work he praises three times in the article.
McHarg suggests the use of town houses with courtyards as a design strategy to return people to the city. He makes a number of very specific suggestions for court house design, based on an analysis of both designed exemplars and vernacular spaces. He displays his familiarity with and his interest in modern architecture. He also draws on his travels during World War II through North Africa, Italy, and Greece.
In the article, McHarg mentions that he taught design studios in which his students explored the court house concept. He also was involved as a practitioner in urban renewal projects in Society Hill in Philadelphia and in Southeast Washington, D.C. In both instances McHarg promoted a low-rise, town house design with public and private green spaces. Visiting Southeast Washington, D.C., today one can see how his work provides a strong contrast to the more dominant style of public housing during the 1950s, the high-rise apartment.
There is considerable optimism for design, urbanism, and modernism in "The Court House Concept." McHarg was still hopeful that modern designers could solve urban ills. This optimism about modernism had faded by 1970 and had been replaced by a faith in ecology and the possibilities of ecological design.
McHarg is a storyteller. One of his favorites is the Generals Overkill story. Because he has spent seven years in the army, it is a tale told with enthusiasm and rebellion. His criticisms of the Generals Overkill and the "lesser Philistines" also reveals his concern about atomic weapons as well as his criticism to the war in Vietnam. Clearly, McHarg was declaring himself a retired warrior who wanted to ban the bomb and make peace, not war.
In the case of "Architecture in an Ecological View of the World" the Generals Overkill story was adapted to an audience of architects initially at an American Institute of Architects (AIA) convention in Boston then in a 1970 AIA Journal article. This version differs somewhat from others, including the one published in A Quest for Life. Employing the device of a fantasy, McHarg attempts to explain to architects "how the world really works and demonstrates that it is an interdependent, interacting biosphere."
In this article, McHarg refutes his modernist design education and especially challenges the goal of modern architecture that simplicity should be achieved at any cost. By embracing ecology, McHarg also urges us to view the world as complex and diverse. By seeking the simple, important aspects are omitted or ignored. Complex and diverse designs have a greater chance of being sustainable.
Instead of "less is more" McHarg states that a role of design is to fit buildings to the site, which requires greater involvement with a place, a more complex approach, rather than abstraction. He notes, "Architecture should not be called architecture; it should be called fitting," or "adaptation for survival."
McHarg also addresses the subject of form, which is fundamental in design. "Every form," he states, "reflects processes engaged in the business of creative adaptation toward the end of survival; form is only one superficial expression of the processes." In short, he advocates "adaptive architecture."
His brief 1990 "Nature Is More Than a Garden" addresses the topic of garden design. Gardens are a basic design type of the landscape architect. A garden is symbolic nature: A peaceful symbol for phenomena that can be violent. As a result, gardens are a simplification. Again in this article, McHarg promotes complexity in design.
He uses the coral reefs of the South Pacific as an exemplar for designers of gardens. I am reminded of Umberto Eco's protagonist in The Island of the Day Before (1995). The seventeenth-century mariner marooned on an abandoned ship adjacent to an uninhabited Pacific Island discovers a new world, an alternative reality, in the coral reef between the ship and the island. The new life, forms, and patterns of this underwater living landscape forces the castaway to see the world anew, to discover new possibilities for his existence, his place in his environments. Designers need a similar alternative vision. Perhaps, a visit to a coral reef should become a requirement for a degree in design.
Garden design is but one type of landscape design. Landscape architects are capable of much more, as McHarg discusses in his 1997 "Landscape Architecture." In this essay the rise of landscape architecture as an art and as a profession is summarized. It is a discipline where "so few persons have accomplished so much." Landscape architecture is also a field which could contribute much more. To achieve its potential, landscape architects should adapt an ecological approach to "justify a more central and consequential role."
McHarg returns to the theme of the English landscape school which he had addressed in his "Ecological Determinism" essay in 1966. Again, he suggests that the work of Kent, Brown, and Repton should be viewed as "an important precursor" for ecological design.
In his 1997 "Ecology and Design" he challenges designers to use ecology in their work. Important contributions are the definitions of ecological planning and ecological design, the latter being the selection "for creative fitting revealed in intrinsic and expressive form."
McHarg clearly expresses sadness that the art of ecological design has not progressed more rapidly. He identifies only three landscape practices advancing ecological design: Andropogon and Coe Lee Robinson Roesch of Philadelphia and Seattle's Jones & Jones. All three practices include principals educated in both landscape architecture and architecture. This combination of an environmentally based design education with form-based education appears crucial for the development of ecological design.
I would add to McHarg's list a few architects engaged in ecological design or, at least, advocating it. Susan Maxman, Bill McDonough, Sim van der Ryn, and Peter Calthorpe are among the few ecological architects or "adaptive architects:" From my own region I suggest the architects and/or landscape architects Steve Martino, Christine Ten Eyck, Laura Bowden, John Douglas, and Joe Ewan. The influence of Design with Nature on planning, environmental management, and geographic information systems is indisputable. Its influence on design has been stronger in theory than in practice. No doubt the reader can add a few more names. But, the list would remain short. We need many more designers who are capable of making creative fittings which are revealed in expressive forms. With more architects, engineers, and landscape architects advocating sustainable, and even regenerative, design, perhaps this will change.
# References
Eco, Umberto. 1995. The Island of the Day Before. New York: Harcourt Brace & Co.
Ellin, Nan. 1996. Postmodern Urbanism. Cambridge, Mass.: Blackwell.
McHarg, Ian L. 1996. A Quest for Life. New York: Wiley.
Van der Wal, Coennard. 1997. In Praise of Common Sense, Planning the Ordinary, A Physical Planning History of the New Towns in the IJsselmeerpolders. Rotterdam, The Netherlands: 010 Publishers.
# 11
# The Court House Concept (1957)
A Penn landscape architecture student in the late 1950s was much more likely to design a town house in Ian McHarg's studio than a regional plan. McHarg had become fascinated with the court house concept in Scotland, where he had been granted permission to build a separate house, rather than live in an apartment, because of his tuberculosis. He designed a 1,200-square-foot, L-shaped house. Although he never built it, McHarg became intrigued with the concept that he later pursued with both Philip Johnson and his graduate students.
McHarg's interests in housing and open space are interwoven. His investigations into open space sought to recognize the rate and value of both public and private spaces. Privacy is critical for the latter. While this can be achieved by using house walls, free-standing walls, fences and hedges, court house design can also result in privacy. The benefits of court houses include improved microclimate and the efficient, economic use of building materials in addition to privacy. McHarg first published his ideas in the 1957 Architect's Year Book, then later the same year in this 1957 Architecture Record article.
The 1957 article seems to have lasting value. In 1997 an architecture masters thesis student pursuing housing design for Phoenix, Arizona, was given a copy. The ideas in the article influenced his design for a complex of three models of court houses ranging in size and price range. Certainly, court houses are an appropriate, environmentally responsive alternative to conventional tract houses in the Southwest United States. They offer an option for urban living in other regions, too.
The free-standing, single-family house in suburb and countryside from such hands as Frank Lloyd Wright, Mies van der Rohe, Le Corbusier, Philip Johnson, or Richard Neutra undoubtedly represents the greatest contribution of modern architecture to art and environment. Bear Run and the Farnsworth House, Villa Savoie, and the Johnson New Canaan House offer a superlative environment for the exurbanite but what of the city dweller? Is not the urgent challenge to modern architecture to provide a new town house, as humane as those designed for the suburbs and country, as urbane as is consonant with the forms and values of the city?
A glance at the palette of urban housing types will only assure us of their conventionality—and their continuing inadequacy—the multi-story flat, no more than a bigger Roman insula with water, sewer, heating, and elevator, the walk-up apartments but the insula without disguise, the terrace house shorn of nobility. This vocabulary sufficed until the city was befouled by industry, congested by traffic, despoiled by philistinism, and abandoned to dereliction. These traditional housing types have been exploited since industry changed the nature of the city—and without success. No single present housing type is capable of arresting the flight to the suburbs, no single housing type offers the prospect of a new recentralization, no single present housing type offers an environment which can equal the salubrity of the suburb. Is there an urban house which, exploiting the advantages of city life, offers in addition a residential environment at least equaling the sum of advantages and disadvantages of the house in suburb or country? Can the essence of the free standing modern house be given an urban form?
Both questions can be answered in the affirmative. As supported by a number of projects during the past twenty years the court house, the introverted house facing upon one or more internal courts can provide the essence of the best free standing country houses; it can provide a residential environment as humane as it is urbane.
Moreover its range of application is surprisingly wide. Obviously, it permits the creation of a new town house of great luxury and elegance but it also is adaptable to the creation of a popular town house at relatively high densities. The former house has the capacity to effect a selective recentralization—the return of the civilized to the cities, while the latter can provide for a recentralization of more significant proportions by providing a milieu for family life within the city.
The claims made for the court house concept are not puny, yet in the following pages can be seen their substantiation in a number of precursory projects. These are from distinguished architects in several countries. The unanimity of their concern for this concept and their projects within its formal expression deserve a generic title—the court house concept—and an analysis of the essential quality, the range of application, the advantages and disadvantages of this concept of urban housing.
# The Essential Quality of the Court House
The court house is overwhelmingly distinguished by its introversion; the house turns its back to the street and faces upon a private, internal court or courts. The historical derivation from the peristyle and atrium of Rome and Greece, the traditional Moslem house, the traditional Chinese and Japanese house is patently obvious and these plans indicate the geometric possibilities within the court house concept. The rectangular house and single walled court, the "L" house with one court, the square house with a single internal space, "H" and "T" plans each with two courts, the "Y" plan with three courts, the cruciform house having four courts and finally the possibilities of asymmetrical plans with numerous courts, these plans are united by the single fact that the quality of experience is oriented to the private delectation of the occupants.
The major problem of modern housing and its most conspicuous failure lie in the distribution and design of open space. Modern architecture has produced classic models for the two extreme conditions and no intermediate solutions. On the one hand, as can be seen in the Farnsworth House by Mies van der Rohe and the New Canaan House by Philip Johnson, the free standing house is placed in an area of beautiful natural landscape where its geometric purity contrasts vividly with the organic sculpture and texture of ground and trees. On the other hand Le Corbusier has provided the alternative model of the tall slab rising free from a lush romantic landscape. But for the intermediate problems of central redevelopment, urban renewal, or a new town there are no adequate solutions. Indeed, in the case of the latter it is the inadequacy of the distribution and design of open space which is the key to the failure of the new towns. There open space, offered as a boon, is in fact a liability; it contributes to neither amenity nor society while it disrupts any attempt to create "town," and finally it contains no art. Open space is over-provided, usable open space is all but absent and the end product is a scene of insignificant houses lost in neutral open space.
Yet we have seen historically from Ancient Greece to the eighteenth century that good standards of open space are not antithetical to urban architecture. The eighteenth-century squares represent a convincing defense of this position. Certainly there are intermediate solutions to be discovered appropriate to the problems and expression of modern architecture.
It is interesting to observe that most of the problems of open space are solved integrally within the plan of the court house. Open space in courts is provided internally and disassociated from the street scene. These courts are private and this characteristic is the most valuable aspect of open space in housing. The courts can associate with internal functions to provide a living court, dining court, bedroom court, service court, etc. and provide a value to open space absent in all other plan types. By bringing the open space inside the house the scale of the street can be determined by traffic consideration and the height of the houses rather than being an uncontrollable sum of street and external open space. Finally the individual has a proclivity and a right to self-expression in the design of open space. In current housing where this is freely expressed the result is often anarchy, vulgarity, or both, where, as in the new towns, it is suppressed, a neutral mediocrity is the product. In the court house the individual has freedom for personal expression which cannot obtrude on the public.
The court in the Orient has long been given a formal expression representing the essence of nature and is so realized with the philosophical assumption that man and nature are indivisible, and that contemplation, soliloquy, and calm can best occur in nature, its essence if not its image, and the court then is made to represent the essence of nature. Again the Moslem court, based upon the Moslem concept of Paradise, has a religious basis for its formal expression in the Island of the Blessed and the four rivers of Paradise. The West has no comparable basis for the formal expression of courts within houses and this must yet emerge as indeed the philosophy for the formal expression of a modern landscape architecture. One direction of this expression may be seen in the court of the Rockefeller House in New York, designed by Philip Johnson. In this as in other projects by Johnson one sees an esthetic consonant with the main stream of modern architecture and perhaps the precursor of the modern expression of the court in the modern court house. In the Western court as in the Oriental, the value of this space is not alone in its functions but rather in its evocative use of natural and inert materials to induce soliloquy, calm, and an attachment to permanent values—in a humane environment.
The court house requires the expenditure of a perimeter wall to create internal open spaces. The courts are the product. These are justified by the intimate relationship established between indoor and outdoor spaces. Indeed the courts become rooms outdoors. This being so the degree of privacy accorded to internal rooms is the minimum necessary for the courts.
In every historical epoch save our own the inevitable residence for the cultivated and influential, the civilized, polite, and urbane as persons or families was the city. Today only a determined minority persists in the central city from choice and the family has been all but banished. The majority of the urban population lives in the city only because it lacks the economic mobility to escape and perforce lives in the slums, the meanness or the sanitary order of nineteenth- and twentieth-century environments. The major population movement in this century has been the flight from the city, the greater the economic mobility the further the residence from its center. The civilized, who traditionally made the city their own, the repository and the artifact of their culture, have today fled to the outermost suburbs. In Western Europe and the United States where they persist in cities they live in the reservoir of earlier historical town houses—in the Georgian and Regency squares and mews of Bloomsbury and Chelsea in London, the new town of Edinburgh, the seventeenth-century "Grachtenhuizen" of Amsterdam, the fifteenth-century Palazzo Farnese, Quartiere del Rinascimento, and via Giulia in Rome, the Marais and Isle de St. Louis in Paris, in Beacon Hill in Boston, Rittenhouse Square and Society Hill in Philadelphia, and Greenwich Village in New York. Such town houses are no longer appropriate as prototypes, changed mores have caused their adaptation to contemporary residential needs while they are continually pre-empted for office space. Yet no modern town house has been produced to replace the traditional seventeenth- and eighteenth-century town house.
The metamorphosis necessary to create the mid-twentieth century city, stimulating yet humane, representing the zenith of the regional or national art, architecture, technology, social and cultural expression, this task poses many problems, but at their heart lies the necessity of returning to the city and accommodating there that segment of society most concerned —with the problems of the city and best able to solve them. Not least then among the problems of the new city is the provision of a new town house. Such a house, while exploiting the existing advantages of city life must offer in addition a residential environment, at least equaling the sum of advantages and disadvantages of suburban environment. Without the introduction of such a new town house, that recentralization, which is the essence of the city's survival, cannot occur. Only by the provision of an urban residential environment at least equal in salubrity to that of the suburbs will the civilized, polite, urbane as persons and families return to the city to ensure its survival.
The court house represents such a town house, its merits are patent, they lie not in modernity of the concept (unless modern is used as a synonym for good) but in an intrinsic excellence which its historical persistence substantiates. The court house does meet the criteria which have been advanced. It can provide a humane urban environment, an urbane architecture, a milieu for family life within the city; it does promise that recentralization which is the prerequisite to the survival of the city.
# Court House Plans
During the past five years, recurring sporadically but with increasing frequency, the court house has appeared in the architectural press. It would appear that these projects constitute a precursory movement towards widening acceptance of the court house as the new town house.
As a modern housing solution the court house concept can be loosely identified with four principal sources, only one of which is a direct extension of a historical form. The prime source stems from Mies van der Rohe and includes L. Hilberseimer and Philip Johnson; the second group, a logical extension of its Mediterranean origins, is located in Italy and, starting from the studies of Pagano, Diotallevi, and Marescotti, reached fruition in the housing at Tuscolano, Rome by Adalberto Libera. The third group, also deriving from the Mediterranean is identified with José Sert, his partner, Wiener and their collaborators in many projects for Latin American cities. The fourth group is a direct extension of the Moslem tradition in North Africa directed today by French architects. Each of these studies originated before or during the war. In the postwar period, and particularly during the past two or three years the court house concept has gathered many new advocates in a wide range of countries—Mexico, Britain, the United States, Denmark, Sweden, Germany, and Switzerland, and consequently is now advocated for other than Mediterranean and South American locations.
# Mies van der Rohe—Hilberseimer—Johnson
Perhaps the first court house from the hands of a modern architect is a study by Mies van der Rohe developed in 1931. This is essentially a row house and is so described but each house is an "L" enclosing a private open court. This concept was elaborated and in 1938 Mies developed a more complex and princely group of three court houses associated in a block, each house distinct and each having a varying number of courts. This represents in the author's opinion the zenith of court house designs. Although at a density of approximately three houses to the acre, and consequently conspicuously open for urban conditions, there can be no doubt but that such development could fulfill the vital role of proffering an urban house equaling the best conditions of a suburban environment yet additionally offering the convenience of city life. A more recent project by Mies van der Rohe, the Gratiot in Detroit, falls between the studies of 1931 and 1938. In this housing complex court houses are interposed between garden apartments and multi-story flats. With one three-bedroom court house, these dwellings are conceived as middle-class, family houses. The court houses are designed in groups of six, arranged around two entrance courts which are in turn entered from a covered carport. If the 1938 project represents an urban alternative to the expressive suburban or rural house, the court houses of Gratiot offer a middle-class equivalent. These projects by Mies van der Rohe are perhaps the best evidence for the court house as a middle- and high-cost urban house. Yet the quality of the court is still not indicated.
In 1942 Philip C. Johnson, biographer and collaborator of Mies, built a simple court house in Cambridge, Massachusetts. This house did not aspire to the luxury of the court house project by Mies but, with a small and elegant structure and one private court, there was developed a conjunction of structural space and social use impossible to achieve in such an area with a conventional plan. The court in this house remained simply as lawn, shaded by large trees, but in 1949, for the Rockefeller town house in New York, Johnson created a court with one pool, a tree, a fountain, terrace, and three stepping stones. The impact of these few elements in this small space, is overwhelming in contrast to the heat, fumes, noise, overpowering scale, and tension of midtown Manhattan. This court, as a demonstration of the quality which can be achieved in a small urban space, is one of the most powerful exhibits as evidence of the validity of the court house concept and supplements the statements of Mies van der Rohe in which the court itself remains an enigma.
Prior to the projects of Philip Johnson, L. Hilberseimer, colleague of Mies van der Rohe at the Bauhaus and the Illinois Institute of Technology, had developed the court house concept as a form of popular housing. Utilizing the L-shaped, single-story house, he achieved densities of 120 persons or 20 houses per acre. This demonstration shows the court house as a popular housing solution applicable to central residential areas. In this Hilberseimer is supported by the historical precedents of Ancient Greece and Rome, ancient and modern North Africa and China.
# The Court House in Italy
In 1940 the Italian architects Pagano, Diotallevi, and Marescotti made an extensive study on the court house concept which was published in Costruzioni Casabella (Pagano et al. 1940). The results of this examination are reflected in the writing of Eglo Benincasa today and more concretely in the housing project in the Tuscolano quarter of Rome, designed by Adalberto Libera and constructed, as low-cost housing, by Ina Casa. This project appears to be the first modern example of the court house used as a popular urban house. It has an approximate net density of 30 houses per acre and, with seven beds per house, an envisaged net density of over 200 persons per acre. The repeated block plan, which consists of four L-shaped houses, is a novel one. Three of the courts face inward while the remaining one opens to the pedestrian footpath. The courts are extremely small for the seven occupants of each house but this is in part compensated by common open space provided in the development.
The appropriateness of the court house concept to the Mediterranean would seem evident. Its appropriateness as a more general solution is indicated by Eglo Benincasa in L'Architettura: "Foreigners admire the 'picturesqueness' of the south, but underlying it there is a livability which the modern architect overlooks in popular housing . . . The sheltered space constitutes a challenge to modern architecture. It is a free creation linking the house to nature" (Benincasa 1955).
# The Court House in South and Central America
The third line of court house development is represented by José Sert, his partner Wiener, and certain Mexican architects, notably Victor de la Lama. Sert, a Spaniard, has adapted the Moslem heritage of the court house to his native land and made this the prototype popular house in his planning and housing studies for Central and South America. In the context of Latin America with a high sun angle, high temperatures, and the Latin demand for privacy and protection the court house, its tradition, remains the most valid housing solution. In the hands of Sert and his collaborators the court house also represents an admirable means of ameliorating the micro-climate, a resource inherent in the plan but more vital in high temperature locales. Wiener and Sert are particularly distinguished by their long-established attachment to this concept and by the skill with which they have utilized it as the basis for larger planning scales—nucleus, neighborhood, and town. Indeed the distinctive contribution of these architects is that they have created a humane city in which the court house concept is the unifying element, appropriate to the climate, technology, art, and mores of South America.
The typical court house plan utilized by Wiener and Sert is the "L" house with a service court cut from the exterior angle. This house is associated in groups of four to form a rectangle with the living courts located internally. This attitude to orientation, acceptable in low latitudes, makes such a juxtaposition possible whereas it might well be questioned for other climates.
Whereas Wiener and Sert have utilized the court house as a low-cost popular house, its exploitation in Mexico has been on a more lavish scale. The projects of Victor de la Lama particularly show the court house in terms analogous to the Mies van der Rohe 1938 court house. Each house contains a number of courts with swimming pools and lush sub-tropical vegetation. These demonstrate a standard of environment, albeit for a climate distinct from either Western Europe or the United States, which compares favorably with the free-standing suburban or rural house.
# The Court House in North Africa
In the eastern hemisphere the court house has persisted as a continuous concept only in the Moslem world. This housing type constitutes a vernacular form, but it has also been utilized and exploited by modern architects as a contemporary solution. Numerous French architects practicing in North Africa have used the court house as the prototype for Moslem housing projects. These developments are generally low cost with minimal standards; indeed it is apparent that while normally bedrooms, kitchen, and bathroom are provided in the house, the living room, the only living space, is the open court. The court house concept in the hands of the architect J. De La Roziere has been used not only as a house plan but as the basis for a neighborhood plan with a hierarchy of open spaces extending from the private court to larger open spaces shared by nuclei to the final open space shared by the entire community (De La Roziere in Ecochard 1955, p. 37).
# Recent Court Houses
Developing from the court houses of these four groups are a large number of recent projects from a wide range of countries. From a Mediterranean and Hispano-American base the concept has entered throughout Europe and penetrated North America. Almost all the major European countries are represented in court house projects, and this housing type has been advanced for countries as distant from the Mediterranean as Denmark. These projects include L- shaped court houses by Custer, Escher, Gisel, and Weilenmann proposed as an extension to the Neubuhl in Zurich, the "atrium house" by George Schwindowski of Berlin, studies by Eske Kristensen, and John Utzon in Scandinavia, projects by Netherland students, the Smithson house for the Ideal Homes Exhibition 1956 with which a written explanation reads "all rainwater is collected and runs through a gargoyle to a container in the internal garden!" These projects, while amenable for use in urban situations, are not so proposed. In contrast, a project by the architects Chamberlin, Powell, and Bon, utilizing the court house concept aims at central urban locations and high densities. This solution, with an average height of one and a half stories provides a net density of 160 persons or about 30 houses per acre. The aim to maximize the applicability of the court house by maximizing density is laudable and the solution is extraordinarily ingenious. Particularly noteworthy is the success achieved in either eliminating or minimizing the degree of overlooking into private courts while employing a two-story bedroom unit. Yet there is some reason to believe that the density may be excessively high for this house type and that with the low sun angle of Britain the courts might well become insalubrious holes. Further, where the courts are so small it would appear vital to provide a secondary area of open space which this proposal fails to grant. With the private court provided for the family, yet another area should exist as focus and venue for social intercourse in the community. Yet this is perhaps the best example to date of the court house as a popular urban house. Densities of 160 persons per acre or even 100 persons to the acre, supplemented by tall buildings can, as the architects indicate, provide densities perfectly compatible with urban land values and building values.
Beyond the Mediterranean and Latin America, it is in the United States that the court house concept has taken the strongest hold. This is assuredly not unconnected with the presence of Mies van der Rohe, Philip Johnson, and L. Hilberseimer and their attachment to this concept. It may well also be influenced by the existence of a Hispano-American tradition in California and the southwestern United States. The preoccupation with the court house is evident but remains to date in the project phase. In addition to those architects mentioned above, Serge Chermayeff, Oskar Stonorov, Ralph Rapson, Donald Olsen, and Morse Payne have all produced court houses or court house complexes. Donald Olsen (1948) has designed a swastika house with four courts; that by Ralph Rapson has a single internal court (1948, p. 119).
The project by Morse Payne exploits the court house concept in a distinct way. The lot is a long rectangle which is divided into three or four courts—by the interposition of building volumes, usually, two or three in number—the living-dining-kitchen volume, the bedroom volume or the bedroom function divided into the master bedroom volume and the children-guest bedroom volume. By raising certain of these volumes the entire space can be continuous or divided into distinct courts associated with specific functions. This plan might well be described as the in-line court house and represents an admirable addition to the vocabulary. The density is approximately 15 houses or 90 habitable rooms per acre.
Serge Chermayeff, Oskar Stonorov, and Mies van der Rohe have during the past year developed not only court houses but have associated these in small groups. In the study which Chermayeff has conducted with students of the Harvard Graduate School of Design, the house, with a "U" plan has three courts—two external and one internal. This prototype house is associated in groups of six, but could be exploited as a row house. Before Mies van der Rohe designed his Gratiot project Stonorov prepared a scheme for this development in which he utilized court houses. This project aims for a higher net density for the court houses than does the subsequent scheme by Mies van der Rohe. To achieve this, two-story quatrefoil court houses were developed. Given the absolute necessity of building two-story structures this design is admirable, but it must be observed that all courts suffer from overlooking and the value of the court is depreciated.
The court house as a house plan has been thoroughly demonstrated and in the projects by Mies van der Rohe, Stonorov, and Chermayeff can be seen small nuclei made with four to six court houses. In the Gratiot project developed by Mies the group of houses depend upon a single-car court and each trio of houses share a common entrance court. However, save in the single project of Adalberto Libera in Rome there is no evidence of the court house as an element in a community. Toward the end of exploiting the court house as an element in the small urban community, graduate students of the University of Pennsylvania under the direction of the author made studies for a central Philadelphia city block site.
In this study the intention was not only to exploit the court house but to create a community which would be as salubrious at the community level as would be the court house at the dwelling level. The program required houses from 1,500 to 3,000 square feet with provision for one car per house. To create the court house community it was required, as in the subsequent Gratiot projects, that groups of six to ten court houses depend upon one common court acting as forecourt and that beyond that a central open space with small playground and swimming pool be provided as the venue for social intercourse in the block. The net density averaged twenty houses per acre, the gross density within the block was half that figure.
It is submitted that this final stage of utilizing the court houses around a sequence of open space as venue for social life constitutes an advance in the exploitation of the court house, that it does present an environment as urbane as it is humane, consonant with the forms and values of the city. The author, in collaboration with Philip Johnson, is in the process of designing such a project in central Philadelphia.
# Conclusion
The housing solutions which have been advanced or utilized in this century fall into three main categories; the two extremes of the multi-story flat and the free-standing, single-family house, and, between these, the terrace-row semi-detached house. The multi-story flat has particularly been espoused by the leaders of modern architecture. From Le Voisin, La Ville Radieuse, The Siemensstadt, "De Plaslaon" in Rotterdam, and their derivatives to the Unites d'Habitation, modern architecture had advanced the multi-story apartment as the prime urban house type. Yet it hardly justifies this devotion and dependence on the grounds of its modernity. Some reaction to this preoccupation and devotion to the flat has appeared within C.I.A.M., founder and prime protagonist of modern vertical living. At C.I.A.M. 10 the English members W. and G. Howell, J. A. Partridge state that "Even if and when we have built up a successful tradition and practice of multi-level living, which we are very far from having done—we feel that—there will always be a demand for a considerable proportion of (town) houses. And if the program demands it we must find ways of using them as elements in the city."
Although having a recurring role for two thousand years, the multi-story apartment has not built a successful tradition; it has found only limited acceptance and far from offering the key to a new and selective recentralization, is unable to arrest decentralization.
The realization must be all the more disheartening to the protagonists of vertical living in view of the fact that the concept has been advanced during a severe housing shortage when choice was severely restricted. The Unites, Barbican, High Paddington, the Smithson concept, Gratiot all point towards improvement in multi-story development, and the tall apartment will continue to play an important role in central city housing. However, it is not a sovereign cure. The vocabulary urgently needs to be expanded; the court house constitutes an invaluable addition to the vocabulary of central housing. The court house can return the civilized, the urbane, and polite to the cities, offer the milieu for family life within the city, and provide an environment as urbane as it is delightful.
# References
Benincasa, Eglo. 1955. "L'arte di abitare nel mezzogiorno: il colore." L'Architettura Cronache e Storia 1(BO1, May-June):58–63.
Ecochard, Michael. 1955. "Habitat Musulman au Maroc." L'Architecture D'Aujourd' hui 26 (60, June):36–39.
Olsen, Donald. 1948. "The Contraspatial House." Arts and Architecture (April):32-37.
Pagano, P. G., I. Diotallevi, and F. Marescotti. 1940. "La casa unit=EO." Costruzioni Casabella XII (B0148, April):11–14.
Rapson, Ralph. 1948. "Case Study House No. 4." Interiors (September):116, 117–19.
# 12
# Architecture in an Ecological View of the World (1970)
Ian McHarg spoke frequently at national architecture conferences and major architecture schools. Here, employing the device of fantasy, he attempts to show the architect "how the world really works" and demonstrates that it is difficult to separate what is environment from what is human. This AIA Journal article is based on excerpts from an address delivered during the "Day of Awareness" at the 1970 American Institute of Architects meeting in Boston.
The first proposition I have to make is that the attitude of man to environment which permeates the Western tradition is a fantasy. It has no correspondence with reality, no survival value, and it is the best guarantee of extinction.
Once upon a time, I met a scientist who was engaged in an experiment to send a man to the moon with the least possible luggage. This experiment was conducted ten years ago when it was thought that going to the moon was worthwhile and would take quite a long time. The experiment then had to be a recirculating process.
It consisted of a plywood capsule simulating a real capsule, in the lid of which was a fluorescent tube simulating the sun. Inside the capsule were some algae—microscopic, photosynthetic plants—some bacteria, some air, and a man. The man breathed in air, consumed oxygen, and breathed out carbon dioxide which the algae breathed and gave out as oxygen which the man breathed. The man became thirsty, drank some water, which, when he urinated, went into the water solution with the algae-bacteria. The algae transpired, the transpirations were collected, and the man drank. The man became hungry, ate some of the algae and defecated. The excrement went into the water medium with the algae and bacteria, the bacteria reconstituted the excrement into forms utilizable by the algae which grew, which the man ate. Now that's how the world works.
What we need to do is to recognize that that capsule is a beautiful simulation of the world-at-large. There is only one input: sunlight. There is only one output: heat. There is a closed cycle of water and food. We are plant parasites. We don't know how to accomplish photosynthesis; we have to depend upon microorganisms which know how to reconstitute the wastes of life. Our requirements are oxygen and food.
This little experiment absolutely changed my life ten years ago. It can change more lives. Let us use it here for instructional purposes.
What we need is a celebratory event which I will describe as "Fireworks at Cape Kennedy." We require a cast of characters for our Independence Day event.
We need to assemble a cast of arch-destroyers. We collect all of the ossified, calcified, implacable Generals Overkill, those people in the Defense Departments of all countries, for whom it is not enough to be able to kill every man, woman, and child in the world a thousand times, but who must devote their energies and half the treasury of the United States to killing every man, woman, and child in the world two thousand times over. We assemble them all and measure each one for an algal-bacterial waistcoat-capsule.
We also assemble the mutational retrogressionists of Atomic Energy Commissions—those people who attack the world at its gonads, who are unmoved by the knowledge that every increase in radiation is an increase in mutation. They began life, in my imagination, pulling wings from flies, gravitated to cherry bombs, made their way to high explosives and realized full gratification only by the accomplishment of atomic explosions. We must assemble them at Cape Kennedy and give them their algal-bacterial capsules.
Then we collect those particular putrescences, the scientists engaged in biochemical warfare who are not satisfied with the bubonic plague but must cultivate more virulent forms. And we move down to the "reign of death" people, those engaged in the business of selling death in the form of herbicides, pesticides, and toxins to satisfy stockholders, the great captains of industry who carelessly, cynically, void their excrement into the environment, who engage us in enormous gluttony by which six percent of the world's population consumes 60 percent of the world's nonrenewable resources. And we assemble, too, those negligent automobile manufacturers of Detroit who give the greatest amount of pollution for the least amount of transportation—those people who are engaged in the aphrodisiac business but should be in the business of transportation.
And so we have this vast array of all the destroyers, every one of them with a Saturn rocket and with an algal-bacterial waistcoat because they are going off into space. With 500,000 cheering schoolchildren waving American flags and with bands playing the national anthem, we send the lot of them into space—all these animated, planetary diseases who masquerade as men and whose best efforts are to inflict lesions on the world body. Bang! Bang! Bang! They are off in space.
If one listens intently, he will begin to hear an Aeolian hymn of regeneration. All the lesions inflicted by men on the world body will begin to heal; every single wound will generate new cells. But we can't wait because we have to follow the path of General Overkill; we have to follow him in space because his conversion is our conversion. We carry within ourselves, right from our mother's milk, this same implicit, explicit view of man and nature. We may not be to the same extent the destroyer General Overkill is, but even architecture has within itself a profound sterilizing power, an inordinate ignorance, malice, and arrogance.
General Overkill spends lonely weeks in space, and the earth is a distant orb. He is extremely lonely with only two companions to address: the algae and the bacteria. "I am divine," he says. Man is made in the image of God—no atoms, no molecules, no cells, no unicellular organisms, no plants, no animals, save one, is made in the image of God. This means there is only one moral arena: the acts of man to man. God looks carefully if you commit adultery; if you covet your neighbor's wife, He will rap you straight across the knuckles. But if you kill every whale in the world, kill every Ponderosa pine, devastate great areas, make Lake Erie septic, apparently neither God, the churches, the priests, nor the courts care about the acts of man to nature.
General Overkill says, "I am divine." The alga doesn't say anything, just holds its hand up to the sunlight. The General says, "I have dominion over you." That's what the Book of Genesis says, that man is given dominion over life and nonlife. And if you have any doubt about the intended meaning, the text continues, man shall multiply and subjugate the earth. Subjugate.
Five or six weeks pass, and the General, who went to West Point and knows about probability theory, realizes that this is a recirculating system. He realizes that the biomass of the algae and bacteria is equal to the biomass of the General. There's going to be a time, according to probability theory, when everything that had been algae and bacteria at the onset will be General; and everything that had been General at the onset will be algae and bacteria.
What's true of the capsule is true of the world—the same interacting, recirculating process. The General realizes it is an interacting system, and there is going to be a point in time when everything that had been algae and bacteria will be General. He contemplates the assumptions about exclusive divinity, dominion, and subjugation. He concludes that divinity is pervasive, not exclusive. The algae know how to accomplish photosynthesis; the General and modern science do not. The bacteria know how to reconstitute waste; neither the General nor modern science does. Any act of dominion or subjugation on the part of the General can only inhibit the capacity of the algae and bacteria to perform that which they can perform uniquely. If the General subjugates them, he will kill them. This, of course, involves him not only in self-mutilation but in suicide.
And so the General learns the lesson we all must learn: Any relationship which concludes that man is exclusively divine and everything else is rubbish is an illusion and a fantasy. The acts of man to man are sacramental, and the acts of man to nature must be sacramental also. There is absolutely no place for subjugation. The exercise of dominion and subjugation extended to the world can only mean self-mutilation, suicide, genocide, biocide—an absolutely profound lesson.
We have to know the way the world works. Architecture is a device to deny the student any possibility of understanding human physiology, psychology, human behavior, or the realities of biophysical processes. This is a horrifying thing to say but absolutely true. I speak as one who teaches in an architectural school and who has visited every important school of architecture in the United States.
Let us return to the General. He has learned that there is one system: nature. It is an extension of ourselves; we are an extension of nature. There is no division between man and environment; we are one thing. There is one world, one biosphere which shares one history of which we are present, sensate expression. There is only man/nature.
Recall that the Defense Department has an alliance with the Atomic Energy Commission. Yet it would be calamitous if this ally left some radioactive material in the capsule. Now any increase in radioactivity inside the capsule is likely to cause mutations in either the General, the algae, or the bacteria. So the General says, "In my capsule, no increase in the level of radiation." If that's good enough for the General it is good enough for us. Every increase in atomic radiation whether from atomic testing, reactors, Gas Buggy, or Alaskan harbors is going to show up in increased numbers of cases of leukemia, skin cancer, bone cancer, mutation deformations, still births; in sum, mutational retrogression.
It would be calamitous if any of the General's cohorts in biochemical warfare who are destroying Vietnam happened to leave some pesticides or herbicides in the capsule because the herbicides would kill the algae, the pesticides, the bacteria—and that would take out the General. The General says, "We've got a great thing going in Vietnam, but in my capsule there will be no agents of biochemical warfare, no pesticides, no herbicides." If it's good enough for the General, it is good enough for the world.
The question is, under what conditions do we let this man and all the others back? (There are tens of thousands of these arch-destroyers going through this same experience in space in my fantasy.) It's a great experiment because if you lose them, what have you lost? Nonetheless, we have to be decent enough to say, "No, you are arch-destroyers, but if you are indeed converted, we will let you back." Under what conditions? I would require a plain prayer.
The General would address the sun and say, "Shine that we may live." He would address the earth simply as "home." He would say to the oceans, "ancient home." He would speak to the clouds, rain, rivers, and say, "Replenish us from the sea, we erstwhile sea creatures who have escaped from the sea by only the length of a single cell." He would say to the atmosphere, "Protect and sustain us," knowing that the life-giving oxygen within that atmosphere was derived from all the breaths of long dead and now living plants. Then he would address the plants with inordinate deference, saying, "Plants, live, breathe, grow that we may breathe, eat, and live." He would say to all the decomposers—carrion-eaters, worms, and grubs—"Decomposers, please reconstitute the waste of life in life and the substance of life after death in order that life may endure."
When he says these things with understanding and deference, we can say to the General, "Come on home; welcome to the blue, green, glorious, wonderfully rich, diverse, ancient, and enduring world, where the evolution of life has persisted for 2 ½ billion years. Welcome back with this new deference, born of understanding, which allows us to say that you may now exercise your creative will upon this earth, and we may give you this freedom in the full certainty that your interventions will contribute to our possibility of survival and the more distant possibility of fulfillment."
In the capsule is a beginning of another view which says this is an interdependent, interacting world, and it is very difficult to decide what is environment and what is man and the difference between them. The only differences among the algae, the bacteria, and man are the apertures of the genetic code. They are united.
Almost all architects, planners, and landscape architects (although the latter are less criminal than the rest) should be handcuffed and their licenses taken away until they learn the way the world works. At the moment, one can toss a coin and decide the consequences of the acts of almost any architect or planner. The consequences of their interventions, honorable and passionate men though they be, have an equal chance of being neutral, detrimental, or beneficial. This is calamitous. So we must find another view which guarantees that simple, decent, and honorable men work within a context which corresponds to reality so that their small, modest interventions accrete toward something which is creative and enhances life.
There is something called creativity. Moreover, creativity has nothing to do with precious art. Creativity in fact has permeated all matter and all life in all time and does so now.
Creation can be defined as the employment of energy and matter in order to raise matter and energy to higher levels of order. Its antithesis is reduction or entropy which consists of matter and energy going from a higher to a lower level of order. You can envisage this reduction by thinking of a forest and measuring everything in it. Measure all the oxygen, carbon, nitrogen, phosphorus, macronutrients, micronutrients—everything. Then burn it and nothing will have happened. No matter has been created or destroyed, but the matter has gone from highly developed and ordered biological elements to inert and simple ones, heat, carbon dioxide, and water.
Evolution, of course, is the reverse of this, and so is creativity in which matter is raised from a lower to a higher level of order, like taking the inert world and suddenly covering it with a bioskin. All the creativity which has engaged all matter and all life in all time is represented by the orderings of matter, represented by the atoms in the periodic table, the evolution of compounds, life forms, ecosystems, and the biosphere. All of the ordering accomplished by all life in all time, all the potential represented by them in cells, tissues, organs, organisms, ecosystems, their apperception, symbiotic relationships, and a genetic potential—these constitute the sum of creativity which all evolution has accomplished in all time.
This conception, that matter and energy are creative, have been creative, and are now creative is a very different view from the one which assumes that nature is a sort of backdrop to a human play in which man plays his uniquely creative role. In the ecological view, we are uncertain about what creative role man has. We know that he is an arch-destroyer of geological dimensions, but his creative role in the biosphere is hard to discern.
We can't possibly believe in a malevolent God who would deny the possibility of any creative role to man. There must be a creative role for man. That is our challenge, prospect, and future. The conception of the biosphere as a creative process which engages all matter in all life in all time is a new, and for me, profoundly important conception.
Evolution has been both creative and retrogressive, but the sum of all the processes has been positively creative. This allows us to look at all the creatures which surround us and all the processes in quite a different way as creative process. If evolution has been creative, let us see what attributes evolution has demonstrated because these then reveal criteria for creativity. Evolution has proceeded from greater to lesser randomness, from simplicity to complexity, from uniformity to diversity, from instability to stability, from a low to a high number of species, from a low to a high number of cooperative mechanisms. Indeed, in sum, from the tendency to degradation, which is entropy, toward a tendency to increasing order, which is negentropy. That is a marvelous model.
When I went to Harvard, Dean Joseph Hudnut used to say of modern architecture, "simplicity at any cost." This was indeed the name of modern architecture. Simplicity in the biological world is a pejorative term because simplicity is the antithesis of complexity and evolution moves from simplicity to complexity. So when an architect designs a simple building, this probably reveals not the complexity of the situation he is trying to solve but simply his simple-mindedness.
We have a model of creativity which applies to any system whether it is a house, an individual, a family, a community, or an institution like the American Institute of Architects, or cells in an organism, or the ecosystems in the biosphere. We have a conception about creativity and the dynamics of the process. If we see a trend from complex to simple, it is retrogressing. If it goes from instability to stability, it is evolving.
There is another term which indicates the degree to which any process is evolving or retrogressing is being creative or reductive. And this term is the conception of fitness. Whether you know it or not, you are engaged in the business of fitting. Architecture should not be called architecture; it should be called fitting.
This verb "to fit" is of profound importance. Charles Darwin said that the surviving organism is fit for the environment. On the other hand, Lawrence Henderson has said that the actual world, with all the variability of environments constitutes the fittest possible abode for all life that has existed, does exist, or will exist.
You can think of yourself, your cells, your tissues, your organs, your institutions and consider all available environments for them. Among the multiplicity of environments, there are most-fit environments. There is a requirement, not only to find the most-fit environment but also to adapt that environment and/or yourself in order to accomplish fitting. The fit survive, according to Darwin.
So we are engaged inextricably in the process of finding fit environments and adapting them and ourselves. We are in this business of adaptation for survival. That is the real definition of architecture.
Where you find that most fit environment and adapt it and yourself to accomplish a fitting, you accomplish a creative fitting in thermodynamic terms. This is real creativity in which every organism and every ecosystem is intensely involved throughout all life.
You are engaged in a creative process, which has nothing to do with long hair, whether you wear sandals, whether you wash or don't wash. This is the implacable test which engages all creatures in all time and must engage all men in all time in order to ensure survival. When done, it is creative, and the measure of its creativity is survival of the process.
Because this whole system is in fact one system, only divided by men's minds and by the myopia which is called education, there is another simple term which synthesizes the degree to which any intervention is creative and accomplishes a creative fitting. And that is the presence of health. Wherever you find health—physical, social, mental in human societies, or physiological in nonhuman ecosystems—you have found absolute, explicit, irrefutable evidence of creative fitting. Any process which has found the fittest environment, which has been able to adapt that environment and itself to accomplish a creative fitting, is healthy. Wherever pathology is found at any level, in cell, tissue, organ, organism, institution, or ecosystem, there is evidence of a misfit, a reductive misfit; and the extension of this pathology will lead to the death of the species, the institution, or the ecosystem.
There is also the subject of form. Form and process are indivisible aspects of a single phenomenon. There is no such thing as abstract form; there is no such thing as capricious form or unmeaningful form. Form and process are indivisible. If one wishes to describe an atom, molecule, crystal, or compound, one can describe it only in formal terms. If one wishes to describe a cell, tissue, organ, organism, or ecosystem, one can do so only in formal terms. All form is meaningful. The degree to which meaning can be perceived is a function of the ability of the observer to perceive the meaning which is intrinsic. One can only understand what is in terms of evolutionary history—evolution of form and process.
The only way to understand the Appalachian Mountains is in terms of the fact that they once were an ancient sea; the only way to understand the molehills of the piedmont is to know that they were once 27,000 feet high. One can understand that which is only in terms of that which has been. That which is, has been and is in the process of becoming. It was process-form, is process-form, and will become process-form.
Every form reflects processes engaged in the business of creative adaptation toward the end of survival; form is only one superficial expression of the processes. Butterflies don't want to be pretty to make us happy. Sphagnum moss doesn't want to look like sphagnum moss. Sphagnum moss is, and as a process is, absolutely appropriate to the business of surviving while being sphagnum moss. Form and process are indivisible. Moreover there is generic form to cells, crystals, vascular processes, nervous systems, plants, and animals. All we have been talking about are processes—processes that have been subject to evolutionary tests over unimaginable periods of time and have been refined by the test of survival. The processes survive not only in terms of process but also in generic form.
There must be generic form in architecture, which leads us to something which must be called adaptive architecture—not architecture to gratify the muddy psyches of architects but architecture as a device by which man can adapt toward survival and the more distant prospect of fulfillment.
One has to think about the client, whether an individual, an institution, or a society. We must identify the organism and the environment because we are in the business of trying to accomplish a creative fitting, which creative fitting is inescapably involved in the business of form. There will be a form of fitting which is most fitting.
Homeostasis is a device which, without the intervention of the brain, is able to deal with the environment of energies and interpose various membranes and processes between these energies and the internal system. Without thought, we can maintain 98.6° F body temperature, and when we fail by a degree or two, we have a fever. Homeostasis is able to deal with environmental energies and through homeostatic controls maintain equilibrium. Architecture is in the same business but is dealing with a larger array of environmental energies. So we have the conception of the organism, its own homeostatic devices, and all the other energies which constitute the environment. We have all of environment variability, the organisms and their proclivities. We seek to find the most-fit environment of all.
That's not enough. We must adapt the environment and ourselves. We have to adapt in order to ensure survival and fulfillment of the system. We have to identify the environment, which brings me to my bailiwick because I am engaged in trying to find fit environments, to identify regions, as a range of opportunities and constraints for all prospective land uses and to fit these demands to available resources.
What I really do is called ecological planning and it simply consists of inventorying the environment so that one understands the way the world works, not only as biophysical process but also as opportunities and constraints for all prospective land uses.
First, you have to assemble those people who are competent. This is an outrageous novelty which architects don't ever consider. The great problem with ecological planning is that you are not allowed to speak in the absence of evidence. We'll start at the beginning, which, first and most important, is bedrock geology. That gives us 500 million years of evidence. We employ the man who knows about bedrock geology, and he describes the geological history and geomorphology of the region.
Then we hire the biometerologist who understands climate. Then we ask the two to get together because the interaction of climate and bedrock geology over the past million years is reflected in surficial geology. If you understand surficial geology, you then are in the process of understanding hydrology. That enables you to understand why rivers are and where they are; whether there is underground water or not.
Once you understand about groundwater and surface-water hydrology, you are able to understand about soils because they are only a byproduct of a process which can be explained in terms of the interaction of climate, bedrock geology, surficial geology, physiography, and hydrology. If you understand about soils, then you understand about plants because plants are variable with respect to environments, which variability is comprehensible in terms of climate, geomorphology, physiography, hydrology, and soils. If you understand about plants, you understand about animals because all animals are plant related, which leads you to understand about that special animal called man.
So one has to assemble those people to identify the region under study as phenomena, at the end of which you know why escarpments and caves and kettles and deltas are where they are. They are not only there, but their form and type explain what they are and what they have been doing and what they are in the process of becoming. Once you have identified the area as phenomena, you ask the same natural scientists to go over the same ground to reconstitute the region as processes. Then you assemble them all together to reconstitute the area as one single, interacting biophysical process, and then you have an ecological model.
Once understood as interacting biophysical process, the same data can be reconstituted as a social-value system. It is now possible to identify and locate all of the most propitious and the least detrimental attributes of air, land, water, life, and location for all prospective land uses.
So if you can identify what the land use is—whether it is a new town, an atomic reactor, a sewage plant, a highway, a single house, or garden—you can identify what is most propitious for that thing or that person, community, or institution in terms of factors of climate, bedrock geology, surficial geology, physiography, hydrology and limnology, soils, plants, animals, and land use. If you can identify all these needs, you can find the most propitious location. It can be done in a handcraft way or we can ask the computer to find these places where all these most propitious factors coexist. By this method we can solve the problem of location.
We can do this for every prospective land use—for agriculture, for urbanization, for recreation, for forests, all by type. Man identifies the most propitious factors which he requires and asks nature for the locations which provide them.
But we must identify those who ask nature for the most fitting environment. We have to find out what their needs and proclivities are. We must identify these in terms of social phenomena, social processes, and interacting social process. This allows us to enter the orthodoxy of city planning to assess the present and make projections into the future. These can be constituted into growth models, each associated with explicit hypotheses linked to the generating forces of growth. These are, then, demands in terms of land and resources. When demands can be matched to the opportunities and constraints which the region represents, there is then the basis for a plan.
Thus you have a creative fitting. And the most important conclusion for architects is that there are implications of form in fitting. There is no capriciousness in nature. The architect who believes that the white paper represents a site upon which he is going to invest his professional skill is mad. Written upon that white paper, whether he sees it or not, are 4 ½ to 6 billion years of physical evolution, 2 ½ billion years of biological evolution, a million years of human evolution and perhaps some thousand years of cultural evolution. All are written upon that land in biophysical and cultural processes having intrinsic form with implications for the form which he must give.
We are on the edge of time when we could, if we wished, feel the world's pulse, where we could use this ecological view to monitor the world. If instead of being so absolutely destructive, if we could abandon this cultural inferiority complex which is the base of our attitudes to nature—these poor, whimpering, puny creatures who were mute and defenseless in face of an implacable nature and in this impotence developed that bile of vengeance which explains the lesions and lacerations inflicted upon the life body of the world—if we could abandon all this and say we would like to be good stewards, we can develop ecological planning and design so that every honorable and decent architect may add his modest interventions to an enduring, cumulative, creative process.
We can, in fact, monitor the world from satellites continuously. We can augment this with high-level aerial photography. We can supplement this with ground truth. We can take this information from these sources and digitize it through high-speed scanners immediately to computers. And we can have an ecological model of the sort I have described, which actually simulates the ongoing biophysical processes. These can be consistently enriched and corrected. We can write programs asking for intrinsic suitabilities for all prospective land uses—for urbanization, for agriculture, for recreation. Anything we want to ask, we can have the world reveal to us.
This is a view of a working world of which we are a part. We can use this unique gift of human consciousness and be able to act as an enzyme, as an intelligent steward of nature. That possible destiny allows us to transform ourselves from geological destroyers to incipient creators.
Now this is a fantastic dream, and architecture must find a place in it. Architecture must absolutely reconstitute itself, as must all of society. You have traditionally assumed the leadership before. You have to reject the metaphysical symbol which has been ours for 2,000 years and replace it with another one: the ecological view. It exists for our use. It is marvelous because it does correspond to reality. It does offer the possibility of survival and a dream of fulfillment beyond dreams.
# 13
# Nature Is More Than a Garden (1990)
Randy Hester of the University of California-Berkeley invited McHarg to read this paper at a conference on garden design. It appeared in the book The Meaning of Gardens edited by Hester and Mark Francis of the University of California-Davis and published by MIT Press. McHarg views the garden as a device for human reassurance, a place for peace and tranquillity, an oasis in nature.
I have found the aspiration for reassurance to be a useful consideration in examining the garden. In the home furniture, memorabilia, and books provide a familiar and reassuring environment. I believe that the garden is also a vehicle for reassurance. I have observed that the familiar is an important component of reassurance and often derives from the ancestral landscape. The English Quakers who left Buckinghamshire in England found familiar environments in Bucks and Chester counties in Pennsylvania. There they proceeded to make their environment even more familiar.
Reassurance may also derive from familiar conventions. Traditional English suburban gardens contain a formal vocabulary—lawn, herbaceous border, rose garden, rock and water garden, wild garden—annually celebrated at the Chelsea Flower Show. The equivalent in Philadelphia adds Colonial Williamsburg overtones and is annually presented at the Philadelphia Flower Show. Often, however, the convention is in conflict with the natural landscape, such as when East Coast and European gardens erupt in the desert.
Affirmation of values is linked to reassurance. I believe that the garden combines both explicit and implicit statements of affirmation. Perhaps the most dominant statement is that nature is benign. Usually docile, tractable, and floriferous plants are arrayed. Poisonous plants, animals, and weeds are stringently excluded.
The next affirmation of the garden is that nature is bountiful. Here vegetables, fruit, and herbs give testament. Within the garden, nature is represented as orderly. There is an accompanying belief that work outdoors, preferably in a garden, touching soils, plants, water, stone, confers not only physical but also mental health. And, finally there are the sentiments in doggerel: "A garden is a lovesome thing, God wot" and "A kiss from the sun for pardon, a song from the birds for mirth, I feel nearer God's heart in my garden, than anywhere else on earth." I have long believed that the garden is a better symbol of peace than the dove.
However true they may be of gardens, these representations are illusory when applied to nature. Nature is not always benign, inflicting volcanoes, earthquakes, hurricanes, tornadoes, floods, and drought. Nor is nature uniformly bountiful. Arctic, Antarctic, tundra, taiga, and deserts provide meager sustenance. Nature is indeed orderly, but with a complex expression quite different from garden forms, which surely represent an illusory order. Then is God more accessible in a garden than elsewhere, than in nature perhaps?
Far be it from me to deprecate the creation and cultivation of gardens. All congratulations to professionals and amateurs alike who create islands of delight, tranquillity, and introspection. These are surely among the most successful testaments to man's humanity, often islands in anarchic, cheerless environments.
My purpose is not to discount but to add. I have been moved by Ryoan-ji, Saiho-ji, the Alhambra, Vaux, but I seek more. Gardens, of necessity, are simplifications. They exclude much, not least time and change. They contribute only a fragment of what is known. Indeed, they have more in common with aquaria and terraria than with nature.
The most beautiful gardens I have ever seen are pristine coral reefs in the South Pacific. They are artifices, created by coral, transforming oceanic deserts into the richest biotic environments in the world. They are also benign, bountiful, and orderly. They are dynamic and they are natural. Perhaps coral should be exemplar to men who make gardens.
However, in terrestrial environments I prefer nature. Nature contains the history of the evolution of matter, life, and man. It is the arena of past, present, and future. It exhibits the laws that obtain. It contains every quest that man can pursue. It tells every important story that man would know. Therein lie its richness, mystery, and charm.
So for me it is not either garden or nature. I can accept the imperfect reassurance, simplicity of order, even the elimination of time and change that gardens represent. I accept the great pleasure that gardens give, but I will reserve my deeper quest and larger fulfillment for nature.
# 14
# Landscape Architecture (1997)
In 1996–97, Ian McHarg undertook a speaking tour on behalf of the American Society of Landscape Architects (ASLA). Jim Dalton, then executive vice president of ASLA, sought to promote membership in the society but, even more, wanted to elevate the aspirations of the existing members. He selected McHarg for this task. As a result of the speaking tour, McHarg wrote this reflective essay for ASLA.
I know of no other profession which escalated as swiftly from oblivion to international significance nor one where so few persons have accomplished so much. Landscape architecture is unique in both respects. This should give us inspiration, confidence, and courage.
Law had its origins in courtiers selling access to power and spawned an army of lawyers; the master builders bequeathed us architecture, what a gift; artificers, sappers, and miners led to engineers; witch doctors, shamans, and in the West, barbers with leeches, led to medicine. Originally this was a mixed blessing. Until this century interventions were just as likely to be detrimental as beneficial. Medicine was no better than chance. Not until the Flexner Report, when medical schools were induced, perhaps better, bribed, to espouse biology, did medicine rise to its present august position. The medical experience contains a powerful lesson. Do we wish our efforts to be random, inconsequential, or indeed malevolent, or should we follow medicine and espouse science, biology, perhaps ecology and anthropology, and justify a more central and consequential role? I hope so.
The precise date when landscape architecture emerged is clouded. There were transitions. But in sixteenth-century Italy there appeared a singular expression. Popes and cardinals chose to leave Rome in summer and reside in Frascati and other nearby towns, where they built villas. These were not remarkable, but their gardens and outdoor spaces certainly were, particularly Villa D'Este and Villa Lante, works of excruciating beauty, a major threshold. Were Pirro Ligorio, Jacopo Barozzi da Vignola, or Donato Bramante members of a new breed, or were they simply green architects? I think not, the originality and the dominance of the gardens demands a new signature. How proud we can be to have these illustrious founders as our own antecedents. Landscape architecture would seem to have developed in full bloom.
The locus of power, invention and art moved to France where a single figure, André Le Nôtre, dominated the next century. First with Vaux-le- Vicômte for Nicolas Fouquet, later for Louis XIV at Versailles, Le Nôtre designed the largest exercise of garden art in history. It still excites our astonishment. Masses of tractable, docile, and colorful plants, with clay, sand, and gravel, were assembled into a great symmetrical composition. Its purpose was to portray the sovereignty and power of Le Roi Soleil, king by divine right, demonstrating his dominion over all men and nature, his power to subjugate them. The composition was an enormous genuflection to his majesty and power, it consumed half of the royal treasury. Would that other monarchs and princes had emulated him with the creation of gardens, or, even better, protecting and cherishing nature. But that was not the message of Versailles which sought to demonstrate man's dominion over nature. This constitutes the worst possible admonition to those explorers who were then about to discover and colonize the earth. Anthropomorphism, dominion, and subjugation are better suited to suicide, genocide, and biocide than survival and success.
Was Le Nôtre a green architect? I think not, the sheer dominance of the garden suggests a different scope of concerns, interests, and interventions. The fact that the palace was simply an exclamation point in the composition, suggests a new role and profession. Le Nôtre was truly a landscape architect. His works were the zenith of art on the seventeenth century. Consider his accomplishments or of the Italian trio of Ligorio, Vignola, and Bramante.
The English landscape movement began as Versailles was being finished. The handful of men who would proceed to transform an entire nation were dedicated to the expression of a harmony of man and nature. Of them there is no doubt. A new view was born in the eighteenth century. Whence its antecedents? I have not been able to discover. But here was Walpole saying of Kent, "he leap't the wall and discovered all nature to be a garden" and again "he found in nature a new creation." In my opinion, these are most powerful statements, evidence of a new view, indeed the emergence of the modern view.
William Kent, Capability Brown, Humphry Repton, Uvedale Price, Richard Payne Knight, and William Shenstone accomplished a total transformation of what was then an impoverished agriculture, deforested landscape, attenuated fields, and created the fairest, most productive, and beautiful landscape in Europe in less than a century. This, I suggest, is the finest accomplishment of art in the Western tradition. It deserves our most profound admiration. It should be revered as an important precursor.
These six men led England to accept and display the unity of man and nature and in so doing, transformed a nation. Persuasive and effective as was this new view it, was unable to withstand the power of its successor which led to the industrial revolution. Coal, iron, steel, mines, factories, railways, canals, industrial cities assumed ascendancy. No longer was nature to be cherished, nurtured, and emulated but rather to be exploited. The world was now seen as a storehouse inviting plunder.
But the landscape ideal survived, absorbed by the father of the profession, that great American, Frederick Law Olmsted, who visited England, observed the first public park at Birkenhead by Joseph Paxton, and proceeded to apply and expand the eighteenth-century principles into nineteenth-century America. Certainly the creation of a system of national parks was an original conception—the identification, protection, management of regions equivalent to nation-states. The invention of the urban park was comparable—Cen—tral Park remains unmatched; Riverside and the American subdivision, Stanford and the college campus, even the highway overpass falls within the inventions of Olmsted who single-handedly equaled the entire production of the professions of architecture, planning, and engineering during the nineteenth century.
This extraordinary paragon, in addition to his other virtues, had the wit to identify his successor—Charles Eliot, son of Harvard's president. In 1880 Eliot determined to spend a productive summer. A student at Harvard, he enlisted six classmates who entitled themselves the Champlain Society and proceeded to undertake an inventory of Mount Desert Island in Maine. Over six years they performed what was the first ecological inventory ever done. It produced two products, "Outline of the Geology of Mt. Desert Island" and "Flora of Mt. Desert, Maine."
But Eliot expanded this purpose to invent the first ecological planning study in the United States or the world, for the Boston metropolitan region, the 1893 study which, to date, has neither been equaled nor surpassed. It included the entire region from Blue Mountains to the sea, it included rivers, forests, marshes, the offshore islands. Eliot and his invention—regional planning—could not have arrived at a better time. Here was the appropriate attitude to lead the colonization of the country, here was a visionary plan for the land, and the world. But, it was not to be, for Eliot died of cerebrospinal meningitis prematurely at 37, to the loss of the nation, the world, and to nature.
The eighteenth-century view envisaged an ideal landscape, first expressed by the painters Salvator Rosa, Nicolas Poussin, and Claude Lorraine, observed in the campagna. This effort of restoration required and received massive energies. This was not necessary in America. Here the pioneers had discovered a virgin continent. The task was not to represent harmony that existed in the systems which were being discovered. Here the task was to recognize, cherish, and protect first and also to develop with discrimination, employ art in accomplishing felicitous adaptations. This was a simpler task, but it was at odds with the explorer mentality, the extraction morality, be it of gold or silver, beavers and bison, timber or wheat, exploitation was its name.
Olmsted and Eliot had presented a better view, but Eliot had died even before Olmsted, and to our astonishment and loss, these extraordinary paragons did not reproduce themselves. The concept died in England at the end of the eighteenth century. Eliot died in 1897, Olmsted in 1903, and their movement stalled. It should have been enthroned at Harvard, spread to other leading institutions, and permeated the new profession of landscape architecture. To my continued chagrin, it did not. Indeed it declined from the elevated status achieved by Olmsted, promised by Eliot.
The time has come to rediscover and celebrate the accomplishments of our predecessors, not least Olmsted and Eliot. They should be enthroned at Harvard, represented by bronze statues, studied, admired, and emulated worldwide.
The traditions continued into this century. First in the person of the Brazilian polymath, painter, naturalist, designer of jewelry, floral decorations for state ceremonies in Brasilia, architect but, most of all, designer of gardens: Roberto Burle Marx. He was a man of such fame as to be known to cab drivers in Rio. He was also renowned as a belly dancer and singer of bawdy songs, and in several languages. Not least, he was a successful plant collector and created an elaborate arboretum.
Yet another is Lawrence Halprin, very much alive at 78, productive, notably in the United States and Israel. His reputation began with modern gardens, catapulted after the Portland Fountain, expanded with Sea Ranch, and had a fitting climax with the FDR Memorial in Washington, D.C., just completed and resoundingly appreciated. Halprin deserves inclusion in this pantheon.
These are emergents, hints of successors to this great lineage. First are the partners of Andropogon, Carol and Colin Franklin, Leslie Jones and Rolf Sauer; Jones & Jones of Seattle; Edward Stone of Florida; EDAW in San Francisco, Atlanta, and elsewhere; Jon Coe, Gary Lee, and their partners in Philadelphia; Design Workshop of Aspen, Tempe, and elsewhere; and, of course, there is A. E. Bye who disclaims scientific ecology, but whose works are of great beauty and very appropriate ecology. There is another figure, Jack Dangermond, President of ESRI in Redlands, California, who is the leader in spatial computation. Dangermond has drawn on his multidisciplinary academic training in landscape architecture, planning, and geography to create a new industry and lead it.
The giants who led the profession, Ligorio, Vignola, and Bramante, Le Nôtre, the English Six, Olmsted and Eliot, Burle Marx and Halprin have been augmented not by practitioners, but by science and technology. Concern for nature in the Renaissance was limited mainly to princes and pontiffs. In England it was the province of the landed aristocracy, artists, and scholars. The audience widened in the nineteenth century but exploded at the end of the twentieth. Newspapers, magazines, film, but mainly television, contributed to this effervescence. PBS, National Geographic, The Planet Earth, Nature, the Discovery and Learning Channels, brought rich, sophisticated, and persuasive insights into nature to a rapt public attention.
Science, long immured in subatomic particles and molecular biology, emerged into the sensate world with plate tectonics, atmospheric physics, chemosynthesis, and a new understanding of microbes and of the importance of environment. Surely the greatest benefit from the diminution of the nuclear threat was the recognition of the environment as having primacy on the global agenda. In this evolution emerged new technological marvels, sensors, satellite imagery, geo-positioning systems, computers, and geographic information systems. We can feel the world's pulse. We can undertake not only national, but also global inventories. We can monitor the planet.
Who would have thought that the Christian church would address man's relation to nature through the garden at D'Este?
Who would have expected that the major energies of the most powerful monarch in Europe would be employed to create a metaphysical symbol of his rank and role as at Versailles?
Was there reason to expect that the most compelling philosophical investigation would be a new and benign view of the relationship of man and nature in eighteenth-century England?
Who could have expected that the environment would emerge as the most compelling subject confronting the world at the end of this century?
The profession of landscape architecture has a client, the earth and its creatures. In order to meet this challenge, to respond to our client in a sustainable manner, the profession must ensure that it forms an alliance with the environmental sciences and that we come to be seen by them and the public as their agents for achieving felicitous, ecological adaptations.
Professional education must recognize this relationship and incorporate environmental science. The American Society of Landscape Architects must become a spokesman for the environment and appropriate adaptations. We must become leaders, not only in understanding the environment, but also in planning ecological restoration, management, and design, that is, in sustainable development.
To this day, neither architects nor city planners have been required to study environmental science and thus bring only innocence and ignorance to bear. Engineers traditionally study physical environmental sciences, notably geology, and hydrology, but learn nothing of life or people. Landscape architects have a long association with botany, horticulture, and, more recently, ecology.
There is a vacuum of environmental competence which the profession should fill. Success is contingent upon landscape architects learning enough environmental science to be unerringly selected to answer all questions relating to its planning and design. When this comes to pass the profession will receive the appreciation, adulation, and rewards befitting to the descendants of our illustrious ancestors. We have increased our numbers a thousandfold. We have inherited an unimaginable expansion of capabilities. While problems have escalated with population, our scientific understanding and technological competence have more than kept pace. The canvas has expanded from site to region, to nation to now embrace oceans, atmosphere, and the planet earth.
The only necessary elements are confidence and conviction. I have long ago concluded that knowledge is the essential ingredient. Given this, then confidence will follow. Consider the example of Frederick Law Olmsted. Has anyone ever equaled his conviction of the salubrious effect of nature, particularly necessary for urban man?
What an extraordinary lineage we have inherited. It gives us example, confidence, and courage. Let us follow their example and proceed into the twenty-first century.
# 15
# Ecology and Design (1997)
In 1992 Arizona State University (ASU) initiated a new degree in landscape architecture. In the process a Department of Planning became the School of Planning and Landscape Architecture. To celebrate the event George Thompson of the Center for American Places and Frederick Steiner organized a symposium and a book on landscape architecture theory. They received a grant from the National Endowment for the Arts for the symposium and the book.
The concept was to pull together the senior and junior theorists in the discipline around the theme of ecological design and planning. The senior figures had pioneered the use of ecology in landscape architecture. Several younger theorists had been critical of this use and noted that ecology had drained some of the creative juices from design.
Ian McHarg was the central figure at the symposium. Each of the other theorists referred to his influence in some way or the other. McHarg responded to the criticisms of ecology with this spirited essay. It appears in the book Ecological Planning and Design, edited by Thompson and Steiner and published by John Wiley & Sons.
I am unabashedly committed to the imperative design with nature, or ecological design and planning. Indeed, I conceive of nonecological design as either capricious, arbitrary, or idiosyncratic, and it is certainly irrelevant. Ecology, the study of environments and organisms, among them people, is totally inclusive. What falls outside this definition? Not content, perhaps only attitude. Nonecological design and planning disdains reason and emphasizes intuition. It is antiscientific by assertion.
There is no doubt about my attitude toward this topic. I invented ecological planning during the early 1960s and became an advocate of ecological design thereafter. This was explicit in Design with Nature; it was not only an explanation, but also a command.
Ecological planning is that approach whereby a region is understood as a biophysical and social process comprehensible through the operation of laws and time. This can be reinterpreted as having explicit opportunities and constraints for any particular human use. A survey will reveal the most fit locations and processes.
Ecological design follows planning and introduces the subject of form. There should be an intrinsically suitable location, processes with appropriate materials, and forms. Design requires an informed designer with a visual imagination, as well as graphic and creative skills. It selects for creative fitting revealed in intrinsic and expressive form.
The deterioration of the global environment, at every scale, has reinforced my advocacy of ecological design and planning. Degradation has reached such proportions that I now conclude that nonecological design and planning is likely to be trivial and irrelevant and a desperate deprivation. I suggest that to ignore natural processes is to be ignorant, to exclude life-threatening hazards—volcanism, earthquakes, floods, and pervasive environmental destruction—is either idiocy or criminal negligence. Avoiding ecological considerations will not enhance the profession of landscape architecture. In contrast, it will erode the modest but significant advances that ecology has contributed to landscape architecture and planning since the 1960s.
Yet, you ask: What of art? I have no doubt on this subject either. The giving of meaningful form is crucial; indeed, this might well be the most precious skill of all. It is rare in society, yet it is clearly identifiable where it exists. Art is indispensable for society and culture.
Does art exclude science? Does art reject knowledge? Would a lobotomy improve human competence, or is the brain the indispensable organ?
There is a new tendency by some landscape architects to reject ecology, to emphasize art exclusively. This I deplore and reject. Such an approach is tragically ironic when so many world leaders are calling for sustainable development, when architects are issuing green manifestos, and professional associations in architecture and engineering are refocusing their attention on the environment.
We have been at this impasse before; it was not beneficial, and the result was calamitous. We have only to remember that, by the end of World War I, landscape architecture was firmly established at Harvard University. It was the world center for this subject. It had inherited the concepts and accomplishments of Frederick Law Olmsted and Charles Eliot, but all was not well: There was dissension.
It transpired that there were opposing camps within landscape architecture. The Olmsted disciples wished to emphasize conservation and regional and town planning. They included Henry Hubbard and Theodora Kimball, Harland Bartholomew, John Nolen, Warren Manning, and, later, Howard Mennhenick. The remainder were oriented to Beaux Arts; they were self-proclaimed aesthetes, interested in designing estates for the rich and famous: Bremer Whidden Pond, James Sturgis Pray, Steven Hamblin, and Robert Wheelwright. The aesthetes defined landscape architecture design between the world wars at Harvard and elsewhere. Meanwhile the Olmsted disciples founded the field of planning during the 1920s, and inadvertently created a schism between design and planning that persists to this day. Brains and knowledge abandoned landscape architecture, which experienced a massive decline from the peaks of Olmsted and Eliot to an abyss with little intelligence, skill, or passion.
This antagonism between art and science, as well as between design and planning, has lasted too long. It is now a serious obstruction to education and the earth's well-being. Both art and science have their antique, prepared positions, their mandarin advocates, their lines of competence defined, and their proprietary jargons. Yet, when stripped of pomp and pretensions, at root art merely means skill and science means knowledge. Can we imagine, in the challenging environment we occupy, the rejection of either art or science? Surely knowledge needs skill to give form and significance to our landscapes and our adaptations. Surely skill needs knowledge just as a solver needs a problem. Surely, once and for all time, art and science, skill and knowledge, ecology and design and planning should unite.
What the world needs from landscape architecture is an enlarged vision, greater knowledge, and commensurate skill. Landscape architects are engaged in the challenge of adaptation. They must acquire the accomplishments that can make a significant contribution to preserving, managing, planning, and restoring the biosphere, to designing human environments.
And, thanks to Charles Darwin and Lawrence Henderson, we have a theory. Darwin said: "The surviving organism is fit for the environment." Henderson, another biologist and author of The Fitness of the Environment (1913), wrote that Darwin's assertion was insufficient. He concluded that, as there are infinite environments and organisms, the evolutionary challenge for every participant is to seek and find the fittest available environment, to adapt that environment and the self to accomplish a better fit. Moreover, Henderson's writing defined a fit environment as one in which most of a consumer's needs exist in the environment as found, requiring less work for adaptation than for any competitor. The thermodynamic challenge implies that successful adaptation is creative.
What are the instruments for adaptation? The universal adaptations—mutation and natural selection—while manipulated extensively for food, plants, and animals, are not widely applied to human breeding (thank goodness). Mutation and natural selection are slow processes. And innate behavior in living organisms is similarly difficult to manipulate. Cultural adaptation is the more pliant and useful. Language, philosophy, science, art, and technology are such instruments. If one is to ask which aspect of cultural adaptation most clearly meets the Darwin-Henderson challenge—to find the fittest environment and then to engage in adaptation—surely the most direct response is landscape planning and design.
But we need an appropriate criterion for guiding and evaluating landscape architecture in the twenty-first century. Art critics evaluate the contribution of painting, sculpture, dance, photography, and other art forms. Can their appreciation extend from the product to the source of creation, the painter, his or her brain, eye, hand, bones, sinews, arteries, and veins? Beauty alone is an inappropriate criterion for evaluating art or the organs that create it, so why should beauty be used exclusively to evaluate a landscape? Fitness, as explained by Henderson, for me is a thoroughly suitable criterion. We have, then, not only a theory, but also criteria for performance and fitness.
If one is fit, then one is healthy, and this applies equally to a landscape. Furthermore, if being healthy enables one to seek and solve problems, or provides the ability to recover from insult or assault, then fitness confers health. Therefore, fitness is an index of health. Extending this thought to landscape architectural theory and practice, do we have a method for accomplishing creative planning and design and, more simply, an ecological method for adaptation?
We live in a physical world, a biological world, and a social world, and our investigations must include them all. As matter preceded life and the human species was late in biological evolution, we can employ chronology as a unifying force. We can recapitulate events and retrace time. Thus, when we design and plan, we should begin with the geological history of a landscape, working in concert with an understanding of climate. Bedrock and surficial geology as well as climatic processes can be reinterpreted to explain geomorphology and hydrology. These processes set the stage for soil formation. Now, the relationships among the constituent parts of a landscape become clear: The past informs the present, and each feature is only comprehensible from understanding its earlier layers. After we learn about a landscape's geology, climate, hydrology, and soils, then vegetation patterns become more apparent, as does the resultant wildlife. At which point we can ask the human occupants who they are, how they distinguish themselves from others, how they view the environment and its occupants, and what are their needs and desires, their preferences and aversions.
I wish to emphasize my belief that ecological study includes natural and cultural processes. We will find that discrete value systems are associated with distinct human constituencies, and we can associate these groups with their needs and values. This approach allows landscape architects to interpret all phenomena in the light of these systems. With such vision and knowledge, we can plan, because we have developed the context for planning and design.
This is the biophysical model I developed more than three decades ago for ecological design and planning. This is the model I live by. It can be reinterpreted to explain social values, technological competence, an ethnographic history of human settlements—urban as well as rural. We then are able to see the primeval landscape successively modified, through history, in order to arrive at the present.
Ecological study is indispensable to planning, but it also produces a context for a regional design vocabulary. The settlements of Dogon, the Berbers in Morocco, the settlement of the Greek islands, Italian hill villages, and pueblo communities in New Mexico and the Colorado Plateau are fitting examples of ecological responses in planning and design. Contemporary examples of ecological designers are too few. I am able, however, to select three firms.
The first is Andropogon in Philadelphia, whose principals are Carol and Colin Franklin, Leslie Jones, and Rolf Sauer. These landscape architects possess unchallenged primacy in ecological design and restoration. Their science is impeccable, their applications cross the threshold of art, and their realizations of design and planning are wonderfully effective, fitting, and, even for the uninitiated, beautiful. Coe Lee Robinson Roesch in Philadelphia is the second example; the third is Jones & Jones in Seattle. Both firms have built sterling reputations designing and building appropriate exhibits for zoos. Their science, too, is elegant, employing inventories of the native environments of animals, replicating these environments with consummate skill. Their creations include ecological and psychological factors never before employed, to my knowledge, for human habitations. It is more difficult to find other examples. These three exceptions simply prove the rule. Ecological design is still an aspiration, not yet a practice, within landscape architecture.
A major obstruction to ecological design is the architecturally derived mode of representation drawings. This is paper-oriented, two-dimensional, and orthogonal. In contrast nature is multi-dimensional, living, growing, moving with forms that tend to be amorphous or amoebic. They can grow, expand, interact, and alternate. Field design would be a marvelous improvement over designing on paper removed from the site. Yet new representations must be developed to supersede the limitations of paper-oriented, orthogonal investigations with their limited formal solutions. We should be committed in our work to designing living landscapes in urban, rural, and wild settings. Yet there are infinite opportunities afforded to those who would study natural systems, their components, rules, succession, and, not least, their forms. This should be the basis for an emerging ecological design.
This system does not need invention. Forms, materials, and processes have evolved by trial and error over eons of time and represent the finest solutions of materials and form that nature could invent. Such ecosystems are exquisitely adapted and provide an example for people to design and plan.
Strangely, the area where art and science have been most successfully employed is neither architecture nor landscape architecture. It is in the creation of prostheses. Here biological knowledge is indispensable, but skill produces the adaptation. The purposes are to amplify the performance of a biological function—to see small and far, we have the microscope and the telescope; to speak far, we have the telephone and the microphone; to move far and fast, we have the plane and the rocket. Our major prosthesis is the expansion of power from muscle to tools, mechanical equipment to atom bomb.
Consider the steps to improve sight—spectacles, bifocals, magnifying glasses, laser operations for cataracts and glaucoma, and soft contact lenses—a miracle of adaptive design. Or walking—first a shoe, then a crutch, a walker, the wheelchair, now titanium hips and Teflon knees. The computer may be the prosthesis for the brain.
Should we pursue this track we would reconsider membranes. The giant clam and nudibranch have incorporated chloroplasts; those animals can now photosynthesize. Why not build membranes, walls, and roofs? Could they do so? Consider vegetation on walls and roofs to add carbon, fixing and minimizing carbon dioxide. Consider creating calcium carbonate, with electric charges in seawater.
Membranes modulate freshwater and saltwater. They facilitate the transport of nutrients. Consider the adaptations to cold in plants, reduction in water content, hairy surfaces, corrugations, and color. Wastes in nature are often nutrients, not problems. The route that Andropogon, Coe Lee Robinson Roesch, and Jones & Jones have developed is superb but needs to be augmented. The example of prosthetic development is a fitting analog. It is indisputably ecological, without caprice.
Of course, the best example of successful adaptation is the coral. The ocean floor is generally sterile. Coral transforms such areas into some of the most fertile and beautiful environments on earth. Would that landscape architects could equal this.
When the training of landscape architects includes at least an introduction to all of the relevant natural and social sciences, as well as design proficiency, then a new proposal can provide them with an exceptional opportunity to design and plan ecologically and artfully.
There are moves afoot to initiate a National Biological Survey. I recommended this in 1974 to Russell Train, then of the U.S. Environmental Protection Agency (EPA). When William Reilly became EPA administrator in 1990, he asked me to advise him on the reorganization of the EPA, and the conduct of the Environmental Monitoring and Assessment Program (EMAP), the proposed inventory. EMAP was proposed to be a broadly conceived ecosystem inventory integrating regional and national scales, allowing for monitoring and assessment, and designed to influence decision making. With my colleagues John Radke, Jonathan Berger, and Kathleen Wallace, I produced a document, A Prototype Database for a National Ecological Inventory (1992). This recommended a three-part process, a national inventory at 1:2,000,000 with all natural and social regions delineated; a regional scale inventory at 1:250,000; and, finally, samples at the scale of 1:24,000.
Our first assertion was that the resolve to undertake a National Ecological Inventory should be recognized as the most important governmental action ever taken in the history of the American environment. This resolve contains implicit resolutions: That inventory, monitoring, and modeling are indispensable to the EPA and essential to the fulfillment of its regulatory roles.
This first requires the reconstitution of the EPA to include leadership from all environmental sciences. To provide such leadership, we recommended the creation of an executive committee composed of leading officers in all of the scientific societies. Regional groups of environmental scientists and planners would be assembled to undertake regional inventories. And landscape architects, being among the most competent professionals to decide on the appropriate data to be incorporated and its contribution to planning and design, would participate in the content of these inventories. For the crystalline piedmont and adjacent coastal plain in Pennsylvania and New Jersey, for example, we would assume that landscape architects and regional planners at the University of Pennsylvania, Rutgers University, and Pennsylvania State University would be part of a central group to that association of regional scientists. This model would be replicated throughout the country with other university landscape architecture faculties, professional designers, and planners contributing to the regional groups. By such an involvement the landscape architecture profession would vastly enhance its social contribution and academic reputation.
It is then necessary to produce a plan for the process to undertake the inventory as well as to monitor its progress. This would involve the recommended scientific committee, an expanded staff representing all of the environmental sciences, and an appropriate organizational structure. When the plan is completed, we advocate a massive public relations and advertising campaign to inform the general public, conservation groups, the scientific community, and government officials at all levels of the significance of the enterprise.
Our proposed inventory must include physical, biological, and social systems. The last is indispensable for understanding human environments. We suggested a demonstration ethnographic survey to illustrate that these are crucial for understanding human ecology. In our recommendation to the EPA, we observed that the greatest problem lies not in data collection, but rather with integration, synthesis, and evaluation.
Now, there is a new development. Secretary of the Interior Bruce Babbitt announced, shortly after his appointment by President Clinton in 1992, his intention to undertake a National Biological Survey (NBS). This may supersede EMAP. There is another possibility: NASA, the Defense Mapping Agency, and the federal departments of defense and energy have collected global environmental data since the onset of the Nuclear Age, employing instrumentalism that is unavailable to civilians. There is discussion of declassifying these data, which, of course, could be included in a National Ecological Inventory or a National Biological Survey. There is, also, the possibility that this capability could be invested in the United Nations as the U.S. arm of the U.N. Environmental Program and employed for a global inventory. Others are recommending the creation of a National Institute of the Environment in which the inventorying and modeling might be invested.
Clearly massive data will soon become available, ultimately in digital form, globally and locally. Whoever has access to this cornucopia will have immense power. Landscape architects should become advocates for such ecological inventories and become primary users of these data for planning and design. Landscape architects must learn to lead.
In 1992 I received a signal honor from President George Bush, the National Medal of Art. The noteworthy aspect of this act was the inclusion of landscape architecture as a category eligible for this high honor. As preface to the award President Bush stated, "It is my hope that the art of the twenty-first century will be devoted to restoring the earth."
This will require a fusion of science and art. There can be no finer challenge. Will the profession of landscape architecture elevate itself to contribute to this incredible opportunity? Let us hope so. The future of our planet—and the quest for a better life—may depend on it. So let us resolve to green the earth, to restore the earth, to heal the earth with the greatest expression of science and art we can muster. We are running out of time and opportunities.
# Notes
Ian L. McHarg. 1969. Design with Nature. Garden City, N.Y.: Doubleday, Natural History Press. 1992. Second edition, New York: Wiley.
Frederick Law Olmsted was an amateur farmer, familiar with science, and Charles Eliot consorted with the leading scientists of his time, but their influence was bred out of city planning, which became the applied social science department that I encountered as a student at Harvard University in 1946. The natural sciences were banished. It is barely credible that the accomplishments of that giant, Olmsted, and the opportunities presented by Eliot were disregarded—first, by landscape architecture and, then, by city planning at Harvard. Nor were either replaced by a superior doctrine. We have yet to rediscover their significance.
I use the word consumer because the roots of ecology and economics both lie in the Greek word, oikos, or "a dwelling place." Henderson used terms such as consumer close to this root meaning.
See Carol Franklin, "Fostering Living Landscapes," in George F. Thompson and Frederick R. Steiner, eds., Ecological Design and Planning. New York: Wiley, pp. 263-92, as well as Leslie Jones Sauer, The Once and Future Forest. Washington, D.C.: Island Press, 1998.
It is worthwhile to remember that Charles E. Little and David Brower, among other leading conservationists, have for decades advocated the necessity of employing advertising and public relations techniques in the cause to inform the public, industry, government, and university officials about environmental matters. Also of note is Richard Beamish's important book, Getting the Word Out in the Fight to Save the Earth (Baltimore: Johns Hopkins University Press, 1995). Beamish is one of the unsung heroes of the modern environmental movement, especially for his work on saving the Adirondacks.
# Part IV
# Revealing the Genius of the Place: Methods and Techniques for Ecological Planning
Ian McHarg provided an orderly procedure for ecological planning. Suitability analysis is the process of determining the fitness of a given tract of land for a defined use or set of uses. This procedure involves overlaying mapped information to reveal opportunities and constraints for potential land uses. "Consult the Genius of the place in all," Alexander Pope advised. Suitability analysis is a logical framework for making such a consultation.
Landscape architects began using hand-drawn, sieve-mapping overlays in the late nineteenth century. Pennsylvania State University professor emeritus Lynn Miller credits Charles Eliot and his associates in the office of Olmsted, Olmsted, and Eliot with pioneering overlays through sun prints produced on their office windows. Both Miller and McHarg acknowledge the early contributions of Charles Eliot, an Olmsted protégé, who worked with scientists systematically to collect and map information to be used in planning and design. Warren Manning, a protégé of both Olmsted and Eliot, used soil and vegetation information with topography and their combined relationship to land use to prepare four different maps of the town of Billerica, Massachusetts, in 1912. Manning's Billerica Plan displayed recommendations and changes in the town's circulation routes and land use (Steinitz et al. 1976; Manning 1913).
Eliot left the most explicit explanation about why and how the overlays were employed. After his death his father Charles W. Eliot, the president of Harvard, wrote a biography-autobiography. Charles Eliot, Landscape Architect (1902) was the father's interpretation of his son's work. This work provides perhaps the first account of the overlay technique. The Boston Metropolitan Park work undertaken by Olmsted, Olmsted, and Eliot, with Eliot in charge, involved six months of "diligent researches" (1902, p. 496). Eliot used a variety of consultants including a Massachusetts Institute of Technology professor, as well as Olmsted staff, such as Manning, to conduct surveys of the metropolitan region's geology, topography, and vegetation. These maps provided the basis for the overlay process which Eliot describes as follows:
By making use of sun-prints of the recorded boundary plans, by measuring compass lines along the numerous woodpaths, and by sketching the outlines of swamps, clearings, ponds, hills, and valleys, extremely serviceable maps were soon produced. The draughting of the several sheets was done in our office. Upon one sheet of tracing-cloth were drawn the boundaries, the roads and paths, and the lettering . . . ; on another sheet were drawn the streams, ponds, and swamps; and on a third the hill shading was roughly indicated by pen and pencil. Gray sun-prints obtained from the three sheets superimposed in the printing frame, when mounted on cloth, served very well for all purposes of study. Photo-lithographed in three colors, namely, black, blue, and brown, the same sheets will serve as guide maps for the use of the public and the illustration of reports.
Equipped with these maps, we have made good progress, as before remarked, in familiarizing ourselves with the "lay of the land" (1902, p. 496).
After Eliot and Manning there are several studies in which the use of the overlay technique is apparent, but a theoretical explanation about the rationale for using the technique as an orderly planning method was missing. The city plan for Dusseldorf, Germany, in 1912 the Doncaster, England, regional plan in 1922 (Abercrombie and Johnson 1922) the 1929 regional plan of New York and its environment (Regional Planning Staff 1929) and the 1943 London County plan (Forshaw and Abercrombie 1943) incorporate typical characteristics of the overlay process (Steinitz et al. 1976). Thomas Adams, who directed the extensive 1929 New York regional planning study, addressed suitability in his 1934 The Design of Residential Areas, but mostly from an economic perspective. An academic discussion of the overlay technique did not surface until 1950 with the publication of the Town and Country Planning Textbook with an article by Jacqueline Tyrwhitt that dealt explicitly with the overlay technique. The book consists of a series of articles dealing with planning and design methods and issues and provides the first explicit discussion of the overlay technique (Steinitz et al. 1976, see also Collins et al. 1998).
In an example given by Tyrwhitt, four maps (relief, hydrology, rock types, and soil drainage) were drawn on transparent papers at the same scale, and referenced to common control features. These data maps were then combined into one land characteristics map which provided a synthesis, interpretation, and a judicious blending of the first four maps (Tyrwhitt 1950, Steinitz et al. 1976). This sieve-mapping, overlay method was widely accepted and incorporated in the large-scale planning of the British new towns and other development projects after World War II (Lyle and Stutz 1983, McSheffrey 1999). At the end of the war McHarg took a correspondence course offered by Tyrwhitt and others. McHarg was also involved in new town planning in the early 1950s in Scotland, so he was quite aware of the British new town program. As a result, he was introduced to the concept of suitability analysis early in his career.
George Angus Hills' plan for Ontario Province (1961) is a pioneering North American example that employed a well-documented data-overlay technique (Belknap and Furtado 1967, 1968; Naveh and Lieberman 1994; Ndubisi 1997). Hills was on the staff of the Ontario Department of Lands and Forests. His technique divides regions into consecutively smaller units of physiographic similarity based on a gradient scale of climate and landform features. Through a process of comparing each physiographic site type or homogenous land unit to a predetermined set of general land-use categories and rankings of potential or limitation for each use or activity, the resulting units were regrouped into larger geographic patterns called landscape units and again ranked to determine their relative potentials for dominant and multiple uses. The land-use activity with the highest feasibility ranking within a landscape unit was recommended as a major use (Belknap and Furtado 1967).
The year after Hills' Ontario plan, Philip Lewis, a landscape architect at the University of Wisconsin-Madison and principal consultant to the Wisconsin Department of Resource Development, applied an overlay analysis technique to evaluate natural resources for the entire state of Wisconsin. This work was a direct response to the growth and demand for outdoor recreation across the state. According to Ndubisi, "Unlike Hills, whose work was based primarily on examining biological and physical (biophysical) systems such as landforms and soils, Lewis was concerned more with perceptual features such as vegetation and outstanding scenery" (1997, p. 21). Lewis stressed the importance of the patterns, both natural and cultural within the landscape. He combined individual landscape elements of water, wetlands, vegetation, and significant topography through overlays onto a composite map depicting Wisconsin's areas of prime environmental importance (Belknap and Furtado 1967, 1968; Steinitz et al. 1976). By combining resource inventory data and soil survey data, Lewis was able to create maps that identified intrinsic (natural) patterns. Once additional resources were grouped by patterns and mapped, points were assigned to major and additional resources and totaled to identify relative priority areas. Demand for planned uses and limitations of each priority area for specific uses were then combined to assign specific uses to each priority area (Collins et al. 1998).
In another early example of the overlay technique, Christopher Alexander and Marvin Manheim (1962) mapped a set of twenty-six decision criteria for highway location (Steinitz et al. 1976). The study was conducted at the Civil Engineering Systems Laboratory, Massachusetts Institute of Technology. Alexander and Manheim propose a hierarchical tree as a formal procedure for combining factors. Each map is shaded separately to show the better or worse areas for highway alignment. They proceed to identify areas of similarly shaded patterns and superimpose these different maps photographically on one print. In doing so, they assigned weights to each set of decision criteria that were then represented on the final composite print.
McHarg, Lewis, and Hills refined their approaches during the 1960s and built their work on all these earlier efforts. McHarg especially advanced previous methods significantly by linking suitability analysis with theory. He provided a theoretical basis for overlaying information. McHarg's approach focused on mapping information on both natural and human-made attributes of the study area and photographing them initially as individual transparent maps (Belknap and Furtado 1967, 1968; McHarg 1969; Gordon 1985). The transparent prints of light and dark values were superimposed over each other to construct the necessary suitability maps for each land use. These x-ray-like composite maps illustrated intrinsic suitabilities for land-use classifications, such as conservation, urbanization, and recreation for the specific planning area. These maps were then combined with each other as overlays to produce an overall composite suitability map (McHarg 1969).
McHarg's inventory process provides one of the first examples of methodological documentation for the overlay technique (with those by Hills and Lewis). McHarg was also the first to advocate the use of the overlay technique to gain an ecological understanding of a place. He noted that "a region is understood as a biophysical and social process comprehensible through the operation of [natural] laws and time. This can be reinterpreted as having explicit opportunities and constraints for any particular human use. A survey will reveal the most fit locations and processes" (1997, p. 321). As a result, he was explicit about the range and sequence of mapped information to be collected. McHarg also observed that the phenomena represented by the maps were valued differently by various groups of people and thus could be weighted differently, depending on the circumstance.
Ecology is a language for reading landscapes. McHarg notes, "written on the place and upon its inhabitants lies mute all physical, biological and cultural history awaiting to be understood by those who can read it." His 1967 Landscape Architecture article, "An Ecological Method for Landscape Architecture," outlines an ecological method and is one of McHarg's earliest published explanations of suitability. He advocates inventorying all unique or scarce features and environmentally sensitive areas. He also advances the notion of "communities of land use," that is, their ecologies or how uses interact with one another, how compatible or incompatible they are with each other. A place may be intrinsically suited for many uses, but some of these uses may fit with neighboring uses better than others.
According to McHarg, his method can be used in both urban and rural areas. He explains that "successive stages of urbanization" are adaptations to the environment, "some of which are successful, some not." He provides a model for urban human ecology by comparing urban systems regressing from health or evolving towards health. McHarg suggests that his model be used both for diagnosing the health of the city and for prescribing appropriate interventions.
McHarg's 1968 "A Comprehensive Highway Route Selection Method," is a specific example of the application of his method to highway route selection. According to McHarg, his method enables social and aesthetic values to be incorporated into the route selection process. His highway route selection work during the 1960s provided the methodological foundation for environmental impact statements and other types of environment assessments mandated by the National Environmental Policy Act (NEPA) and many similar laws enacted by the federal, state, and local governments in the United States during the "environmental decade" of the 1970s. Environmental impact assessment documents also have become mandatory in many other nations since the 1970s. NEPA's purpose is
To declare a national policy which will encourage productive and enjoyable harmony between man and his environment; to promote efforts which will prevent or eliminate damage to the environment and biosphere and stimulate the health and welfare of man; to enrich the understanding of ecological systems and natural resources important to the Nation; and to establish a Council on Environmental Quality.
NEPA required federal agencies "to consider every significant aspect of the environmental impact of a proposed action" and federal agencies to "utilize a systematic, interdisciplinary approach which will ensure the integrated use of the natural and social sciences and the environmental design arts in planning and in decision making which will have an impact on man's environment." Furthermore, NEPA instructed federal agencies to "identify and develop methods and procedures . . . which will ensure that presently unquantified environmental amenities and values may be given appropriate consideration in decision making along with economic and technical considerations."
NEPA was exactly what McHarg had been advocating during the 1960s, and he had been refining for a decade the "methods and procedures" for assessing environmental impacts. His best known pre-NEPA suitability work involved highways. During the 1950s and 1960s, interstate highways had been built at a rapid pace, providing an efficient transportation system but destroying many communities, much farmland, and significant amounts of environmentally sensitive lands in the process. Because of President Dwight Eisenhower's admiration for Hitler's autobahn, the interstate highway system turned its back on the earlier American, more gentile tradition of parkways and turnpikes (such as the Pennsylvania Turnpike). Whereas the parkways had been designed, at least in part, by landscape architects, civil engineers dominated the planning of the interstates, a situation that McHarg criticized vehemently.
McHarg sought to recapture the involvement of landscape architects. In this quest, he had a powerful ally in Lady Bird Johnson. His work on Interstate 95 between the Delaware and Raritan Rivers in New Jersey is an explanation of this alternative method for highway location. He suggested that "presently unquantified environmental amenities" be considered in the route selection process. Specifically, he advocated that traditional cost-benefit analysis be replaced with one that assessed "maximum social benefit and the minimum social cost." To accomplish this, the role of the highway needed to be expanded from its single-purpose function of moving traffic to a multipurpose perspective that contributed to community development.
Balance sheets were developed by McHarg for the location of 1-95. These balance sheets reveal the highway alignments with the most social benefits with the least social costs. Maps were used to represent these benefits and costs which were then superimposed for interpretation to reveal the most suitable routes.
McHarg has been critical of both sanitation and traffic engineers. In 1976 in "Biological Alternatives to Water Pollution" he suggested alternative techniques for ensuring clean water supplies. He noted that, "The entire subject of water must be seen to be a problem of biology rather than engineering." Biologically based techniques to alleviate water pollution are purposed. In this case, he relied much on the knowledge and expertise of his colleague Ruth Patrick. Dr. Patrick, a leading limnologist, conducted research at the Philadelphia Academy of Natural Science and taught for many years in McHarg's department at Penn.
The article is also based on work he undertook for the U.S. Environmental Protection Agency (EPA) to help the agency implement NEPA. McHarg suggested an administrative structure to plan, manage, and regulate environmental quality, including water quality. Essentially, he suggested using physiographic regions in tandem with watersheds as the geographic basis for planning. McHarg's suggestions were consistent with EPA's response during the 1970s to clean-water legislation. Section 208 of the Clean Water Act mandated an "areawide" approach. In spite of the withdrawal of the federal commitment to Section 208 planning during the 1980s (in fact the overall hostility to the environment and planning by the Reagan administration), areawide, or regional, approaches continued to evolve within EPA. With the Clinton administration the agency was advocating the use of watersheds for planning, much as McHarg had advised two decades earlier.
It is interesting to note that in his 1976 biological alternatives paper McHarg had become a critic of environmental impact statement (EIS) procedures. His criticism was that EISs had become ad hoc project exercises divorced from the comprehensive understanding of regions. His approach for impact analysis was to build the ecological database on the scales of physiographic regions and watersheds, then to draw on that information to assess the consequences of possible actions.
One of the strengths of suitability analysis is its applicability to a broad range of planning situations. In the early 1970s, Wallace, McHarg, Roberts, and Todd was engaged in the planning of The Woodlands new community in Texas. In 1979 McHarg published an explanation of the method used in The Woodlands with his Penn colleagues Arthur Johnson and Jonathan Berger. The article, "A Case Study in Ecological Planning," is perhaps the best single description of suitability analysis and was included in a comprehensive book on land-use planning methods.
In addition to maps, matrices are used in suitability analysis. McHarg and his associates explain that these matrices can be used to explore relationships, such as those between land uses and development activities and those between development activities and the landscape. Maps and matrices are used to reveal both opportunities and constraints for various land uses.
For a new town, such as The Woodlands, the range of land uses is broad. The Woodlands site presented many constraints to development. It was flat, forested, and wet. The development principles that guided the new town planning included the minimization of disruption of the hydrological regime, the preservation of the woodlands, the preservation of wildlife habitats, the minimization of development costs, and the avoidance of hazards to life and health. Suitability analysis addressed all these principles.
The most serious constraint was the threat of flooding. New development was certain to alter an already delicate flood regime. The solution was to recognize natural drainage systems and to fit the development accordingly.
Flooding is a serious issue and new development frequently exacerbates the problem. The application of suitability analysis at The Woodlands by McHarg and his associates illustrates how new development can occur without endangering the health and safety of people or the environmental quality of a place. Health and safety are central to the methods and techniques for ecological planning developed by McHarg. He illustrated that decisions can be made in an environmentally responsible manner with methods that are replicable.
# References
Abercrombie, P., and T. H. Johnson. 1922. The Doncaster Planning Scheme. London: University Press of Liverpool.
Adams, Thomas. 1934. The Design of Residential Areas. Cambridge, Mass.: Harvard University Press.
Alexander, Christopher, and Marvin Manheim. 1962. The Use of Diagrams in Highway Route Location: An Experiment. Cambridge, Mass: MIT Civil Engineering Systems Lab.
Belknap, Raymond K., and John G. Furtado. 1967. Three Approaches to Environmental Resource Analysis. Washington, D.C.: The Conservation Foundation.
————. 1968. "The Natural Land Unit as a Planning Base." Landscape Architecture 58(2):145-147.
Collins, Michael G., Frederick R. Steiner, and Michael Russman. 1998. "Land-Use Suitability Analysis: Retrospect and Prospects." Under revision for the Journal of Planning Education and Research.
Eliot, Charles. 1902. Charles Eliot, Landscape Architect. Boston: Houghton, Mifflin.
Forshaw, J. H., and Patrick Abercrombie. 1943. County of London Plan. London: Macmillan.
Gordon, Steven I. 1985. Computer Models in Environmental Planning. New York: Van Nostrand Reinhold.
Hills, G. A. 1961. "The Ecological Basis for Natural Resource Management." In The Ecological Basis for Land-use Planning. Ontario, Canada: Ontario Department of Lands and Forests Research Branch.
Lyle, John, and Frederick Stutz. 1983. "Computerized Land Use Suitability Mapping." In W. J. Ripple, ed. Geographic Information Systems for Resource Management: A Compendium. Falls Church, Va.: American Society for Photogrammetry and Remote Sensing and American Congress on Surveying and Mapping.
Manning, Warren. 1913. "The Billerica Town Plan." Landscape Architecture 3:108-18.
McHarg, Ian L. 1969. Design with Nature. Garden City, N.Y.: The Natural History Press.
————. 1997. "Ecology and Design." In George F. Thompson and Frederick R. Steiner, eds. Ecological Design and Planning. New York: Wiley.
McSheffrey, Gerald. 1999. Planning Derry, Planning and Politics in Northern Ireland. Liverpool, England: Liverpool University Press.
Naveh, Zev, and Arthur S. Lieberman. 1994, 1984. Landscape Ecology: Theory and Applications. New York: Springer-Verlag.
Ndubisi, Forster. 1997. "Landscape Ecological Planning." In George F. Thompson and Frederick R. Steiner, eds., Ecological Design and Planning. New York: Wiley.
Regional Planning Staff. 1929. The Graphic Regional Plan, Regional Plan of New York and Its Environs, Vol. 1. New York: Regional Plan Staff.
Steinitz, Carl, Paul Parker, and Lawrie Jordan. 1976. "Hand Drawn Overlays: Their History and Prospective Uses." Landscape Architecture 9:444–55.
Tyrwhitt, Jacqueline. 1950. "Surveys for Planning." In Town and Country Planning Textbook. London: Architectural Press.
# 16
# An Ecological Method for Landscape Architecture (1967)
The editor of Landscape Architecture magazine during the 1960s and 1970s was Grady Clay, a friend and admirer of McHarg. Clay published several of McHarg's early works, including this article proposing that landscape architects embrace an ecological model. No other discipline besides landscape architecture provides a bridge between the environmental sciences and the design and planning professions. McHarg observed that this situation offers landscape architects a unique opportunity for leadership, a role that they have only partially pursued.
In many cases a qualified statement is, if not the most propitious, at least the most prudent. In this case it would only be gratuitous. I believe that ecology provides the single indispensable basis for landscape architecture and regional planning. I would state in addition that it has now, and will increasingly have, a profound relevance for both city planning and architecture.
Where the landscape architect commands ecology he is the only bridge between the natural sciences and the planning and design professions, the proprietor of the most perceptive view of the natural world which science or art has provided. This can be at once his unique attribute, his passport to relevance and productive social utility. With the acquisition of this competence the sad image of ornamental horticulture, handmaiden to architecture after the fact, the caprice and arbitrariness of "clever" designs can be dismissed forever. In short, ecology offers emancipation to landscape architecture.
This is not the place for a scholarly article on ecology. We are interested in it selfishly, as those who can and must apply it. Our concern is for a method which has the power to reveal nature as process, containing intrinsic form.
Ecology is generally described as the study of the interactions of organisms and environment which includes other organisms. The particular interests of landscape architecture are focused only upon a part of this great, synoptic concern. This might better be defined as the study of physical and biological processes, as dynamic and interacting, responsive to laws, having limiting factors and exhibiting certain opportunities and constraints, employed in planning and design for human use. At this juncture two possibilities present themselves. The first is to attempt to present a general theory of ecology and the planning processes. This is a venture which I long to undertake, but this is not the time nor place to attempt it. The other alternative is to present a method which has been tested empirically at many scales from a continent, a major region, a river basin, physiographic regions, sub-regional areas, and a metropolitan region to a single city. In every case, I submit, it has been triumphantly revelatory.
First, it is necessary to submit a proposition to this effect: The place, the plants, animals, and men upon it are only comprehensible in terms of physical and biological evolution. Written on the place and upon its inhabitants lies mute all physical, biological, and cultural history awaiting to be understood by those who can read it. It is thus necessary to begin at the beginning if we are to understand the place, the man, or his co-tenants of this phenomenal universe. This is the prerequisite for intelligent intervention and adaptation. So let us begin at the beginning. We start with historical geology. The place, any place, can be understood only through its physical evolution. What history of mountain building and ancient seas, uplifting, folding, sinking, erosion, and glaciation have passed here and left its marks? These explain its present form. Yet the effects of climate and later of plants and animals have interacted upon geological processes and these too lie mute in the record of the rocks. Both climate and geology can be invoked to interpret physiography, the current configuration of the place. Arctic differs from tropics, desert from delta, the Himalayas from the Gangetic Plain. The Appalachian Plateau differs from the Ridge and Valley Province and all of these from the piedmont and the coastal plain. If one now knows historical geology, climate, and physiography then the water regimen becomes comprehensible—the pattern of rivers and aquifers, their physical properties and relative abundance, oscillation between flood and drought. Rivers are young or old, they vary by orders; their pattern and distribution, as for aquifers, are directly consequential upon geology, climate, and physiography.
Knowing the foregoing and the prior history of plant evolution, we can now comprehend the nature and pattern of soils. As plants are highly selective to environmental factors, by identifying physiographic, climatic zones, and soils we can perceive order and predictability in the distribution of constituent plant communities. Indeed, the plant communities are more perceptive to environmental variables than we can be with available data, and we can thus infer environmental factors from the presence of plants. Animals are fundamentally plant-related so that given the preceding information, with the addition of the stage of succession of the plant communities and their age, it is possible both to understand and to predict the species, abundance or scarcity of wild animal populations. If there are no acorns there will be no squirrels; an old forest will have few deer; an early succession can support many. Resources also exist where they do for good and sufficient reasons—coal, iron, limestone, productive soils, water in relative abundance, transportation routes, fall lines, and the termini of water transport. And so the land-use map becomes comprehensible when viewed through this perspective
The information so acquired is a gross ecological inventory and contains the data bank for all further investigations. The next task is the interpretation of these data to analyze existing and to propose future human land use and management. The first objective is the inventory of unique or scarce phenomena, the technique for which Philip Lewis is renowned. In this all sites of unique scenic, geological, ecological, or historical importance are located. Enlarging this category we can interpret the geological data to locate economic minerals. Geology, climate, and physiography will locate dependable water resources. Physiography will reveal slope and exposure which, with soil and water, can be used to locate areas suitable for agriculture by types; the foregoing, with the addition of plant communities, will reveal intrinsic suitabilities for both forestry and recreation. The entire body of data can be examined to reveal sites for urbanization, industry, transportation routes, indeed any human land-using activity. This interpretive sequence would produce a body of analytical material, but the end product for a region would include a map of unique sites, the location of economic minerals, the location of water resources, a slope and exposure map, a map of agricultural suitabilities by types, a similar map for forestry, one each for recreation and urbanization.
These maps of intrinsic suitability would indicate highest and best uses for the entire study area. But this is not enough. These are single uses ascribed to discrete areas. In the forest there are likely to be dominant or co-dominant trees and other subordinate species. We must seek to prescribe all coexistent, compatible uses which may occupy each area. To this end it is necessary to develop a matrix in which all possible land uses are shown on each coordinate. Each is then examined against all others to determine the degree of compatibility or incompatibility. As an example, a single area of forest may be managed for forestry, either hardwood or pulp; it may be utilized for water management objectives; it may fulfill an erosion-control function; it can be managed for wildlife and hunting, recreation, and for villages and hamlets. Here we have not land use in the normal sense but communities of land uses. The end product would be a map of present and prospective land uses, in communities of compatibilities, with dominants, co-dominants, and subordinates derived from an understanding of nature as process responsive to laws, having limiting factors, constituting a value system, and exhibiting opportunities and constraints to human use.
Now this is not a plan. It does not contain any information of demand. This last is the province of the regional scientist, the econometrician, the economic planner. The work is thus divided between the natural scientist, the regional planner-landscape architect who interprets the land and its resources, and the economics-based planner who determines demand, locational preferences, investment, and fiscal policies. If demand information is available, then the formulation of a plan is possible, and the demand components can be allocated for urban growth, for the nature and form of the metropolis, for the pattern of regional growth.
So what has our method revealed? First, it allows us to understand nature as process insofar as the natural sciences permit. Second, it reveals casuality. The place is because. Next, it permits us to interpret natural processes as resources, to prescribe and even to predict for prospective land uses, not singly but in compatible communities. Finally, given information on demand and investment, we are enabled to produce a plan for a continent or a few hundred acres based upon natural process. That is not a small accomplishment.
You might well agree that this is a valuable and perhaps even indispensable method for regional planning, but is it as valuable for landscape architecture? I say that any project, save a small garden or the raddled heart of a city where nature has long gone, which is undertaken without a full comprehension and employment of natural process as form-giver, is suspect at best and capriciously irrelevant at worst. I submit that the ecological method is the sine qua non for all landscape architecture.
Yet, I hear you say, those who doubt, that the method may be extremely valuable for regional rural problems, but can it enter the city and reveal a comparable utility? Yes, indeed it can, but in crossing this threshold the method changes. When used to examine metropolitan growth the data remain the same but the interpretation is focused upon the overwhelming demand for urban land uses, and it is oriented to the prohibitions and permissiveness exhibited by natural process to urbanization on the one hand and the presence of locational and resource factors which one would select for satisfactory urban environments on the other. But the litany remains the same: historical geology, climate, physiography, the water regimen, soils, plants, animals, and land use. This is the source from which the interpretation is made although the grain becomes finer.
Yet you say, the method has not entered the city proper; you feel that it is still a device for protecting natural process against the blind despoliation of ignorance and Philistinism. But the method can enter the city and we can proceed with our now familiar body of information to examine the city in an ecological way. We have explained that the place was "because" and to explain "because," all of physical and biological evolution was invoked. So too with the city. But to explain "because" we invoke not only natural evolution but cultural evolution as well. To do this we make a distinction between the "given" and the "made" forms. The former is the natural landscape identity, the latter is the accumulation of the adaptations to the given form which constitute the present city. Rio is different from New Orleans, Kansas City from Lima, Amsterdam from San Francisco, because. By employing the ecological method we can discern the reason for the location of the city, comprehend its natural form, discern those elements of identity which are critical and expressive, both those of physiography and vegetation, and develop a program for the preservation and enhancement of that identity. The method is equally applicable when one confronts the made form. The successive stages of urbanization are examined as adaptations to the environment, some of which are successful, some not. Some enter the inventory of resources and contribute to the genius loci. As for the given form, this method allows us to perceive the elements of identity in a scale of values. One can then prepare a comprehensive landscape plan for a city and feed the elements of identity, natural process, and the palette for formal expression into the comprehensive planning process.
You still demur. The method has not yet entered into the putrid parts of the city. It needs rivers and palisades, hill and valleys, woodlands and parkland. When will it confront slums and overcrowding, congestion and pollution, anarchy and ugliness? Indeed the method can enter into the very heart of the city and by so doing may save us from the melancholy criteria of economic determinism which have proven so disappointing to the orthodoxy of city planning or the alternative of unbridled "design" which haunts architecture. But here again we must be selective as we return to the source in ecology. We will find little that is applicable in energy system ecology, analysis of food pyramids, relations defined in terms of predator-prey, competition, or those other analytical devices so efficacious for plant and animal ecology. But we can well turn to an ecological model which contains multi-faceted criteria for measuring ecosystems and we can select health as an encompassing criterion (table 3). The model is my own and as such it is suspect for I am not an ecologist, but each of the parts is the product of a distinguished ecologist. Let us hope that the assembly of the constituents does not diminish their veracity, for they have compelling value.
The most obvious example is life and death. Life is the evolution of a single egg into the complexity of the organism. Death is the retrogression of a complex organism into a few simple elements. If this model is true, it allows us to examine a city, neighborhood, community institution, family, city plan, architectural, or landscape design in these terms. This model suggests that any system moving toward simplicity, uniformity, instability with a low number of species and high entropy is retrogressing; any system moving in that direction is moving towards ill health.
Table 3. Ecological Model for Healthy Systems
Retrogression | Evolution
---|---
Ill health | Health
Simplicity | Complexity
Uniformity | Diversity
Independence | Interdependence (symbiosis)
Instability | Stability (steady state)
Low number of species | High number of species
High entropy | Low entropy
Conversely, complexity, diversity, stability (steady state), with a high number of species and low entropy are indicators of health, and systems moving in this direction are evolving. As a simple application let us map, in tones on transparencies, statistics of all physical disease, all mental disease, and all social disease. If we also map income, age of population, density, ethnicity, and quality of the physical environment we have on the one hand discerned the environment of health, the environment of pathology and we have accumulated the data which allow interpretation of the social and physical environmental components of health and pathology. Moreover, we have the other criteria of the model which permit examination from different directions. If this model is true and the method good, it may be the greatest contribution of the ecological method to diagnosis and prescription for the city.
But, you say, all this may be very fine, but landscape architects are finally designers—when will you speak to ecology and design? I will. Lou Kahn, the most perceptive of men, foresaw the ecological method even through these intractable, inert materials which he infuses with life when he spoke of "existence will," the will to be. The place is because. It is and is in the process of becoming. This we must be able to read, and ecology provides the language. By being, the place or the creature has form. Form and process are indivisible aspects of a single phenomenon. The ecological method allows one to understand form as an explicit point in evolutionary process. Again, Lou Kahn has made clear to us the distinction between form and design. Cup is form and begins from the cupped hand. Design is the creation of the cup, transmuted by the artist, but never denying its formal origins. As a profession, landscape architecture has exploited a pliant earth, tractable and docile plants to make much that is arbitrary, capricious, and inconsequential. We could not see the cupped hand as giving form to the cup, the earth and its processes as giving form to our works. The ecological method is then also the perception of form, an insight to the given form, implication for the made form which is to say design, and this, for landscape architects, may be its greatest gift.
# Notes
Australia; Rhodesia; the United Kingdom; the Gangetic Plain, the Potomac River Basin; Allegheny Plateau; Ridge and Valley Province; Great Valley Province; Piedmont; Coastal Plain; the Green Spring and Worthington Valleys; Philadelphia Standard Metropolitan Statistical Area; and Washington, D.C. See Ian L. McHarg and David A. Wallace, 1965, "Plan for the Valleys vs. Spectre of Uncontrolled Growth," Landscape Architecture (April). (Reprinted in chapter 2.)
See Philip H. Lewis, Jr., 1964 "Quality Corridors for Wisconsin," Landscape Architecture (January).
"Simplicity, Complexity; Uniformity, Diversity; Independence, Interdependence; Instability, stability" thesis by Dr. Robert McArthur; "Stability, Instability," thesis by Dr. Luna Leopold; "Low and High Number of Species," thesis by Dr. Ruth Patrick; "Low and High Entropy," thesis by Dr. Harold F. Blum; "Ill-Health, Health," thesis by Dr. Ruth Patrick.
# 17
# A Comprehensive Highway Route Selection Method (1968)
This paper was presented at the 47th Annual Meeting of the Committee on Roadside Development in 1968. It was published in the Highway Research Record by the Highway Research Board of the National Academy of Sciences-National Academy of Engineering.
McHarg's method for highway route selection had considerable influence in the 1960s. In particutar, it attracted the attention of Lady Bird Johnson. Mrs. Johnson presented McHarg's ideas to senior federal transportation officials. She even suggested that McHarg become director of the Bureau of Public Roads. McHarg helped locate several highways, including Interstate 95 through part of New Jersey, and the Richmond Parkway on Staten Island. Although some road locations were influenced by McHarg or his method, too many highway projects remain insensitive to environments and to human communities.
# Abstract
The major deficiency in prevailing highway route selection method has been the inability to include social values, including natural resources and aesthetic values, within the criteria utilized. In this study, an attempt has been made to identify components of social value, natural resources, and scenic quality, and to locate these geographically. It is presumed that the area of lowest social value, if transected by a highway, incurs the least social cost. The normal determinants of highway route selection, topography, soils, etc., have been expanded to include management or impairment of groundwater and surface water resources, susceptibility to erosion, etc. When highway corridors of minimum social cost and minimum physiographic obstruction were revealed, they were tested against their effect on scenic values. The objective of providing an excellent scenic experience was considered as a social value created by the highway. The corridor of least social cost was next tested against the degree to which it could create new and productive land uses where these would be necessary and welcome. The sum of least social cost and highest benefit alignment was identified. It is described as the rate of maximum social benefit.
The method revealed the route of least social cost and least physiographic obstruction as a funnel of land parallel to Jacobs Creek, proceeding along the base of Pennington and Sourland Mountain and widening into a trumpet as it meets the Raritan River. This alignment has the added merit of permitting management of the aquifer-recharge area which it parallels to the benefit of groundwater resources in the Delaware-Raritan area. It provides an excellent scenic experience. It should additionally induce new and productive land uses in the West Trenton area and the Raritan River communities in New Jersey.
# Introduction
The importance of highways is incontrovertible. They are essential components of the economy. The modern highway incorporates high levels of engineering skill in its geometric characteristics and traffic performance. Yet the increasing opposition to proposed highways reveals a fundamental weakness. The route-selection process has, to date, been unable to recognize and respond to important social and aesthetic values. As a result, the major highway is increasingly viewed as a destroyer, carving remorselessly through the hearts of cities, outraging San Francisco and Boston, humbling Faneuil Hall, destroying precious urban parks as in Philadelphia, offending great sectors of metropolitan population, and threatening resources and values such as Rock Creek Park and the Georgetown waterfront in the nation's capital. Examples abound. The major highway is increasingly regarded as a tyrant, unresponsive to public values, disdainful of the people it purports to serve.
The cause can be simply identified. It lies in the inadequacy of the criteria for route selection. These have been developed for rural areas where social values are scarce and engineering considerations paramount. These criteria have not been enlarged to deal with situations where social values hold a primary importance and engineering considerations are accorded a much lower value. The remedy lies in inverting the criteria to obtain a balance between social-aesthetic values and the orthodoxy of engineering considerations.
This study seeks to develop a more comprehensive route-selection method including social, resource, and aesthetic criteria for a section of proposed Interstate 95 (1-95) between the Delaware and Raritan Rivers.
# Social Values in Highway Route Selection
The method utilized by the Bureau of Public Roads for route selection involves calculating the savings and costs derived from a proposed highway facility. Savings include savings in time and in operating costs and reduction in accidents through construction of a new highway facility. The costs are those of construction and maintenance necessary to obtain these savings with a minimum ratio of savings to costs of 1.2 to 1.0. Any qualitative factors are considered descriptively after the conclusion of the cost-benefit analysis.
The Bureau of Public Roads seeks to enlarge this method and develop a land-use model in which more parameters can be included in both cost and benefit calculations. The bureau was instrumental in initiating the Penn-Jersey Transportation Study which sought to advance the science of highway location. It seems reasonable to assume that it would welcome any serious effort to improve highway route selection method.
The objective of an improved method should be to incorporate social, resource, and aesthetic values in addition to the normal criteria of physiographic, traffic, and engineering considerations. In short, the method should reveal the highway alignment having the maximum social benefit and the minimum social cost. This poses many difficult problems. It is clear that many new considerations must be interjected into the cost-benefit equation and that many of these are considered nonprice benefits. Yet the present method of highway cost-benefit analysis merely allocates approximate money values to convenience, a commodity as difficult to quantify as health or beauty.
The relevant considerations will vary from location to location, with land and building values preponderant in central urban areas and scenic beauty dominant in certain rural areas. It should, however, be possible to identify a number of critical determinants for each area under consideration.
# Criteria for Interstate Highway Route Selection
Interstate highways should maximize public and private benefits by (1) increasing the facility, convenience, pleasure, and safety of traffic movement; (2) safeguarding and enhancing land, water, air, and biotic resources; (3) contributing to public and private objectives of urban renewal, metropolitan and regional development, industry, commerce, residence, recreation, public health, conservation, and beautification; and (4) generating new productive land uses and sustaining or enhancing existing land uses.
Such criteria include the orthodoxy of route selection, but place them in a larger context of social responsibility. The highway is no longer considered only in terms of automotive movement within its right-of-way, but in context of all physical, biological, and social processes within its area of influence.
The highway is thus considered as a major public investment that will transform land uses and values and that will affect the economy, the way of life, health, and visual experience of the entire population within its sphere of influence. It should be located and designed in relation to this expanded role.
Positively, the highway can
* Facilitate safe, convenient traffic movement
* Provide key investment for urban renewal, and metropolitan and regional development
* Maximize the visual quality of movement
* Locate new, productive land uses
* Reveal the physical and cultural identity of the region it transects
* Provide access to existing inaccessible resources
* Link scenic and recreational resources
Negatively, the highway can
* Constitute a health hazard due to stress, anxiety, lead, hydrocarbons, ozone, carcinogens, positive atmospheric ionization
* Constitute a nuisance due to noise, glare, fumes, dust, danger, ugliness
* Destroy existing resources—scenic, historic, ground and surface water, forests, farmland, marshes, wildlife habitats, areas of recreational value, and areas of natural beauty
* Reduce property values by any of the above, plus division of property, creation of barriers, and loss of access
* Destroy existing communities and their hinterlands by transecting them, denying access, inconsiderate alignment, and dissonant scale, in addition to the preceding deleterious effects
It is clear that the highway route should be considered a multipurpose rather than a single-purpose facility. It is also clear that when a highway route is so considered, there may be conflicting objectives. As in other multipurpose planning, the objective should be to maximize all potential complementary social benefits and reduce social costs.
This means that the shortest distance between two points meeting predetermined geometric standards is not the best route. Nor is the shortest distance over the cheapest land. The best route is that which provides the maximum social benefit at the least social cost.
The present method of cost-benefit analysis as employed for route selection has two major components: the savings in time, operating costs, and safety provided by the proposed facility; and the sum of engineering, land and building purchase, financing, administrative, construction, operation, and maintenance costs.
On the credit side it seems reasonable to allocate all economic benefits derived from the highway. These benefits accrue from the upgrading of land use, frequently from agricultural to industrial, commercial, or residential uses. On the debit side, highways can and do reduce economic values; they do constitute a health hazard, a nuisance, and danger; they can destroy community integrity, institutions, residential quality, scenic, historic, and recreational values.
This being the case, it appears necessary to represent the sum of effects attributable to a proposed highway alignment and distinguish these as benefits-savings and costs. In certain cases these can be priced and can be designated price benefits, price savings, or price costs. In other cases, where valuation is difficult, certain factors can be identified as nonprice benefits, savings, or costs.
A balance sheet in which most of the components of benefit and cost are shown should reveal alignments having the maximum social benefit at the least social cost (see tables 4 and ).
Considerations of traffic benefits from the Bureau of Public Roads can be computed for alternative alignments. The cost of alternative routes can be calculated. Areas where increased land and building values may result can be located, if only tentatively, in relation to the highway and prospective intersections. Prospective depreciation of land and building value can also be approximately located. Increased convenience, safety, and pleasure will presumably be provided within the highway right-of-way; inconvenience, danger, and displeasure will parallel its path on both sides. The degree to which the highway sustains certain community values can be described, as well as the offense to health, community, scenery, and other important resources.
Table 4. Components of Benefits and Costs Attributable to a Proposed Highway Alignment
BENEFITS AND SAVINGS | COSTS
---|---
Price benefits | Price costs
Operating economies: Function of distance × volume × constant for operating economies and safety | Costs: Function of distance x engineering, land and building acquisition, construction, financing administration, operation, and maintenance
Economic values created: Increment of increased land and building values attributable to the highway within its sphere of influence | Economic values lost: Increment of reduced land and building values due to the same cause
Nonprice benefits | Nonprice costs
values added: Increased convenience, safety, and pleasure provided to drivers on the facility | Values lost: Decreased convenience, safety, and pleasure to populations and institutions within the sphere of influence; health hazard, nuisance, and danger
Price savings | Price costs
Construction economies: Propitious conditions of topography, geology, drainage foundations, and minimum structures | Construction costs: Inordinate costs due to difficult topography, geology, drainage, foundations, and necessity for many and/or expensive structures
Nonprice savings | Nonprice costs
Social values saved: | Social costs:
Community | Community values lost
Institutional | Institutional values lost
Residential | Residential values lost
Scenic | Scenic values lost
Historic | Historic values lost
Surface water | Surface water resources impaired
Groundwater | Groundwater resources impaired
Forest water | Forest water resources impaired
Wildlife Wildlife | water resources impaired
# Application of the Comprehensive Method to 1-95 between the Delaware and Raritan Rivers
This area under consideration is defined by a discontinuous ridge composed of Baldpate, Pennington, and Sourland Mountains to the northwest, and the Delaware-Raritan Canal to the southeast; the Delaware and Raritan Rivers form the remaining southwest and northeast boundaries. Urbanization is concentrated in and around Trenton to the southwest, New Brunswick-Bound Brook-Manville-Somerville to the northeast, with the intervening area substantially rural piedmont landscape with localized settlement in the town of Princeton and the villages of Pennington, Hopewell, and Griggstown. The majority of the area falls within the Stonybrook-Millstone watershed.
The entire area constitutes a segment of the Philadelphia–Trenton–New York corridor running southwest–northeast in the piedmont, paralleling the fall line dividing piedmont from the coastal plain.
This is an attractive landscape defined by wooded hills, drained by pleasant streams, and graced by decent farms. Princeton is its spiritual capital, and the quality of this institution and its surrounding landscape constitutes a prime locational value for the area. While terminated by industrial locations in Trenton and New Brunswick–Bound Brook and Somerville, the intervening valley is predominantly a rural-residential area. In such an area certain social values can be described, if only in normative terms—scenic beauty, residential quality, the integrity of institutions, towns, and villages. The diminution or impairment of these values can be described, again only in normative terms, as social costs.
Toward this end a number of maps have been prepared. They share the description of approximations as follows:
1. Sequence related to construction savings and costs
1. Topography
2. Land values
2. Sequence related to social values
1. Urbanization
2. Residential quality
3. Historic value
4. Agricultural value
5. Recreational values
6. Wildlife value
7. Water values
8. Susceptibility to erosion
3. Scenic value
4. Physiographic obstructions
In every case three values are allocated to each parameter. The presumption is that the corridor of maximum social utility will transect the zones of lowest social value and provide the major social benefit (see tables 4 and ).
The method employed uses transparent overlays for each parameter. Each of these has three values, the highest representing the greatest physical and engineering obstruction to highway location. The highest value of obstruction is represented by a dark tone, the intermediate value by a light tone, while the lowest category is the residue and is fully transparent. When all of the parameters are overlaid on the base map, it is presumed that the maximum darkness represents the greatest aggregate physiographic and social value obstruction. In contrast, light areas, representing the least physiographic obstruction and the least social value, offer prospective corridors for consideration.
It is immediately conceded that not all of the parameters are coequal. Existing urbanization and residential quality are more important than scenic value or wildlife. Yet it is reasonable to presume that where there is an overwhelming concentration of physiographic obstruction and social value that such areas should be excluded from consideration; where these factors are absent, there is a presumption that such areas justify consideration.
Table 5. Suggested Criteria for Interstate Highway Route Selection
BENEFITS AND SAVINGS | COSTS
---|---
Price benefits | Price costs
| Reduced time distance | | Survey
Reduced gasoline costs | Engineering
Reduced oil costs | Land and building acquisition
Reduced tire costs | Construction costs
Reduced vehicle depreciation | Financing costs
Increased traffic volume | Administrative costs
Operating and maintenance costs
Increase in value (land and buildings) | Reduction in value (land and buildings)
| Industrial values | | Industrial values
Commercial values | Commercial values
Residential values | Residential values
Recreational values | Recreational values
Institutional values | Institutional values
Agricultural land values | Agricultural land values
Nonprice benefits | Nonprice costs
| Increased convenience | | Reduced convenience to adjacent properties
Increased safety | Reduced safety to adjacent populations
Increased pleasure | Reduced pleasure to adjacent populations
Health hazard and nuisance from toxic fumes, noise, glare, and dust
Price savings | Price costs
| Nonlimiting topography | | Difficult topography
Adequate foundation conditions present | Poor foundations
Poor drainage
Adequate drainage conditions present | Absence of construction materials
Available sands, gravels, etc. | Abundant structures required
Minimum bridge crossings, culverts, and other structures required |
Nonprice savings | Nonprice costs
| Community values maintained | | Community values lost
Institutional values maintained | Institutional values lost
Residential quality maintained | Residential values lost
Scenic quality maintained | Scenic values lost
Historic values maintained | Historic values lost
Recreational values maintained | Recreational values lost
Surface water systems unimpaired | Surface water resources impaired
Groundwater resources unimpaired | Groundwater resources impaired
Forest resources maintained | Forest resources impaired
Wildlife resources maintained | Wildlife resources impaired
This is not yet a precise method for highway route selection, yet it has the merit of incorporating all existing parameters, adding new and important social considerations, revealing their locational characteristics, permitting comparison, and revealing aggregates of social values and costs. Whatever limitations of imprecision it may have, it does enlarge and improve the existing method.
The preceding discussion has emphasized the identification of physiographic corridors containing the lowest social values as the preferred route for highways. In discussion of cost-benefit analysis, reference has been made to the role of the proposed highway in creating new values. This view deserves a greater emphasis. Within limits set by the points of origin and destination, and responsive to physiographic obstructions and the pressure of social values, the highway can be used as conscious public policy to create new and productive land uses at appropriate locations. In any such analysis, cost-benefit calculations would require that any depreciation of values would be discounted from value added. In addition, scenic value should be considered as possible added value. It is, of course, possible that a route could be physio-graphically satisfactory and could avoid social costs, create new economic values at appropriate locations, and also provide a satisfactory scenic experience.
The highway is likely to create new values whether or not this is an act of conscious policy. Without planning, new values may displace existing ones, but even if a net gain results there may well be considerable losses. An example would be a highway transecting a good residential area that then became used for industrial purposes. The increase in land and building values might well be destroyed. The same effect, adjacent to industrial zones in existing urban centers, would accomplish the same benefits without comparable social and economic costs.
# Parameter Analysis
1. Topography. The area under consideration is a broad valley extending from the Delaware to the Raritan, limited by the Baldpate, Pennington, and Sourland Mountains and divided by Rocky Hill. It contains two major drainage systems, the Stonybrook, draining the area to the west of Rocky Hill, and the Millstone, draining the area to the east. As a result of this physiography, fans of tributaries run tangentially across any direct line between West Trenton and the northeast. Rocky Hill represents a major topographic obstruction.
The two obvious topographic corridors are to the north and south of Rocky Hill. The northern corridor follows Jacobs Creek, may proceed either north or south of Pennington Mountain, and then follows the base of Sourland Mountain and widens into a broad swath toward Somerville and New Brunswick. The southern corridor is less defined but falls south of Rocky Hill and parallels U.S. Route 1 and the Pennsylvania railroad.
2. Land Values. Land values reflect urbanization and residential quality in nonurbanized areas. Land values are highest in a swath of land including Trenton along U.S. Route 1 to New Brunswick, with an intrusion of high values in and around Princeton. Values tend to decrease generally from this base line to a band on the summit of Sourland Mountain and rise northward. Lowest land values occur in the northern corridor; highest in the southern corridor, influenced by Route 1.
3. Urbanization. Trenton, Princeton, New Brunswick, Bound Brook, Manville, Somerville, and Raritan constitute the major areas of urbanization. The villages of Hopewell, Pennington, Lawrenceville, etc., constitute the remainder. Thus urbanization is concentrated at the termini of the area under consideration—in Trenton at one end and the broad band of communities united by the Raritan River at the other. Princeton and the remaining villages are islands between these urban extremes. Southern alignments will affect Lawrenceville, Princeton, Rocky Hill, and recent subdivisions on Dead Tree Run Road, Willow Road, and Township Line Road; a northern alignment will affect the village of Hopewell.
4. Residential Quality. Princeton and its environs constitute the major concentration of residential quality as measured by value. Land and building values of $120,000 per acre exist in Princeton alone. An acre to the east contains average values in excess of $60,000. Values fall fairly regularly from the Princeton centroid. Destruction of residential quality will rise as Princeton and its environs are transected and will diminish as distance from Princeton increases. The northern corridor transects areas of lesser residential quality as reflected by value.
5. Historic Value. Only one value has been shown reflecting historic importance. Three areas are discerned: the first, Washington's Crossing, Delaware—Trenton; the second, Princeton and environs; and the third, the Millstone River corridor. Northern and southern routes avoid conflict with historic values. A central route does profoundly affect the historic values adjacent to the Millstone River.
6. Agricultural Value. The value of agricultural land is twofold: as a source of food and (even more so, in this area) as a land use that sustains an attractive landscape and maintains it. There is no prime agricultural land in the area, and intrinsic value is quite low, with the single important exception of the visual landscape quality provided by farming. Second-class agricultural land is limited to a triangle of Trenton-Princeton-Pennington. This is more accurately identified as scenic quality.
7. Recreational Value. Existing open space is a major constituent of recreational value. This is concentrated in the Stonybrook–Rocky Hill-Millstone–Six Mile Run Impoundment areas. These unite a crescent of land bordering Rocky Hill and the Millstone River. Within this area fall the major open spaces (public and private), the major rivers and streams, and the majority of the recreational potential, with the exception of the mountain ridge. These represent a major social value. Avoidance of these resources would divert the highway corridor to either the north or south of Rocky Hill and outside of the mountain ridge.
8. Wildlife Value. These values are concentrated on the forested uplands—Baldpate, Pennington, Sourland Mountain, and Rocky Hill; and the forests and large woodlands, the major rivers, and marshes. There are thus two major divisions of habitat, the forested hills and the water-related system of marshes, streams, and rivers, and their containing valleys. Highways constitute a major hazard to terrestrial wildlife; filling of marshes is a threat to fish-spawning and wildfowl. The forested uplands are concentrated in the defining mountain ridge and Rocky Hill. Both northern and southern corridors can avoid conflict with wildlife resources. The fissures of rivers and streams run tangentially across the major axis of the valley, and any central corridor will adversely affect these resources.
9. Water Values. Water resources fall into two categories: surface water and groundwater. As has been observed, the Millstone River is the major drainage, with the Stonybrook as secondary. Toward these run a number of tributaries, generally tangential to the long axis. A highway that follows the central axis of the valley will interfere with surface runoff and affect the behavior of existing streams. This will be exacerbated when the highway is either in cut or fill. It will have the effect of a trench interceptor drain in cut or of a dam in fill. This can be obviated in two ways. The first is by following ridge lines throughout its length, a proposal barely feasible. The second is to locate the highway near the source of the watershed where the effect is minimized. A northern alignment would accomplish this objective.
The inadvertent function of a highway in behaving like a trench drain or dam has an effect on groundwater resources. It can be utilized as a conscious device for management of groundwater. In the negative sense, the highway will cause least offense by passing over the least valuable aquifer—that of Brunswick shale. It will commit a minor offense by passing over the intermediate aquifer of Stockton sandstone. It will create the greatest offense by paving the major aquifer recharge areas.
Positively, the highway, in fill, acting as a dam, can be used to recharge the aquifer and as such would perform a supplementary service. A route south of, and parallel to, the aquifer recharge area at the foot of Sourland Mountain and north of Pennington Mountain would avoid a destructive role on the water regimen and perform a function in groundwater management.
10. . Susceptibility to Erosion. Highway construction is a process which exacerbates erosion and sedimentation. Excavation and grading combined with hundreds of acres of bare soil guarantee that surface water channels will be surfeited with sediment. This increases the turbidity of streams and reduces photosynthesis, and thus reduces the biotic population. It is often unsightly. It raises flood plains and deposits alluvial silts.
The Stockton sandstone is least susceptible to erosion; Brunswick shale is intermediate; Argillite and Diabase are most susceptible. A southern route crosses mainly Stockton sandstone and Brunswick shale. A central route crosses bands of Lockatong Argillite, Brunswick shale, and Diabase. A northern corridor can follow either Brunswick shale or Stockton sandstone. A central corridor appears to produce the maximum erosion and sedimentation.
# Interpretation of Superimposed Maps
Ten maps were prepared, representing a range in values for the selected parameters. Each of these has been analyzed independently. The method, however, suggests that (1) the conjunctions of social values represent an obstruction to a highway alignment, (2) the conjunctions of physiographic difficulties represent an obstruction to a prospective highway, and (3) the aggregate of these obstructions, social and physiographic, represents the sum of obstructions to a highway corridor.
In contrast, the lightest area, representing the lowest social values and the least physiographic obstruction, indicates the highway alignment of least social cost and least physiographic difficulty.
When maps of social values and physiographic obstructions are superimposed, it is immediately apparent that the concurrence of obstruction occupies a broad belt between the Delaware and Raritan Rivers, Trenton, and New Brunswick. This aggregate obstruction diminishes to the south, but the continuous area containing the least obstruction occurs as a trumpet with its northern limit at the base of Pennington and Sourland Mountains, narrowest at Hopewell, and widening eastward between Sourland Mountain and the Millstone River.
If social values are accepted as criteria for route selection, in addition to normal factors of topography, it is clear that the route of least social cost proceeds parallel to and west of Jacobs Creek, either north or south of Pennington Mountain, and thence along the base of Sourland Mountain. At the tail of this mountain the corridor widens, permitting a route to proceed either to I-187 near Pluckemin or to South Bound Brook.
In addition to providing an alignment of minimum social cost, this route, if located south of the major aquifer recharge, permits the highway to enhance the water regimen by permitting managed aquifer recharge.
# Tests of the Area of Lowest Social Value against Physiographic Restraints
A map was prepared which showed three levels of physiographic obstruction:
* Zone 1. Areas having slopes in excess of ten percent. Areas requiring major rivers, highway, and railroad crossings
* Zone 2. Areas having slopes in excess of five percent but less than ten percent. Areas requiring minor river and highway crossings
* Zone 3. Areas having slopes of less than five percent. Areas without consequential bridge crossings
When the corridor of least social cost was tested against this finer-grain analysis, it was seen to follow a route of least physiographic obstruction.
# Test of the Area of Least Social Cost and Least Physiographic Obstruction against Scenic Value
In order to measure the highway corridor against the scenic experience it could provide, a map of scenic value was prepared. Three values were determined as follows:
* Zone 1. A broad band embracing the Delaware River, Washington's Crossing, Pennington Mountain, Rocky Hill, the Stonybrook watershed, and narrowing to include the Millstone Creek and the Duke Estate
* Zone 2. In general, this zone includes the Sourland and the area of farmland on either side of the Millstone River
* Zone 3. Remainder of area
The valley of Jacobs Creek, Baldpate Mountain, the valley of Stonybrook and the Pennington Mountain, Rocky Hill, the Millstone River Valley, Beden Brook, and Sourland Mountain constitute the major elements of scenic value in the area. They fall into two major elements: the northern ridge as one, and the heartland, composed of Stonybrook, Rocky Hill, Millstone River, and Beden Brook as the second.
Scenic value can be considered in two opposing ways: as a value that the highway should avoid, or as a resource the highway should exploit. If avoidance is the objective, then the corridor should fall south of the Stonybrook and Millstone. It should positively avoid transecting the heartland or paralleling the Millstone River. If exploitation is the criterion, then two possibilities are apparent. The first would select the Rocky Hill Ridge and the Millstone River corridor. The other alternative is the northern corridor at the base of the Baldpate, Pennington, and Sourland Mountains.
The Rocky Hill—Millstone River corridor, while undoubtedly scenic, transects the area of highest social values and high physiographic obstructions. It does not correspond to a reasonable direction for the highway. The alignment on Jacobs Creek, skirting the base of Pennington and Sourland Mountains, equally offers an excellent scenic experience, but avoids areas of high social value and areas of physiographic difficulty—and it corresponds with the general direction of the prospective route.
The proposed corridor skirting the base of the mountain ridge not only satisfies the criteria of low social value and low physiographic difficulty, but offers an excellent scenic experience.
# The Highway as a Conscious Device to Locate New and Productive Land Uses
If one assumes that cannot only the highway avoid unnecessary social costs but also create new productive land uses, then it becomes important to decide where these might best be located.
In the area under consideration, industrial and commercial land uses exist in Trenton and in the band of communities bordering the Raritan River. In the intervening area industry is sparse. The ethos of the heartland, Princeton-dominated, is residential and institutional. The creation of new industrial land uses would be unwelcome in the heartland but welcome at either end, in Trenton, New Brunswick, Bound Brook, Manville, Somerville. This being so, the highway corridor should be selected to locate new productive land uses at the extremities but avoid the creation of conflicting land uses in the central valley. In addition, any direct central route would inflict not only dissonant land uses but also a corridor of noise, glare, dust, health hazard, nuisance, and danger on the rural-residential-institutional land uses of the heartland.
If this analysis can be substantiated, the optimum highway corridor would follow an alignment and locate such intersections as to benefit both the Delaware River and Raritan River termini with new productive land uses and would avoid such conflicts in the intervening area. A northern route could induce new land uses adjacent to West Trenton and the Raritan River communities, where new industrial and commercial land uses would be both consonant and welcome.
# 18
# Biological Alternatives to Water Pollution (1976)
In the early 1970s McHarg established the Center for Ecological Studies in Planning and Design at Penn as the research program for the Department of Landscape Architecture and Regional Planning. The Center became engaged in several projects, including the landmark ecological plan for Medford Township, New Jersey, and the book Biological Control of Water Pollution (1976), co-edited by two former students of McHarg's: Robert Pierson and Joachim Tourbier. Tourbier later directed the Water Research Center at the University of Delaware and is now professor of landscape architecture at the University of Dresden, Germany. McHarg contributed this chapter to Biological Control of Water Pollution, which helped introduce an alternative way to addressing the issue of water pollution.
Any consideration of alternatives must begin from the datum of the status quo. It may then be useful to describe the assumptions and processes which now produce what is euphemistically described as drinking water. First let us consider the assumptions. The implicit assumption, the inalienable right governing water management, is that thou shalt defecate into thy neighbor's drinking water and, further, if for any reasons of delicacy, hygiene, or prudence you prefer not to do so, be assured that thy municipality shall do it for thee. So water treatment and sewage treatment are, for all practical purposes, synonymous. We can observe, parenthetically, that this rule does not hold for all primitive societies where often taboos seek to separate excrement from drinking water. Primitive, then, can also mean the capability of drinking water from rivers, lakes, streams, and springs. This is not the way of advanced technological societies. Beginning from a modest commitment to excrement as an essential component of water supply, with advancing technology there followed an enrichment of the attendant nutrients. The development of industry, not least the chemical industry, ensured a wide variety of new additives. Among the earliest of these were the nitrates and phosphates; the former, derived from sewage and fertilizer, is an essential ingredient for blue babies or cyanosis. But a wider population could be served by DDT, dieldrin, aldrin, and the organic phosphates. These are not benign, nor are arsenic, lead, cyanide, selenium, cadmium, or other benefits from modern industry. Until recently these additives to drinking water were uncoordinated, but with the commitment to regionalization of sewage treatment, industry was invited to contribute its toxins to the nation's resources of drinking water. This is advanced technology.
In the competition among major cities for reducing the time factor between the two critical orifices in the human water cycle, the City of Philadelphia assumes a high position, perhaps even primary. This success derives from the inspired location of the sewage and water treatment plants on the Delaware River. The sewage receives secondary treatment when it does not rain; it is poured into the Delaware River where the U.S. Geological Survey monitors the continued absence of dissolved oxygen, year after year, with meticulous instrumentation. As the Delaware is tidal, the sewage moves downstream until slack tide when it returns upstream, passing the sewage treatment plant, and proceeds to the intake basins of the water treatment plant which can only be filled at high tide. Thence follows water treatment in a fully computerized facility. This, we observe, is a modern facility utilizing these high-technology innovations of caged sedimentation, coagulation, filtration, and, finally, chlorination. Now we all know that sanitary engineers, valiantly fighting their way into the twentieth century, never accomplished water treatment. This was done by microorganisms. These creatures have been successfully engaged in water treatment for hundreds of millions of years. Their commitment to this endeavor is not related to the objective of cleaning water, but to eating and survival. One would assume that these indispensable creatures and their ways would be of the greatest significance to those engaged in treating water. Not so, for the sanitary engineer does not need to know these essential decomposers. His objective is to kill all organisms. How else can you drink a dilute soup of dead bacteria in a chlorine solution? And so hyper-, super-, extra-, ultra-chlorination ensues at such levels as to guarantee that the consumer at the end of the water line receives only dead bacteria with his toxins. And, not least among these toxins, as we have recently learned, are the carcinogens produced by the chlorine. This is modern technology at its best. Only when drinking water is delivered at 98.4°F will there be another significant advance.
But what of modern industry? It clearly operates on a modified version of the implicit inalienable right. Thou shalt poison thy neighbor's drinking water, and, if for any reason of delicacy, hygiene, or prudence, you prefer not to do so, thy regional sewage system shall do it for thee.
Once upon a time I was invited by Fortune magazine to address the presidents of fifty of the largest corporations in the United States at the Four Seasons Restaurant in New York. "Gentlemen," I said, "I wish to make a bargain with you on behalf of the American people. I know that this is presumption on the part of a Scottish immigrant, but hear me out. I have no doubt that you all bathed and changed your shirt and underclothes before coming here. I am sure you have a clean pocket handkerchief. I would not be surprised if many of you added a dab of underarm deodorant. Moreover, I have observed you at cocktails and dinner, noted your handling of knives, forks, spoons, table napkins, even finger bowls. I am sure that the American people would agree with me that your standards of personal hygiene are impeccable. But as my bargain requires concessions on your part, the American people, too, then must make concessions to you. On their behalf, I offer you a relaxation in these demanding standards of personal hygiene to which you subscribe. You may forego the changes of underwear, finger bowls, underarm deodorants; you may belch or even fart in public. This relaxation may be welcome to you and yet will not endanger the American environment. For this relaxation the American people require an improvement in your standards of corporate hygiene. We require that you now cease and desist from voiding thousands of millions of gallons and tons of your excrement into the American environment. The time has come for American industry to be toilet trained. You are incontinent. This condition may either be a sign of infantilism or senescence. Should it be the latter, we can only await your demise with impatience. If the former, we can help you into adult continence whereby you control the deposition of your excrement to minimize offense to the American people and the American environment, in particular, that you cease and desist from poisoning the nation's drinking water."
Of course, government is not immune to similar criticism. Indeed, the gravest threats to the nation's health and well-being are accomplished at public expense. Atomic testing, atomic reactors—notably accidents to these—contribute radioactivity to the atmosphere, to the hydrosphere, and to our drinking water. The government also contributes more modest toxins. An analysis of well logs in the Denver metropolitan region disclosed many wells with excessive levels of arsenic, cyanide, lead, and selenium. When these were mapped, they pointed like an arrow to the source and the culprit, the Rocky Mountain National Arsenal. It would be unfair to omit the United States Army Corps of Engineers, entrusted with the enforcement of the 1899 Refuse Act. They failed to initiate prosecutions under the act for over three score years and ten. They must be held responsible for the pollution of America's water systems. In other military societies the appropriate remedy for such celestial negligence is ceremonial disembowelment. Industry and government have been coequally negligent in the area of water treatment. Yet, it would be unfair and ungracious if the positive accomplishment of conventional water treatment and distribution were not recognized and applauded. It is precisely this system which has banished the major water-borne diseases, the pestilences of typhoid, gastroenteritis, cholera, dysentery, and others. It would be just as unfair to fail to note that many primitive societies are scourged by such diseases today and are less successful in solving these problems than the Western countries. Yet, the successes are long past and the processes for water and sewage treatment, long efficacious, are incapable of dealing with the products of modern industry and chemistry and, finally, the finest tool of the system, chlorination, is now accused of contributing carcinogens to our water supply.
Wherein lies the remedy? Surely it must begin with recognition of the fact that clean, potable water is a byproduct of biological systems. The entire subject of water must be seen to be a problem of biology rather than engineering. This would recognize that nature has been in the business of water and sewage treatment since the beginning of life, most effectively, and that solutions to the problems of water lie in understanding aquatic biology and, dominantly, limnology. But, knowing that water systems are affected by terrestrial processes (indeed water quality represents a synthesis of aquatic and terrestrial processes), then the scientific basis for the management of water resources must be ecology. In fact, we can say that the biological alternatives which concern us in this volume are ecological alternatives to nonecological existing methods. The definition of ecology is the study of the interaction of organisms and the environment, which includes other organisms. Applying this concept to the provision of potable water would suggest that, as such, water is ultimately a product of aquatic organisms, and as these organisms require specific environmental conditions to survive and prosper, the conditions of terrestrial and aquatic systems must be preserved or managed to maintain the survival and success of the essential organisms which produce potable water. We do not have to create or discover the model which would govern such an ecological approach. It is the limnological model. One of its creators and most effective practitioners, Ruth Patrick (and her colleagues at the Philadelphia Academy of Natural Science), has devoted several chapters in Biological Control of Water Pollution to an exposition of this theoretical concept. It is enough to say here that this model can identify water quality from the number of species and discriminate between levels of quality using certain organisms—their presence, absence, or relative abundance—as specific indicators. Moreover, the factors which determine these conditions of aquatic "health" are generally known and can be manipulated. I subscribe totally to the limnological theory as the essential perception for ecological planning for water quality. How could such a view be institutionalized? I will simply paraphrase parts of a research project which I undertook as an agent of the American Institute of Planners [now the American Planning Association] for the Environment Protection Agency (EPA) entitled, "Towards a Comprehensive Plan for Environmental Quality" (Wallace, McHarg, Roberts, and Todd 1973).
The first matter concerns an administrative structure for the planning, management, and regulation of water resources. Acceptance of ecology-limnology as the scientific basis for this objective implies an administrative structure. The unit of surface-water hydrology is the watershed, the unit of subsurface hydrology is the groundwater basin, a unit of ecological planning is the physiographic region—an area, homogeneous with respect to geological history and therefore homogeneous with respect to physiography, hydrology, soils, plants, and animals. The United States is subdivided into thirty-four such physiographic regions. The problem of selecting the appropriate unit is not simple. While the major groundwater basins are frequently within physiographic regions, the major rivers often transect several regions. A solution to the complex problem of an administrative structure for water planning and management, consonant with ecology-limnology, should first involve acceptance of the watersheds of the major rivers as the primary planning units. It would recognize the Continental Divide and thus divide the country into the two administrative units of east and west. Within each major subdivision the great rivers—for example, the Hudson, Delaware, Susquehanna, Potomac, and James in the east—would provide the major administrative elements. The next subdivision would be the physiographic region within the watershed or, as in the case of the coastal plain, the physiographic region itself. The smallest administrative unit would be the watershed within a physiographic region.
Applied to the Philadelphia area, the major administrative unit would be the Eastern Office. The Delaware River Watershed provides the next unit, subdivided into the physiographic regions of Allegheny Plateau: Valley and Ridge Province; Great Valley; Reading Prong; Triassic Piedmont Lowlands; Piedmont Uplands; and Coastal Plain. Each of these regions is homogeneous and has characteristic problems associated with hydrology, limnology, and land use. These regional problems cumulatively effect water quality in the watershed. The Allegheny Plateau, with the Catskills and Poconos, is primarily used for recreation and second homes. It has excellent water in lakes and streams. The problem here is preservation and enhancement. In the Valley and Ridge Province, transected by the Delaware and its tributaries and characterized by the location of coal mining, the overwhelming problem is acid mine drainage, and water management must seek to solve this problem. The next province, the Great Valley, is preponderantly limestone with significant groundwater resources. It is the location of extensive agriculture. Here the characteristic problems are pollution from fertilizers, herbicides, pesticides, sedimentation, and nitrates. The Reading Prong has little influence upon water quality, but the piedmont, site of extensive urbanization, contributes massive human wastes and concentrates industrial toxins. Finally the coastal plains constitutes the most valuable water resource of the watershed reposing in the aquifer system. It presents the problem of managing groundwater. This suggests that the characteristic problems of each physiographic region require an appropriate staff of scientists, planners, and administrators. Each region is specific, yet all are concerned with cumulative effects reflected in the entire watershed, its water quality and quantity.
Given an administrative structure consonant with the morphology of hydrologic-limnologic systems, it is then possible to plan and regulate within existing legislation. Sections 201 and 208 of the Federal Water Pollution Control Act Amendments (Public Law 92-500, commonly referred to as the Clean Water Act) provide effective vehicles for water planning and management, particularly the latter. Section 201 is devoted to the provision of sewage treatment facilities; section 208 is concerned with nonpoint sources of pollution and with the effects of urbanization and land use. If this latter legislative power required ecological planning, including the limnological model, and were such planning structured within watersheds and physiographic regions, there would be reason for confidence of ultimate success. However the process today is hampered by arbitrary political subdivisions and by an assumption that the objectives of the act can be satisfied by the orthodoxy of sanitary engineering. Acceptance of ecology as the scientific basis, limnology as the specific science for water management, hydrological units as administrative units, and ecological planning as the instrument for fulfilling the objectives of the Water Pollution Control Act would provide massive improvement but yet would still not be sufficient. The environmental impact statement procedure, while valuable, is ad hoc, adventitious, and can be negative. That is, the sum of environmental impact analyses for projects in a region does not contribute to an understanding of that region as an interacting biophysical system.
Planning and management of water resources require that the workings of the biophysical cultural region, in this case the watershed, be understood so that prediction can be made of the consequences of certain contemplated acts. Moreover, in accordance with the Clean Water Act, these contemplated actions must be assessed for their effect upon water quality and quantity. The primary requirement then is to understand the operation of the system, be it a major watershed or a physiographic region. This suggests that modeling of watersheds and regions must be undertaken to provide the necessary predictive capability. Such inventories and models then permit both positive planning towards established social ends and the avoidance of social costs. The avoidance of negative impacts, toward which the present procedures are oriented, would be complemented by a capability of inducing positive acts, i.e., development of propitious areas and positive steps to ameliorate water quality. It would require only an interpretation of Sections 201 and 208 to undertake such ecological planning studies involving both inventories of natural phenomena and processes, and the creation of predictive models, insofar as science can now provide these.
I would recommend a major improvement to existing planning method, that is, the uniform employment of a planning method having the attributes of being overt, explicit, and replicatable. In other words, the data employed should be available to the public, the weights or values attributed to factors should also be overt and explicit so that the planning process is replicatable. Such a planning method would then be employed east and west, in every watershed and physiographic region.
Yet another improvement to present processes would involve devolution of water planning and management to the smallest units of government, recognizing differences in perception of the environment, values attributed to its several aspects, and alternate forms of remedy. This suggests that municipalities produce ordinances regulating land use and management, based upon the health and welfare provisions of the U.S. Constitution, the provisions of the National Environmental Policy Act, and appropriate state legislation specifically related to place and people. This requires the same understanding and capability necessary for planning watersheds and regions. However, if Section 208 induced ecological inventories and models as central to ecological planning, these data could be made available to municipalities. If performance specifications can be written, local ordinances can be formulated specific to the municipality, the natural system it includes, and the needs, desires, expectations, and aversions of its population.
This returns us to the primary concerns of this book (Toubier and Pierson 1976): the biological control of water pollution. The insistence upon performance specifications is tolerant to alternatives. There are many simple and effective methods to be considered and employed. These range from preservation of existing resources to a wide range of remedial techniques. What distinguishes those techniques presented in this collection is their ecological-limnological basis. They recognize that organisms are essential for potable water in nature or by human management, that water planning and management must be ecological-limnological, that sanitary engineering cannot achieve the objectives of the nation.
The world of water and sewage treatment is dominated by the subject of waste. Yet this is a misnomer. There are no wastes, only matter in various states. Most of that named waste is better described as nutrients. We need to learn how to complete the cycle, notably the return stroke involving the decomposing microorganisms in water and soil and their symbionts. And it is these microorganisms who are the heroes of this book, the creatures who have been thoughtlessly engaged in water treatment for aeons and who continue to be indispensable. The EPA disclosure that carcinogens are produced as a product of chlorination is only the death knell to the concept of killing for water potability. It must be replaced by a biological alternative that utilizes the essential microorganisms. The future of water treatment lies in ecology-limnology, as this book so clearly affirms. Let us formally abandon our right to defecate in our neighbor's drinking water. Let us embark upon a national toilet-training program, and above all, let us make a formal commitment to limnology and ecology as the basis for this endeavor.
# References
Toubier, Joachim, and Robert Pierson, eds. 1976. Biological Control of Water Pollution. Philadelphia: University of Pennsylvania.
Wallace, McHarg, Roberts, and Todd. 1973. Towards a Comprehensive Plan for Environmental Quality. Washington, D.C.: American Institute of Planners for the U.S. Environmental Protection Agency.
# 19
# A Case Study in Ecological Planning: The Woodlands, Texas (1979)
with Arthur H. Johnson and Jonathan Berger
A distinguishing characteristic of the Penn Department of Landscape Architecture and Regional Planning under McHarg's leadership was its interdisciplinary nature. Landscape architects and planners collaborated with ecologists, geologists, soil scientists, and anthropologists. Jon Berger, a planner, and Art Johnson, a soil scientist, were among the most enthusiastic of these collaborators. They were central participants in an introductory interdisciplinary studio that exposed graduate students to ecological planning. Suitability analysis was an important component of this studio.
This paper is one effort to provide a clear explanation of suitability analysis, as it was conceived by McHarg and refined by him and his Penn colleagues, including Johnson and Berger. It was published in Planning the Uses and Management of Land, a comprehensive text on land planning and management, by the American Society of Agronomy, the Crop Science Society of America, and the Soil Science Society of America.
# Introduction
In reviewing the publications and reports in the regional and landscape planning literature and project reports from the profession, one finds a lack of uniformity in methodology, with ad hoc procedures "suited to the particular problem" a common approach. A method is presented in this chapter for determining the inherent suitability of a landscape for assimilating human activities and their artifacts. The approach is suggested in the writings of McHarg (1969) and exemplified by Juneja (1974) and has been applied professionally to a wide array of sites and locations. The method of landscape analysis described here is one part of a more comprehensive planning process which includes the social, legal, and economic factors which must be melded into a comprehensive plan that responds to the needs, desires, and perceptions of the people for whom the planning is being done. In developing an area, one would like to achieve the best fit between each human activity and that portion of the landscape to which that activity is assigned. As a starting point, a landscape may be thought of as being comprised of elements or components which may be labeled geology, physiography, soils, hydrology, vegetation, wildlife, and climate. Each landscape element may provide opportunities for certain land uses, and likewise, there may be constraints to each kind of desired land use imposed by components of the landscape. Areas which are most suitable for a specific use will have the greatest number of opportunities provided by the landscape and the least number of, or least severe, constraints imposed by the landscape on that particular use.
By using the approach of combining analyses of opportunities and constraints, the environmental impacts of the planned uses can be minimized, and the energy required to implement and maintain the proposed uses and artifacts can likewise be minimized. For example, areas where the water table is near the surface frequently or for extended periods provide an obvious constraint to subdivision housing in unsewered areas. This property of the landscape lowers the inherent suitability of such areas for that use. The situation can be ameliorated by the addition of sewers or by other engineering solutions, but costs, either economic or ecological, will be incurred, and additional energy will be required for installation and maintenance. This same area may provide little constraint to a golf course or park. Areas which are on the lee side of vegetative or physiographic barriers to winter winds provide a slight advantage for housing as energy costs for winter heating will be somewhat reduced. This same property of a site produces little opportunity for a park or golf course if the use is confined to the warm seasons. Thus the pattern of land uses assigned to the landscape could be controlled to a large degree by the characteristics and properties of the landscape. To this end, a careful analysis of the physiographs, geology, soils, hydrology, plants, animals, and climate-microclimate of an area should be carried out and the implications for specified land uses determined by trained scientists.
The approach outlined here is designed to be flexible. It has been applied to areas ranging in size from a few hectares to a few hundred square kilometers and to urban, suburban, and rural areas.
There are also mechanisms to incorporate new data which may be generated after an initial plan has been formulated. Although flexible, the method is designed to be as objective as possible. The solutions are replicable and the methods of analysis overt and explicit.
Additionally, the method may be used to derive performance requirements (i.e., conditions which must be met by the developers) for the development of areas of less than prime suitability. The impact of any use on the landscape (or the impact of the landscape on the land use) can be mitigated by engineering to have the same result as the same development in the most suitable areas. The areas of prime suitability thus may become a "meter stick" for specifying what additional measures should be taken to minimize impacts on the land use by the landscape, and on the landscape by the land use.
# Outline of the Methods
A flowchart of the planning process, of which this method of landscape analysis is a part, is given in Figure 1. The stream of landscape analysis identified by the box in Figure 1 is the subject of this chapter. This part of the process is dependent upon the input of natural scientists. Clearly, the assembly of scientific data and its interpretation require the perceptions and expertise of soil scientists, geologists, meteorologists, hydrologists, and ecologists. For a plan to be sound, the interpretations for opportunities and constraints must be suitable for the level of information obtained. The judgment and experience of trained scientists is necessary in collecting and interpreting data from the landscape. It should be the planners' charge to combine the natural scientists' perceptions with those of social scientists and engineers to cast these into a comprehensive plan within a sound legal and economic framework.
The first step in this holistic approach to analyzing a particular landscape is to collect information and map the components of the landscape. Some representative inventory maps which have generally been proven useful are listed in table 6. The level of detail of the data will be determined by the available information, time, and available resources which are related to the size of the area.
Figure 1. Flow chart of the ecological planning process. Box indicates the part of the process treated in this chapter.
Table 6. Some Useful Inventory Maps
1. Physiography | Elevation, slope
---|---
2. Geology | Bedrock or subsurface geology, surficial deposits, geologic cross-sections
3. Soils | Series or phases, drainage classes, hydrologic groups, capability group, depth to seasonal high water table, as applicable
4. Hydrology | Depth to water table, aquifer yields, direction of groundwater movement, recharge areas, water quality, surface waters (lakes, streams, wetlands), flood zones, drainage basins
5. Vegetation | Distribution of associations, communities, and habitats as identifiable, areas important as noise buffers, food supplies, for wildlife, nesting areas
6. Wildlife | Identification of species and their habitats and ranges, movement corridors
7. Climate | Macro- and microclimate parameters (temperature, moisture, wind). Ventilation and insulation may be determined in conjunction with physiography
8. Resources | Mineral or other valuable natural resources
An inventory of the landscape in a parcel of, say, ten hectares can be carried out in considerable detail, whereas for a 1,000-km2 site published reports may constitute the bulk of the useable information. In general, as the size of the area to be planned increases, the level of detail decreases and the uniformity of information across the various categories listed in table 6 decreases.
The next step is to determine how the landscape functions as an interacting system of related components. For this purpose, a two-dimensional array like that shown in figure 2 may be used as a guide. Each element in the matrix identifies the possible interaction or relationship of two of the landscape components, and the sum of the bivariate relationships includes all of the major interactions and processes important in the landscape. Knowledge of how the various components of the landscape affect and are affected by one another leads to an understanding of how the whole system works. This should indicate chains of events which might occur because of some proposed land uses. The completeness of understanding will, of course, depend upon the level of information used and the perceptions and abilities of the scientists who contribute to the understanding of the natural system. It is safe to assume that a complete understanding of a landscape and its processes is never achieved—the planner must deal with incomplete information, and care must be taken that the inferences drawn from the data are justifiable given the detail and completeness of the database from which they are made.
The categories shown in figure 2 may be varied to suit the nature of the landscape and level of information used in the analysis. In a small area with detailed information, one can subdivide the categories to a more detailed level, producing a larger matrix. Increasing the detail of information used allows a better understanding of the processes which may be affected by development or other changes in land use.
To understand the links between landscape elements and proposed land uses the set of matrices shown in figure 3 may be useful if there are a number of land uses which need to be considered. Matrix I describes the relationship between land uses and development activities. Matrix II describes the relationship between development activities and the landscape. Matrix III is the same as figure 2. These arrays are one way of organizing the information which is brought to bear on the final land-use plan, helping to make the assimilation of a large amount of information orderly and explicit.
Figure 2. Simple version of a matrix arraying landscape components. The interactions or relationships identified by each element in the matrix are organized into this format which facilitates systematic evaluation of disruptions of natural processes which may accompany specific human activities. Numbers refer to the example given in the text.
Figure 3. Organization of information for assessing the relationship between a set of desired land uses and the impacts they will have on the landscape.
As an example of how this process may be used, consider the development of an area which requires a substantial amount of paving and impermeable surfaces such as roofs and roads with attendant drainage improvements. Consider high-density housing; one of the major impacts is on the hydrologic regime. Portions of the landscape are rendered impermeable and other landscape components will also change in response to the new conditions. By using figure 2, the changes in other parts of the landscape may be evaluated by considering the components which interact with the hydrologic system. Many of the changes considered in this example are defined quantitatively by Leopold (1968). Groundwater recharge will be reduced and the depth to the water table may be increased (no. 1 in figure 2), marshes and seasonally wet areas may dry up, altering vegetation and wildlife (nos. 6, 7, 10), and surface water discharge will be increased. Flood peaks will be higher and bankfull stage will be more frequent, as will floods of a given magnitude (no. 2). As a result of the stream regime changes, stream channels will enlarge to adjust to the new flow regime (no. 11), suspended solids loads will increase at least during the period of channel enlargement, low flows will decrease, water quality will change, and stream communities will necessarily adjust to the new conditions (nos. 8, 9). The vegetation and wildlife will be changed (nos. 3, 4), and the energy balance of the site may be altered as transpiration may be reduced, the soil moisture balance altered, and the microclimate in the area may change considerably as observed in large conurbations.
The information assembled in the planning process amounts to someone's interpretation and synthesis of information compiled and arrayed in map and matrix form to define the landscape components. Given sufficient information of this type at an appropriate level of detail, there is a basis for interpreting the assembled information to understand the opportunities afforded by the landscape for specified activities, and the constraints imposed by the landscape.
Determination of opportunities begins with a specific set of land uses which are desired by the users. Such uses have optimal or prerequisite conditions for their implementation and these must be defined, i.e., swimming areas require good water quality, appropriate bottom material and topography, and accessibility. Houses optimally need stable material beneath, well-drained soils for on-site sewage disposal, gentle to moderate slopes, and perhaps a good view and protection from winter winds. For each desired land use, the geology, soils, vegetation hydrology, and/or other inventory maps are interpreted for the opportunities they afford, producing a set of opportunity maps which show the best areas for each land use individually based on the landscape components which afford opportunity. For each land use, the individual opportunity maps derived from each of the pertinent landscape components are combined by overlay techniques to produce a composite opportunity map which shows the opportunities afforded by the whole landscape for each desired land use.
In most cases the greater the number of concurrences of opportunities found in a particular environment, the higher the capability of that environment for the defined use. The trade-off between the environments of higher and lower capability will be increased capital costs of design and construction as well as increased energy costs for construction and maintenance if performance requirements are met. Users can decide between the possible trade-offs. Using the method outlined in figure 3, the consultant scientists can demonstrate the attendant environmental costs and benefits of any desired scheme.
Constraints, defined here as adverse impacts of the land upon the land use and adverse impacts of the land use upon the land, are best expressed using the vocabulary of the National Environmental Policy Act and the health and welfare provisions of the states' and federal constitutions (table 7). Some landforms because of the natural processes are "inherently hazardous to life and health." Examples would be flood-prone areas, areas subject to landslides, areas subject to collapse, and areas of fire-prone vegetation.
Table 7. Types of Constraints
Legally Defined Constraints | Rules of Combination with Other Uses
---|---
Inherently hazardous to life and health | Preempts nearly all development
Hazardous to life and health through specific human action | Allows some land uses but not others
Unique, scarce, or rare vulnerable resource | Requires regulation through performance requirements
Other natural factors present "hazards to life and health through specific human action." Examples are the pollution of ground and surface waters from septic tanks in soils with a seasonally high water table, or the pollution of domestic groundwater supplies through construction or waste disposal on an aquifer recharge area. Certain landforms with associated vegetation and land use can be classified as "unique, scarce, or rare." Alteration of these areas through development would mean the loss to society of irreplaceable features. Social scientists (historians, ethnographers, archaeologists, folklorists, and art historians) and natural scientists value such areas. Finally, particular landforms may be "vulnerable resources" which need regulation to "avoid social costs." Depending on the environment under study, these would include prime agricultural soils, high-quality gravel deposits, and scenic features, among others.
Planners and their consultant scientists can evaluate every inventory map and determine from the categories of data the relevant set of constraints. These constraints are mapped. Unlike opportunities, the concurrence of numerous constraints may not be as significant as the existence of one constraint which represents a "hazard to life and health."
The next step in the procedure is a synthesis of opportunities and constraints for a selected land use to produce a suitability map which identifies a gradient of suitabilities for that prospective use. The areas with the greatest number of opportunities and least constraints are the most suitable for the specified land use. The method of combining the opportunities and constraints and ranking the suitability may be arbitrary but is explicit if an array is used to show how decisions of suitability were made. The matrix in figure 4 shows the determination of most suitable land and land of secondary suitability for housing with septic tanks. The example is oversimplified, but the method has been applied successfully to complex landscapes. The reliability and accuracy of the map overlay techniques this method employs are discussed by MacDougall (1975).
The suitability maps for the land uses that the landscape must accommodate are then assessed. Where there are areas which are of primary suitability for only a single use, that use should be allocated to the suitable areas if the other relevant social, economic, and legal factors are favorable. In many instances, multiple suitabilities will arise. That is, some areas will be highly suitable for more than one use. Clearly, prime agricultural lands will be suitable for housing, recreation, and other uses. In these cases, land uses are assigned based on the needs and desires of the users which can be determined by surveys and interviews (Berger 1978) or reflected in local officials or spokespersons for the users. Such allocations are also subject to legal and economic considerations which should also be incorporated into a land-use plan.
Figure 4. Overlay method for determining suitability of a specific land use by combination of opportunities and constraints.
# Application
A simplified example is given below which is a summary of a portion of the plan for the new city called The Woodlands. (Wallace, McHarg, Roberts, and Todd 1974), now being developed just north of Houston, Texas. A site map is shown in figure 5. The site presented many problems for such a development. It was entirely forest—a pleasant place to live, but a difficult environment to build in. It is extremely flat with few slopes greater than five percent. As a result of the topographic and rainfall characteristics, nearly a third of the site is in the 100-year flood plain of Panther, Bear, and Spring Creeks. Drainage of storm runoff was poor. Many depressions exist on the flat terrain which is dominated by impermeable soils, and standing water was common. The determination of housing sites and housing densities in The Woodlands is used as an example of the method of landscape analysis outlined above.
Figure 5. Regional location and site map of Woodlands new community.
# Nature of the Site
The Woodlands is located in the Gulf coastal plain and is underlain by unconsolidated formations comprised of Quaternary and Tertiary age gravels, sand, silt, and clay, in various combinations and proportions. The formations strike northeast, roughly parallel to the coast, and dip southeast at 1 to 2 m/km. Several of the units are good aquifers and are sources of high-quality water. The bearing strength of the geologic units underlying the site is adequate for most development purposes. There are subsidence problems in the Houston area, but subsidence should not affect The Woodlands; groundwater withdrawals and recharge have been carefully determined in the planning for the area, as described in more detail below.
The hydrologic regime was an extremely important consideration in designing the plan for The Woodlands community. There were flooding and storm-drainage problems to be dealt with and a water supply to be developed.
Recharge of water was of primary importance to diminish the risk of subsidence on the site as well as down dip in Houston, which pumps from the same aquifers. Additionally, groundwater was planned as a means of augmenting baseflow in the creeks to enhance the amenity value of artificial lakes to be constructed on the site.
Spring Creek and lower Panther Creek are the only perennial streams within the site. The others are intermittent, flowing during periods of storm runoff. During low flow periods, discharge is low since baseflow is limited, but storm-periods produce large peak flows due to the heavy precipitation events and predominance of impermeable soils.
The soils on the site are mostly paleudults. Surface horizons are generally sandy or loamy with well-developed argillic horizons below. The Woodlands soils were grouped according to permeability and storage capacity in inches of water. The soil will store beyond field capacity above the slowly permeable layers or the seasonal high water table. Figure 6 shows the profiles rated according to the permeability of the horizons, and table 8 summarizes the pertinent properties.
Table 8. Some Properties of the Soil Series Present on The Woodlands Sites
The site is a mixed woodland dominated by loblolly pine. In mature stands, the pines are associated with oaks, sweet gum, hickories, tupelo, magnolia, and sycamore. The woodlands provide an amenity for development as well as limiting runoff and erosion. Additionally, the forest provides habitats for wildlife. Eight major vegetation associations were recognized: (1) short-leaf pine—hardwood; (2) loblolly pine—hardwood; (3) loblolly pine—oak—gum; (4) pine—oak; (5) mixed mesic woodland; (6) pine—hardwood; (7) flood-plain vegetation; and (8) wet-weather pond. Much of the forest has been logged at one time or another.
There are a multitude of types of wildlife present on the site. Those types which are abundant include songbirds, rabbits, raccoon, squirrels, opossum, armadillo, white-tailed deer, and wild turkey. The persistence of most of these types can be promoted by careful management and by maintaining suitable habitats and territories and movement corridors which are large enough to suit the species. Flood-plain areas in The Woodlands provide a diversity of habitats suitable for several desirable forms of wildlife, so that type of vegetation association has a high value for wildlife preservation. Forest edge conditions provide a diversity of habitats and also encourage a diversity of wildlife. These edge conditions occur around wet-weather ponds, which increases their value for wildlife protection. The ponds also serve as sediment traps and temporary shortage basins during storm periods, amounting to a significant value for the ponds.
Figure 6. Soil profiles interpreted for permeability and depth to seasonal high water table. Map unit designations refer to Table 8.
The climate of the area is subtropical with warm, moist summers and mild winters. The climate factors were used in site planning but were not of overriding importance in determining suitability for development and so will not be discussed here.
# Planning for Development
An overall plan for locating best areas for development, including high-and low-density residential, commercial, recreational, municipal, industrial, and open-space land uses was derived from the inventory of the landscape. Economic consultants produced a housing market analysis for the Houston metropolitan region which showed seven feasible housing types for The Woodlands New Town. Engineers and landscape architects described the attributes of each development type in terms of space occupied by buildings, space covered by other impervious surfaces and vegetation cleared (table 9).
Table 9. Development Types with Clearance and Coverage Requirements Determined by a Housing Market Analysis
Housing Type* | Clearing Required (%) | Coverage by Impermeable Surfaces (%)
---|---|---
Single family, 2.5 DU/ha (1 DU/acre) | 37 | 24
Single family, 6.9 DU/ha (2.75 DU/acre) | 70 | 36
Single family, 10 DU/ha (4.0 DU/acre) | 50 | 43
Patio houses, 15 DU/ha (6.0 DU/acre) | 90 | 51
Garden apartments, 37.5 DU/ha (15 DU/acre) | 85 | 45
Townhouses, 25 DU/ha (10 DU/acre) | 95 | 55
Elevator apartments, 100 DU/ha (40 DU/acre) | 70 | 50
*DU = dwelling unit.
# Relationship of Development Activities to Landscape Processes
The desired land uses and their attendant development activities would affect soils, vegetation, groundwater levels, stream flow, and stream-channel form. Modification of these landscape elements would affect the following processes: the balance between infiltration and overland flow, channel deposition and erosion, storage and movement of groundwater, and the regenerative capacity of the forest and wildlife communities. The effects on the hydrology of the site are essentially the same as identified in figure 2.
Given these forecasted impacts of urbanization, several interest groups wished to mitigate potential adverse environmental impacts. The regional water management commission wanted to maintain the recharge of the city of Houston's groundwater supply. The new town developers wanted to maintain a healthy forest as the prime marketing element of the new town. The U.S. Department of Housing and Urban Development had environmental guidelines for the processing of guaranteed loans. In response to these different interests the five guidelines listed below for the plan and design of the new town were established:
1. Minimize disruption of the hydrologic regime by creation of a natural drainage system which allowed removal of the low-frequency event runoff and recharged as much precipitation as possible from high- and low-frequency storms to maintain groundwater reserves
2. Preservation of the woodland environment
3. Preservation of vegetation providing wildlife habitats and movement corridors
4. Minimization of development costs
5. Avoidance of hazards to life and health
# Requirements to Minimize Disruption of the Hydrologic Regime
Since The Woodlands site is a flat landscape with large areas of impermeable soils and streams of low gradient, conventional means of storm-water management called for site drainage through a large and expensive network of concrete drainage channels. These would decrease recharge to the groundwater reservoir, and call for the removal of vegetation. To avoid these environmentally and economically expensive problems a "natural" drainage system was devised.
Calculations of cleared area and impervious area for typical The Woodlands residential clusters indicated the magnitude of development impacts on surface runoff, soil storage capacity, and forest cover. The design aim was to promote infiltration of high-frequency, low-volume storm water (up to 25 mm of precipitation in six hours) to reduce the period of standing water and increase movement into the groundwater reservoir. For purposes of this design storm (25 mm in six hours), the soils were assumed to be at field capacity and flood plains were assumed to be left in their naturally forested state.
As a design tool to promote percolation, the most abundant soils were grouped according to their capacity to accommodate water from the high-frequency storm. Available storage capacity was calculated from the depth of the permeable soil layer, the depth to the seasonal high-water table, and the air-filled pore space at field capacity. The proportion of each soil map unit which needed to be left undisturbed to absorb runoff from the cleared portion was calculated (table 10).
Table 10. Tolerance of Soils to Coverage by Impermeable Surfaces Based on Ability to Store Storm Runoff from a 6-hour, 25mm Storm
With the on-site recharge capacity of the soils known, the capability of any soil environment to handle any development type could be determined. In some cases higher densities on lower recharge capacity soils were possible if adjacent land had a moderate to high storage capacity to handle the storm runoff not recharged by the lower capacity soil. Housing densities could be increased in this case, since all of an area of high recharge capacity soil could be used as a sump for the runoff from adjacent areas developed on soils not suitable for storage of storm runoff.
Given the need to "borrow" recharge capacity from adjacent soils, the juxtaposition of soil types on the landscape became important in addition to the on-site recharge capability of a soil in determining housing suitability. Soil patterns based on drainage relationships were identified for their suitability for the different development types. figure 7 shows one example. For each drainage relationship, management guidelines, housing suitabilities, and siting considerations were specified.
Figure 7. Management guidelines, housing suitability, and siting considerations for group D soils (no ability to store 6 hour 25mm storm).
# Requirements to Preserve the Woodland Environment
Different vegetation types gave rise to different levels of clearing. Based on their desirability to the projected residents, their tolerance to disturbance, the soils on which they grow, and their regenerative requirements, the forest types were rated somewhat subjectively on a scale of allowable clearing. For example, large pure hardwood or nearly pure hardwood stands are relatively scarce in the region, attractive to the residents, intolerant of soil compaction and change in groundwater levels, better landscape shade trees, and slower to regenerate than pure pine stands which abound in the area. Clearing of hardwood stands was considered less desirable than clearing of pine stands. figure 8 is a summary of the recommended clearance percentages based on the amenity value of the various forest types.
The gradient of tolerance from pure hardwood to pure pine is a gradient of opportunity and constraint for the development types. The more tolerant the vegetation to clearing the greater the opportunity for higher density housing. The lower the tolerance to clearing the greater the opportunity for lower housing densities.
# Movement Corridors
Wildlife needs cover, food, and water. The design objective was to provide for wildlife needs so that a maximum number of species present on the site could remain after development. Large areas offering diverse vegetation and water would make suitable wildlife refuges. These refuges would be connected by corridors of vegetation. Vegetation in refuges and corridors would be preempted from development. The corridors were provided by the design of a natural drainage system as described below.
# Requirements to Avoid Hazards to Life and Health and Cost Savings for Construction
The ecological inventory of The Woodlands showed that flood hazard was the only natural hazard to life and health. Development in the area along major streams inundated by the projected 100-year flood (under development conditions) was preempted. In addition to the use of some soils as sinks for high-frequency storm drainage, a system of naturally occurring swales and stream corridors supplemented with man-made swales was designed to carry storm runoff from the low-frequency events. Development in the 25-year flood zone of the smaller drainage ways was also preempted. The swales and stream corridors were left in a vegetated condition which helped preserve the woodland environment and maintain corridors for wildlife movement. Wherever possible, the drainage system was routed over permeable soils to further increase the infiltration of storm runoff. Coupling this with the siting plan for infiltration of storm runoff, the need for conventional storm sewers was eliminated. This saved an incredible $14 million dollars in development costs and raised land values due to the elimination of unsightly concrete ditches, in addition to minimizing the disruption of runoff-recharge relationships, helping to preserve the woodland environment, and helping to provide for the maintenance of wildlife on the site.
Figure 8. Suggested clearance percentages based on vegetation types.
# Opportunities and Constraints
The constraints to housing were the flood zones and restrictions placed on clearance by the vegetation analysis. This meant that areas prone to flooding, wetland areas, and hardwood areas were considered restrictions to development. Opportunities for various housing types were determined from the allowed clearance of vegetation and the impermeable surface permitted by the soil groups so as to allow infiltration of the high-frequency storm. figure 9 shows the distribution of soil groups and the vegetation types mapped with constraint factors for one portion of the new town site. The different soil groups of figure 9 are most suited to different types of housing, since the storage capacity of the soils for the 6-hour 25-mm event must not be rendered inadequate by too much impermeable surface. The suitabilities are identified in table 11. For those areas that are not restricted by the defined constraints, a certain percentage of clearance was allowed, based on the vegetation present, a maximum amount of coverage by impermeable surfaces was set by the water-holding capacity of soils, and a certain type of housing with its characteristic density could be accommodated. figure 10 shows the clearance, coverage, and allowed density for the area in figure 9.
Table 11. Housing Suitabilities Related to Runoff Holding Capacity of Soil Groups A, B, C, and D
Map Code | Housing Suitability
---|---
3A1 | All types and densities suitable
3A2 | A or B soils used for recharge of runoff from C or D soils should not be developed or cleared. C and D soils for which A or B soils have been provided to accomplish recharge have no development restrictions
3A3 | Low-density housing
3B1 | Suited to high-density housing with runoff carried by natural drainage system
3B2 | All types suitable, but drainage will be required
Figure 9. Mapped opportunities (as determined by soils and vegetation). Suitability for the housing types is given in Table 11.
Figure 10. Clearance coverage and housing densities determined for an area.
# Summary
The development will surely have impacts on the landscape and the natural processes occurring within it, but the development scheme allows for minimum disruption of the hydrologic cycle—recharge is maximized, exacerbation of flooding by development minimized, the groundwater and baseflow to streams augmented vis-à-vis conventional drainage, and erosion hazard reduced due to vegetated drainage ways. Desirable wildlife and vegetation are also preserved. Planning for The Woodlands encompassed far more. Site planning and phasing were considered in detail, as were the location of roads and industrial, commercial, and recreational areas. Engineering and economic considerations were incorporated into the overall development plan. Wildlife and the other components of the landscape were treated in much more detail than described here. The scope of the example is limited, as the inclusion of larger areas and more land uses greatly increases the complexity, and would require a great deal more space to describe.
An understanding of the features of a landscape, i.e., the soil, geology, hydrology, vegetation, and wildlife, as well as how they interact or are linked by natural processes, allows some understanding of the effects specified types of development will have on the whole ecosystem. Certain elements of the landscape may therefore become determinants of the pattern of planned land uses so as to minimize the adverse affects on the landscape. In The Woodlands example vegetation, soils, and the nature of the hydrologic conditions of the site were the most important determinants in the siting of residences. In other areas, certain other natural features may be more important determinants of planned land-use patterns. For instance, areas underlain by cavernous limestone bedrock, fault zones, or areas of vertisols may preclude building, or the ameliorative design strategies necessary to protect lives, property, or natural resources will be costly. Ecological planning as it is described here is sound in practice as well as in concept. In the case of The Woodlands, this type of planning saved the development corporation money in construction costs.
Presently [1979] there are 2,500 residents in The Woodlands, and the first phase of development is underway. The ecological plan was submitted to HUD in 1972 and led to a $50 million loan guarantee, the maximum under Title VII provisions.
# References
Berger, J. 1978. "Towards an Applied Human Ecology for Landscape Architecture and Regional Planning." Human Ecology 6 (2):179–99.
Juneja, N. 1974. Medford—Performance Requirements for the Maintenance of Social Values Represented by the Natural Environment of Medford Township, New Jersey. Philadelphia: Center for Ecological Research in Planning and Design, University of Pennsylvania.
Leopold, L. B. 1968. Hydrology for Urban Land Planning—A Guidebook on the Hydrologic Effects of Urban Land Use. U.S. Geological Survey Circular 554. Washington, D.C.
MacDougall, E. B. 1975. "The Accuracy of Map Overlays." Landscape Planning 2 (1):23–31.
McHarg, I. L. 1969. Design with Nature. Garden City, N.Y.: Doubleday, Natural History Press. 1992. Second edition, New York: Wiley.
Wallace, McHarg, Roberts, and Todd. 1974. Project Reports for the Woodlands New Community: (I) An Ecological Inventory, (II) An Ecological Plan, (III) Phase 1: Land Planning and Design Principles, (IV) Guidelines for Site Planning. Philadelphia: Wallace, McHarg, Roberts, and Todd.
# Part V
# Linking Knowledge to Action
An important aspect of Ian McHarg's career is his involvement in numerous actual community design and regional planning projects. Throughout his career, he has been both a theorist and a practitioner. He straddled the academic and the professional worlds, blurring the distinctions between teaching, research, and service. Much innovation is captured in the professional reports produced by Wallace-McHarg Associates; Wallace, McHarg, Roberts, and Todd; and the Penn Department of Landscape Architecture and Regional Planning. These reports contain detailed lists of the project participants. McHarg always sought to give credit to all involved in each undertaking. Such acknowledgments were rare in the 1960s and, unfortunately, remain too infrequent. They reveal a multidisciplinary team approach to planning and design as opposed to a plan or a design produced by a single individual. An emphasis on process rather than a prescribed end-state plan or design is also revealed.
Among McHarg's many articles and papers, there are a few detailed accounts of his projects (I believe too few). Although even in his most theoretical papers actual projects are discussed; there are few detailed, reflective analyses of his plans. Included here are two descriptions of plans, a plan, plus one call for a more comprehensive planning framework. These four works were written in collaboration with professional colleagues.
The first description is McHarg's well-known plan for the Green Spring and Worthington Valleys in Maryland, written with his partner David Wallace. The planning study was not undertaken for a governmental jurisdiction; rather it was privately financed by local residents. In this 1964 study Wallace and McHarg addressed many of the problems resulting from suburbanization that are still prevalent and are likely to continue: How can prime farmlands be retained? Where are the best locations for new developments? How can growth be managed? How can environmentally sensitive lands be protected? How can the health and safety—the sustainability—of the area be ensured for current and future residents?
The Plan for the Valleys was, and remains, a unique combination of graphic presentation and economic analysis. Drawings were used to illustrate the deleterious consequences of uncontrolled growth as well as the prospects for imaginative planned growth. The economic analysis was used to illustrate that planned growth was as profitable as (and more fiscally responsible than) uncontrolled growth.
Central to the plan was McHarg's notion of "physiographic determinism," that is, the prospect that "development should respond to the operation of natural processes." This is, of course, McHarg's central message.
The plan was innovative in numerous ways. The attorney-planner Ann Strong suggested that a "real estate syndicate" purchase and transfer development rights approach be explored to manage growth. This idea was a precursor to both current land trusts as well as purchase and transfer of development rights programs. Today, Maryland leads the nation in farmland protected as a result of state and county purchase of development rights programs. Various zoning suggestions also pioneered subsequent performance-based, exclusive farm-use, large-lot, and environmentally sensitive lands zoning.
The Plan for the Valleys was a turning point for McHarg. He discovered a voice for vivid description of the environmental issues we face. In his PBS film, Multiply and Subdue the Earth, for example, he proclaimed, "if something isn't done—the future metaphysical symbol for the valleys will be a roll of toilet paper."
The Plan for the Valleys is thoroughly documented in Design with Nature. The Vermont and The Woodlands plans followed the publication of McHarg's landmark book. The 1972 Vermont study provides a comprehensive discussion of the ecological approach to planning. The study was conducted for Wilmington and Dover, Vermont; the Windham Regional Planning and Development Commission; and the Vermont State Planning Office. The study was undertaken at the time when Vermont was launching its innovative growth-management requirements.
The period following the first Earth Day and the publication of Design with Nature has been called "the environmental decade," when "a quiet revolution" in land-use regulation occurred (see Bosselman and Callies 1971). Land-use regulation is a state right and states generally had passed this authority on to local governments to protect the public's health, safety, welfare, and morals. During the "quiet revolution" states started to pull back some of their powers from local government and to require that specific environmental elements be addressed in local plans and ordinances. Hawaii was the first state to exert its authority, enacting statewide zoning. Even more far-reaching and influential state initiatives were adopted in Oregon and Vermont.
Ian McHarg influenced the growth control efforts in Vermont and produced one of the first plans in response to the state law. In 1970 he addressed the joint houses of Vermont legislature at the governor's invitation. McHarg entered the Vermont debate about state land-use initiatives and urged legislators to adopt what would become Act 250. Several local leaders from Wilmington and Dover heard McHarg address the Vermont legislature. "They liked what they heard and prevailed upon McHarg to undertake a pilot planning study" (Rushman 1997, p. 3).
The study was conducted with limited funding. As a result, all data were generated by the townspeople, then compiled, analyzed, and synthesized by regional planning graduate students at Penn. The Wilmington and Dover study, thus, is an example of citizen participation in what is normally considered a technical step in the planning process. Environment attorneys Victor Yannacone and Arthur Palmer provided legal assistance. McHarg's firm—Wallace, McHarg, Roberts, and Todd—was involved in this "philanthropic" effort. McHarg directed the study and was assisted by three of his most capable lieutenants from the firm: Michael Clarke, David Hamme, and Narendra Juneja.
Michael Rushman, who was a young planner in the northern Catskills during the early 1970s, has made a critical analysis of the Wilmington and Dover ecological planning study twenty-five years after its publication. Both a planner and a lawyer, Rushman asked, "Has Vermont stayed the course with ecologically-based planning?" He notes that Act 250 contains both a planning component and a regulatory component (Rushman 1997). The plan by McHarg and his colleagues addressed both, but emphasized the planning component. Rushman found that, since its promising beginnings, Vermont has favored the regulatory component over the planning component and, in the end, although developers, towns, and the attorneys of both must go through many procedural hoops, there is little substantive change on the ground. In fact, Rushman finds little difference with neighboring New Hampshire, well known for its local control and antipathy to government regulations.
Michael Rushman also studied the impact of the study on Wilmington and Dover themselves. The study by McHarg and his colleagues had some influence on the town plan of Wilmington but none on Dover. The overall local influence, however, has been minimal. Still Rushman indicates several recent planning initiatives in Vermont where the McHarg influence lingers and is perhaps resurging. These efforts involve the watershed-wide cooperative management efforts in the Connecticut River Valley and bioregionally based planning of the Northern Forest (Rushman 1997). In fact, Rushman holds more hope for ecologically based planning in these efforts "rather than with traditional jurisdiction-based comprehensive planning."
The Woodlands provides a more successful outcome, at least in the short term. The Woodlands new community was introduced in chapter 19 that summarized the suitability analysis. In this section McHarg and Jonathan Sutton provide a more holistic account. Like the Plan for the Valleys and the Wilmington and Dover study, the planning for The Woodlands was undertaken by McHarg's consulting company. Wallace, McHarg, Roberts, and Todd developed the ecological plan for The Woodlands near Houston, Texas, for the Mitchell Energy and Development Corporation in the early 1970s. The Woodlands was one of fifteen Title VII U.S. Department of Housing and Urban Development (HUD) sponsored new towns and the only one not to go bankrupt (Steiner 1981).
The Woodlands and the other HUD Title VII new communities were planned at a climax of American interest in new towns. During the 1960s Reston, Virginia, and Columbia, Maryland, both near Washington, D.C., created considerable interest among elected officials, developers, planners, landscape architects, and architects. New towns were viewed as a solution to America's urban woes, which were brought in sharp focus during the Civil Rights movement of the 1960s. In contrast to the suburbanization that occurred in the United States after World War II, a coordinated program of urban reconstruction and of new town development was undertaken in Europe with considerable success.
The HUD Title VII new communities were also planned following the passage of the National Environmental Policy Act (NEPA). As a result, the Title VII new towns were to address both social and environmental concerns. Private developers, mostly large homebuilders with little social or environmental planning expertise, received guaranteed loans of $50 million to plan and build these new towns. Eventually fifteen HUD Title VII new communities were initiated, including The Woodlands. George Mitchell, the developer of The Woodlands, was a Texas oil baron with deeper financial resources than the other HUD Title VII developers. Mitchell turned to McHarg because of Wallace, McHarg, Roberts, and Todd's environmental planning acumen. After all, who would better address NEPA requirements than the man who had helped to create its intellectual base and methodological framework?
McHarg and his colleagues with other consultants and Mitchell's staff planned the new community around the drainage and, as the name implies, the woodlands of the site. McHarg and Sutton note in their 1975 Landscape Architecture magazine article that the analysis of ecological processes "determined the form of The Woodlands." Computer technology was used in this analysis. In fact, the computerized soil and vegetation surveys conducted in the first planning phase represent one of the first actual applications of geographic information systems technology to a built project.
The Wallace, McHarg, Roberts, and Todd effort linked regional environmental information to detailed ecological design. The Woodlands is an important link in the American tradition of new town planning that began with the Spanish Laws of the Indies in 1573 and continued through the planning of Williamsburg, Philadelphia, Savannah, and Washington, D.C. The Woodlands can be viewed as part of the organic approach of that tradition that began with Olmsted's Riverside, Illinois, and was carried on by John Nolen, Henry Wright, Clarence Stein, and the Greenbelt New Towns of the New Deal. The neotraditional town planners, or new urbanists, of the 1990s (i.e., Elizabeth Plater-Zyberk, Andres Duany, and Peter Calthorpe) have rekindled this approach.
McHarg has undertaken both actual projects, like The Woodlands, as well as conceptual projects. After the publication of Design with Nature, McHarg focused largely on three overlapping interwoven idea projects: the understanding and application of human ecology, the application of computers to ecological planning, and very large-scale ecological inventories. He proposed these inventories, initially, at the national and, eventually, at the global scales. His premise was that ecological planning depended on a sophisticated database. If such comprehensive inventories could be collected, stored, updated, and made accessible, then accurate and fair environmental assessments of future activities could be undertaken.
McHarg's first opportunity to formally propose a national inventory came in 1972 when U.S. Environmental Protection Agency (EPA) administrator Russell Train asked him to propose a comprehensive plan for environmental quality. The EPA gave a grant to the American Institute of Planners (AIP) for the study. Gerald Mylroie then contracted McHarg through Wallace, McHarg, Roberts, and Todd. Mylroie also formed an AIP advisory group of prominent planners for the project, including Robert Einsweiler as chair, Alan Kreditor, James Park, Robert Paternoster, E. Jack Schoop, Paul Sedway, and Harold Wise. Einsweiler had previously helped to sponsor McHarg's ecological study for the Twin Cities region in Minnesota.
McHarg's resulting report Towards a Comprehensive Plan for Environmental Quality was quite influential within the EPA and helped to define the early development of environmental impact assessments. However, McHarg's central recommendation to establish a national ecological inventory was not followed by the EPA. Through the years the EPA did evolve a computer-based inventory that was not as comprehensive as McHarg's proposal.
The EPA's system is known as the Environmental Monitoring and Assessment Program (EMAP). McHarg had a second opportunity to propose his grander vision in 1990 when EPA administrator William Reilly sought McHarg's advice on EMAP. McHarg responded with his colleagues John Radke, Jon Berger, and Kathleen Wallace with a proposal for a National Ecological Inventory. Radke is a geographer who was running Penn's geographic information system (GIS) lab at the time and is currently on the University of California-Berkeley faculty. Berger and Wallace were both graduates of McHarg's regional planning program. Jon Berger also holds a doctoral degree in city planning from Penn. The central chapter of their 1992 report "A Strategy for a National Ecological Inventory" is included here.
The timing of the release of their report in 1992 doomed its implementation. Their champion, William Reilly, left office with the defeat of George Bush. The recommended National Ecological Inventory was not pursued by the EPA. However, President Clinton's Secretary of Interior Bruce Babbitt proposed a National Biological Survey. McHarg continues to consult with the Department of the Interior about the design of the National Biological Survey. He also remains committed to both smaller-scale comprehensive GIS-BASED inventories, such as for the state of Alaska, and larger global-scale ecological databases. Eventually, national and global inventories will no doubt be conducted and hopefully become an ongoing element of planning and decision making at all scales. As McHarg and his colleagues note, such inventories could well be the most important acts to protect the environment in history, "indispensable for intelligent regulation and management, but moreover, capable of transforming both environmental research and education."
# References
Bosselman, Fred, and David Callies. 1971. The Quiet Revolution in Land Use Controls. Washington, D.C.: U. S. Government Printing Office.
Rushman, Michael J. 1997. McHarg's Ecological Planning Study for Wilmington and Dover, Vermont, "Twenty-five Year's Later: Has Vermont Stayed the Course with Ecologically Based Planning?" (unpublished paper). Tempe: School of Planning and Landscape Architecture, Arizona State University.
Steiner, Frederick. 1981. The Politics of New Community Planning. Athens: Ohio University Press.
# 20
# Plan for the Valleys vs. Spectre of Uncontrolled Growth (1965)
with David A. Wallace
Lewis Mumford summed up the importance of the Plan for the Valleys as follows:
This Plan for the Green Spring and Worthington Valley is brilliantly conceived and thoroughly worked out, down to the detailed demonstration of a better community pattern, based on the cluster instead of the row, for the individual housing development. In both method and outlook, this is the most important contribution to regional planning that has been made since Henry Wright's original 1926 report on the Development of the State of New York. McHarg and Wallace have shown by constructing an appropriate many-sided model what great opportunities for improving the human habitat actually exist once the forces that are now blindly despoiling the landscape and depressing every human value are guided with intelligence and imagination to more valid goals. The Plan for the Valleys in both its method of approach and its human aims should serve as a pattern for all future efforts to conserve life values in a growing community, where uncontrolled or misguided developments may, as in so many parts of California, obliterate the very natural advantages that stimulated this growth. This report should guide not only the farsighted Council that brought it into existence, but also communities all over the United States that are confronted with similar problems who have too often been frustrated and deformed by ill-conceived highway and residential settlement plans, and which can be saved or improved only following the strategy that the Plan for the Valleys has worked out.
Mumford's forward to this Landscape Architecture magazine article nicely introduces this important plan. McHarg discusses it more comprehensively in Design with Nature.
The Plan for the Valleys is a privately financed planning study for 70 square miles of Baltimore County northwest of Baltimore undertaken by Wallace-McHarg Associates of Philadelphia. It examines a larger segment of Baltimore's rural hinterland, now sparsely populated and ripe for development. It is a study which has significance to many problems of metropolitan growth and suburbanization.
The basic originality of this study lies in the client and the problem itself. A large number of landowners of this 45,000-acre area constituted themselves into The Green Spring and Worthington Valley Planning Council. This private, nonprofit organization recognized that current planning powers could neither prevent destruction nor ensure wise development. It assumed the initiative for the preparation of a plan and the responsibility, with public powers, for its realization. This assumption of initiative and responsibility on the part of landowners is both remarkable and thoroughly commendable.
The Plan for the Valleys, the report submitted to the Planning Council, exists in two forms, a technical report of some 80,000 words and an illustrated synopsis designed for submission to each participant landowner (figure 11). This report contains five concepts which, while important to the study area, have a wider relevance as conceptual tools for planning for metropolitan growth.
If planning necessitates the posing of alternatives with the costs and benefits of each, it is necessary to be able to demonstrate the physical and financial consequences of the status quo extended into the future. This is the second element in the study with some claim to wider relevance. The existing population of 17,000 was predicted to increase to between 110,000 and 150,000 in thirty years by William Grigsby, housing market analyst for the study. An Uncontrolled Growth Model was developed by which the physical form of the future could be seen if no new planning powers were introduced. The physical form of this product is described as The Spectre. The profits from land sales under this process were calculated as a base line for comparison with other alternatives. The description of this process reveals the commonality with the rural fringes of all metropolitan areas.
One face of the future is revealed in the Spectre Map (figure 12). Uncontrolled growth, occurring sporadically, spreading without discrimination, will surely obliterate the valleys, inexorably cover the landscape with its smear, irrevocably destroy all that is beautiful or memorable. No matter how well designed each individual subdivision may be, no matter if small parks are interfused with housing, the great landscape will be expunged and remain only as a receding memory.
Yet this melancholy process produces enormous profits in land sales and development. In the study area these will total $33,500,000 in development value by 1980. Consequently, any alternative method of development must aspire to equal this level of development value.
The nature of prospective uncontrolled growth was represented in both graphic and financial terms. It was rejected as a spectre. Given the anticipated population to be accommodated and the development potential of the area, what principles can avert spoliation, ensure enhancement, and equal the development values of uncontrolled growth? The Plan for the Valleys depends exclusively upon physiographic determinism to reveal the optimum pattern of development. This concept is yet another aspect of the report having wide relevance to problems of development. In short, physiographic determinism suggests that development should respond to the operation of natural processes. These processes will vary from region to region. The application of the concept in the study area was circumstantial, but the concept is, however, general in its applicability.
Figure 11. Bird's eye perspective as proposed, the Green Valley and the Worthington Valley, Maryland.
Figure 12. How the valleys will look if current sprawl patterns continue.
As the name suggests, the Valleys contain three major features, the Green Spring, the Worthington, and the Caves Valleys. These broad, sweeping lands have been well farmed for 200 years. In pasture, with dairy and horse farms, these great valleys are beautiful and memorable. They are defined by wooded ridges to both north and south with a major, intervening wooded plateau.
The first test used is the genius loci, the genius of the site. This was revealed as the major valleys and their wooded, confining slopes. The initial conclusion suggested that, while these were vulnerable to development, easily destroyed, and attractive to developers, the maintenance of the pervasive character and beauty of the area required that the valleys be exempted from development. Subsequent physiographic analysis involved topography, subsurface geology, surface and groundwater, the 50-year flood plains, impervious soils, slopes in excess of 25 percent, and forest cover.
Geological examination revealed that the valleys were coterminous with the major aquifer in the metropolitan area, groundwater used for wells and feeding city reservoirs. Development over this aquifer would constitute a hazard to health. Consequently development was prohibited in the valleys.
As surface water and groundwater are interacting, development was prohibited on riparian lands 200 feet from the edge of streams. For reasons of safety, development was prohibited on the 50-year flood plain; and impervious soils were identified and exempted from development requiring septic tanks. These features predominate in the valleys. In the next category, development was prohibited on all 25 percent slopes and on unforested valley walls. The last category, forest cover, was most abundant on plateau, ridges, and valley walls. Here development was limited to one house per three acres of wooded valley walls, one house per acre on wooded plateau. No restrictions were imposed upon the open plateau, and all restrictions as to density were waived for promontory sites suitable for tower apartments.
While the application of these criteria suggested prohibition of development in the valleys and restricted development on valley walls, the plateau exhibited few restraints upon development. Examination showed that development, conforming to physiographic determinism, could absorb all prospective population at densities consonant with market preferences. Without violating the genius loci of the great valleys, this development was located in a hierarchy of communities, a country town, and several villages and hamlets.
When this proposal was examined in terms of the development value produced, it was seen to create an anticipated value of $7 million in excess of the uncontrolled growth model.
Given a projection of population, the next question is how to carry out a development conception that satisfies both amenities and development values. The major innovation in this realm is the proposed real-estate syndicate developed for Wallace-McHarg by Ann Louise Strong. This device suggests that the landowners of the valleys constitute themselves into a syndicate and acquire, among other powers, the development rights of the land for either cash or stock. The syndicate is seen as a private planning and/or development instrument supplementary to public planning processes.
The syndicate can both develop land and preserve open space. It may acquire either development rights, options, first refusals, or title to land as a method of ensuring that development be in accord with the plan. For these rights, it pays either in stock or in cash, in full or by installments. It may also be the agent through which bilateral or multi-lateral agreements between landowners are negotiated in conformity with the plan. It may thereafter sell title or rights or lease land for development according to the plan or may act as developer itself, either singly or in cooperation with other agents. The profits from these transactions will be used to reimburse landowners whose property is not planned for development, and to finance additional purchases of rights or title or for outright development.
The basis for this proposal lies in the expectation that planned growth is likely to develop $7 million more land value by 1980 than uncontrolled growth, that land values will appreciate over time, particularly on the plateau, and, that a real estate syndicate can utilize this increasing value as the basis of its operation.
The final aspect of the Plan for the Valleys which may contain some wider relevance is the concept of an accumulation of powers. A sequence of both private and public actions, including the acquisition of new powers, is shown in a timed sequence.
The public powers necessary to realize the plan extend from the vigorous enforcement of present powers to entirely new controls requiring state legislation. It is a primary objective to obtain public acceptance of the plan in principle, reflected in directives to various agencies of county government. A most important defense of the area lies in the county intention to sewer the plateau but not the valleys. In the absence of sewers, the recent state health regulations prohibiting development on the 50-year flood plains, impervious soils, and steep slopes are of vital importance. These regulations should be rigorously enforced. Sewer and highway policy can be used strategically to guide development to the plateau and divert it from the valleys.
In addition, new public powers must be sought, mandatory cluster zoning, subsequently expanded to include deeding, minimum 3-acre zoning, promontory zoning, promontory zoning for highrise development, and minimum 25-acre zoning are all advocated. Natural resource zoning, including compensation where necessary, is recommended and would include flood-plain zoning, forest and woodland zoning, steep slope zoning, and riparian zoning. Special assessment districts and public development corporations are less immediate objectives.
The Plan for the Valleys then contains these five aspects, private planning, the uncontrolled growth model, physiographic determinism, the real-estate syndicate, and the accumulation of powers. It has revealed a process whereby a beautiful landscape can absorb growth without destruction and a parallel process by which this may be realized. The report presents its thesis in the form of propositions from which the sequence of analysis and proposals are revealed:
* The area is beautiful and vulnerable.
* Development is inevitable and must be accommodated.
* Uncontrolled growth is inevitably destructive.
* Development must conform to regional goals.
* Observance of conservation principles can avert destruction and ensure enhancement.
* The area can absorb all prospective growth without despoliation.
* Planned growth is more desirable and as profitable as uncontrolled growth.
* Public and private powers can be joined in partnership in a process to realize the plan.
Fortune magazine reviewed the study and quoted from the report, "The United States awaits a large scale demonstration of a beautiful landscape developed with wisdom, skill and taste." Its comment, "This could be it."
# 21
# An Ecological Planning Study for Wilmington and Dover, Vermont (1972)
with Wallace, McHarg,
Roberts, and Todd
Especially after the publication of Design with Nature, McHarg undertook numerous ecological planning studies for local, state, and federal agencies as well as private businesses and nonprofit organizations. These studies were conducted by the firm Wallace, McHarg, Roberts, and Todd or at Penn or, as in this case, some combination. During the 1960s and 1970s there were many interactions between the firm and the university. The founding partners of the firm were academics, and theories and methods generated at the university were often tested at the firm or vice versa. Many students and alumni worked at the firm. The Wilmington and Dover study was selected for inclusion here because it epitomizes that close relationship. It is republished here without the many detailed, colorful inventory maps that McHarg became well-known for.
The study is also included because of its connection to the innovative state growth management (or growth control, as it was called then) efforts of the early 1970s. The Wilmington and Dover study was a partnership involving local citizens, the regional planning commission, the Vermont State Planning Office, Penn regional planning students, and Wallace, McHarg, Roberts, and Todd. The recommendations in the plan are closely linked to Act 250, Vermont's pioneering land-use and development control law. In total the study resulted in twenty-five recommendations. Some were followed, some not. From these suggestions it was clear that McHarg and his colleagues envisioned the study both as a beginning and as a reference point for decision making. Ecological planning is an ongoing process, one where information about a place is used to chart paths for its futures.
# Preface
It began one snowy day in 1970, when Ian McHarg traveled to Vermont at the governor's invitation to address the joint houses of the legislature on the subject of an ecological planning study for the state. The address was received with mixed emotions, either limitless enthusiasm or equally intense aversion. One small group, already converted, was anxious to proceed immediately with a pilot plan for Dover-Wilmington, and so, in view of the State House, over many cups of coffee, Ellen Reiss, Robert McCafferty, Peter Zilliacus, and others initiated a process which reaches a certain culmination now with the publication of this study two years later.
It was Robert McCafferty of Wilmington who first suggested the study. He, with Ellen Reiss, William Schmidt, the executive director of the Windham Regional Planning and Development Commission, Ted Riehle, Benjamin Huffman, and Bernard Johnson of the State Planning Office, were the cadre which developed the idea and generated support for it. The planning commissions of both towns participated, notably Verne Howe and Merrill Haynes of Wilmington and Elva Turner, Peter Zilliacus, Rodney Williams, and Richard Joyce, all of Dover. After inordinate efforts by all concerned it became clear that a modest study would be financed, employing funds from the U.S. Department of Housing and Urban Development, the state of Vermont, the Windham Regional Planning and Development Commission, and the towns of Wilmington and Dover. A magnificent picnic meeting was arranged to be held, Peter Zilliacus cooked a gourmet meal, and the study was formally authorized. But the problems had only begun. It was immediately clear that the funds were totally inadequate for the study envisaged. It was as clear that the same dedication and generosity which had characterized the preparation must permeate the entire study. It would have to be a philanthropy, both from the citizenry and the consultant. In fact the operation of the study required that all data would be generated by townspeople, gratis. But, it should be noted, that while the data collectors were unpaid, they were not amateurs. As to the consultant, the partners [of Wallace, McHarg, Roberts, and Todd] agreed that the study would be done at cost, it would pay no overhead and produce no profit. But that was still not enough. It was impossible to pay prevailing salary scales and so graduate students of regional planning from the University of Pennsylvania were retained as the work force. Although students, the three staff members were considerably skilled. They were Karen Glotfelty, Richard Notzold, and James Wilson. Other staff engaged in producing the report were Ravindra Bhan, Susan Beatty, Carolyn Jones, and Margaret Dewey. The consultant also relied upon John Edinger of the University of Pennsylvania for his insights on sanitary engineering and water supply. Finally, enormous assistance was provided by attorneys Victor Yannacone and Arthur Palmer who are well known for their devotion to writing and defending laws in support of ecological planning, whose special efforts will make the study especially valuable to the towns. The entire study was under the direction of Ian McHarg, assisted by Michael Clarke, the project manager, David Hamme, and Narendra Juneja.
In Vermont a much larger group had been assembled. The indefatigable Ellen Reiss was the local coordinator; Verne Howe, chairman of the Wilmington Planning Commission, supervised students who surveyed streams. Arthur Ball, a young forester waiting to join the Army, undertook the vegetation survey. Dr. Ralph Haslund, a physics professor, resident of Wilmington, invented a device for measuring stream flow. Dozens of residents from both towns cruised the area day after day recording land-use information. The tedious business of assembling data on property was undertaken by Peter Zilliacus, an erstwhile restaurateur now in public service. Dr. Charles Ratte, a geology professor, and his students assembled data on geology. As a result, a considerable segment of the population was actively engaged in finding out the nature of the region they inhabited. It was at once an educational experience and a significant accomplishment of community involvement in the planning process. While the more normal method of contracting a professional team may have produced more data, it would have produced neither the educational experience nor the planning commitment which did in fact result.
The data are imperfect, and this caused much anguish. Should the study halt and await better data or should it proceed? But, in fact, the study has revealed more and better data for Wilmington-Dover than may exist for any other area of Vermont. These data allow good first approximation judgments to be made on the destiny of the region. The process should continue, more and better information should be developed, but it is important to recognize that much has been accomplished; a small group of people know much of their region, they are competent to discuss their destiny and, not least, they have initiated a most remarkable and commendable planning process.
The objective of the study was to develop and apply an ecological planning method to the areas of Dover and Wilmington. This method should reveal the region to its inhabitants as a natural system which is at once a social value system. So understood, the region should be comprehensible as a system of opportunities and constraints for all prospective land uses. Moreover, the study was required to consider alternative dimensions for future growth, to relate these to the opportunities afforded, and to advise on a development structure which provided the maximum social benefit to the community at the least social cost. The areas hazardous to life and health, where high social values existed or where the environment was inordinately intolerant, were also to be identified as unsuitable for development.
Having identified the region and demonstrated alternative patterns of growth responsive to ecological realities, it was necessary to determine the degree to which its recommendations could be affected. This involved an examination of powers now reposing in state, regional, and local governments, giving particular attention to the extent to which they can be made mutually consistent. The study should be complementary to the objectives of both the Regional Planning Commission and the State Environmental Board. It should be seen as a pilot for the State Land Use Plan as well as for other Vermont towns, and the basis for the promulgation of Town Ecological Laws. We commend this report to your attention.
# Summary of Recommendations
1. The study shall be employed as the basis for a continuing ecological planning process involving data collection and interpretation. However, the present data as identified herein shall be used as the basis for public and private action.
2. The study shall be examined in public hearings and, as amended, be adopted as a public planning document, as prescribed by the Vermont Planning and Development Act and by other legal devices ensuring its use in planning decisions.
3. All future town, regional, and state planning decisions shall be preceded by environmental impact statements of all alternatives considered.
4. In the absence of ecologically relevant local statutes, Vermont Acts 250 and 252 shall be employed as the major instruments for controlling land use.
5. In conformance with Act 252 and Act 250, Section 12a(1), no new locations shall be established for discharge of sewage into streams and lakes.
6. In conformance with Act 252 and Act 250, Section 12a( 1 ), no increase in discharge of effluent volumes shall be authorized at existing sewage treatment plants prior to a detailed investigation of aquatic ecosystems.
7. Before granting a development permit under the water requirement provisions of Act 250, Section 12a(2), a developer shall provide a record of pumping tests as evidence of adequate water resources.
8. As prescribed by Act 250, Section 12a(3), any new development shall provide its own water supply, in that the present public water supply of Wilmington is unsafe, and Dover has no water supply system.
9. As prescribed by Act 250, Section 12a(4), flood plains and muck soils, sand and gravel aquifers, steep slopes over 15 percent, and areas above 2,500 feet elevation shall be prohibited from development.
10. As prescribed by Act 250, Section 12a(8), certain scenic areas, wildlife habitats, and historic sites as identified in this study, shall be exempted from development.
11. The urban suitability and protection maps identified in this study shall be adopted as land capability plans under Act 250, Section 12a(9).
12. Full implementation of the intent of Act 250 shall be achieved at the town level by the enactment of ecologically sophisticated, environmentally responsible, socially relevant legislation, dealing with local development. Such legislation shall include zoning and subdivision regulations under Chapter 91 and a new class of environmental protection acts.
13. The present uniform commercial and one-acre residential zoning shall be revoked, and development be guided by new zoning categories in response to intrinsic suitability as defined by the study.
14. New taxation policies shall be adopted to provide benefits to long-term owners of rural lands through tax deferrals, homestead exemptions, and special assessments for properties under voluntary restrictive-use covenants.
15. A new Highway Route 100 proposed by the Vermont Highway Department shall not be accepted until it is demonstrated that improvements to the existing road system cannot accommodate traffic demands.
16. Sewage planning and management areas (now comprised of Fire Districts 1 and 2) shall be enlarged to include the aquatic ecosystem of Harriman Reservoir and its drainage area, and Rock Creek Watershed.
17. A development tax or a special capital gains tax shall be imposed on profits from the sale of land, the funds from which to be used for environmental protection programs.
18. The Green Mountain National Forest shall acquire certain lands within its jurisdiction.
19. Wilmington and Dover shall acquire or invite the state to acquire property rights to preserve regional resources and other natural and cultural features where private action or use of noncompensatory regulations is not effective.
20. The proposed Vermont Land Acquisition and Development Agency shall be activated and invited to take actions to induce future urban growth into suitable locations in Wilmington and Dover.
21. It is recommended that Wilmington and Dover create their own public corporations to acquire and improve suitable lands, and that they convey these lands to private enterprise for appropriate uses.
22. It is recommended that private foundations, trusts, and other civic organizations acquire and maintain those lands identified in this study as requiring protection.
23. Wilmington and Dover should invite landowners to constitute themselves into real estate syndicates to control the timing and location of future development.
24. Developers whose subdivisions are already platted shall revise their plans wherever possible to conform with the intrinsic suitabilities of their properties.
25. Certain land uses in conflict with intrinsic suitability shall be defined as non-conforming uses.
# The Ecological Planning Method
The method employed is described as ecological planning. Simply, it means understanding Wilmington and Dover as a natural system, recognizing that the natural elements which compose regions are also social values. Certain places are better suited for towns, parks, farms, and ski slopes than others. If the towns can be described as a natural system, and if the elements that compose it can be seen as social values, then it becomes possible to plan. It is then necessary to identify places hazardous to life and health on the one hand, and areas which are intrinsically fitting for all of the prospective uses which are likely in the future. The region can be described under the titles of climate, geology, physiography, groundwater and surface water, soils, plants, and animals. The phenomena in each of these categories are variable: more or less stressful climates, rocks of different strength and stability, slopes obstructive or beneficial, water of varying quality and quantity, soils differing in properties and usefulness, vegetation comprised of different communities having distinct values, and similarly for animals.
But people have been responding to this natural system and continuously adapting it. We must then proceed to identify people, families, institutions as both phenomena and processes. This can best be done by invoking history and examining colonial subsistence agriculture in the last century and the resurgence of the present. So current land use can be seen in terms of cultural history, revealed in the pattern of settlements, villages, roads, farms, schools, and the like.
Natural environments are then variable, comprised as they are of variable rocks, slopes, soils, plants, animals, and microclimates. People, in turn, have modified those processes and phenomena and added variable environments of their own. Similarly, environmental needs also vary. The requirements for crop agriculture are different from those necessary for ski slopes or a new community. If we can identify the place as composed of different environmental attributes more or less suitable for human uses, we can then assemble all of the factors most beneficial for every prospective use. When we find locations which provide all beneficial attributes, and where the major detrimental factors are absent, we can describe such locations as intrinsically suitable for the land use in question. The summation of this exercise is the representation of the place as having variable intrinsic suitabilities for all prospective land uses. Such are its opportunities. The reverse image reveals those areas or processes hazardous or stressful to life or health, where environments are intolerant, or where significant social values exist.
Given this vantage, it next becomes necessary to examine alternative futures. A projection of the most likely future can be made, assuming that all current trends will continue. That may be compared to other options for growth which respond entirely to the towns' intrinsic suitabilities. Social costs and benefits can be approximated for all alternatives. It then becomes necessary for the towns to choose their own futures through public discourse and political and legal instruments. This will require citizens to participate in an active planning process using methods and data such as those employed in this study.
# Geology
The Green Mountains are comprised of metamorphic rocks, most of which are derived from pre-existing mountains of the Pre-Cambrian period, dating back one billion years ago. This very early geologic history consisted of alternating sequences of mountain uplifts, erosions, submergences, sediment depositions, and volcanic activity. The Green Mountains were created at the close of the Ordovician period about 425 million years ago, the same time as the Taconic mountain-building sequence. Structurally, they are an anticlinorium, an arched complex of folds.
Unconsolidated surficial rock material overlies bedrock, derived predominantly from the Pleistocene Epoch, which occurred between 10,000 and one million years ago. During the Pleistocene, Wilmington and Dover experienced two glaciations, separated and followed by times of post-glacial erosion and deposition. Advancing ice scoured large areas over which it crossed, typically removing several feet of bedrock. Many stream valleys transversed by ice were probably filled with glacial debris, whereas parallel valleys were scoured and given a characteristic U-shape. During their advance, glaciers accumulated rock debris by exerting plucking stresses on bedrock underneath and along their margins. When the lower part of the ice became heavily laden with debris, excesses were deposited as ground moraine which were then overridden by the more active ice above. Toward the end of glacial advances, ice became stagnant and began to melt, leaving deposits that took various forms such as kames, kame terraces, eskers, and valley trains. The Pleistocene has thus left clear marks in the area.
# Physiography
Physiography can be best understood from those natural processes responsible for its being. Following the Green Mountain episode, the landscape has continued to evolve through successive glaciations and subsequent fluvial processes. Physiography, then, is a product of the past. But it is continuing to change, although usually at imperceptible rates. Streams are downcutting their beds while reworking sediments in their flood plains. Weathering processes in the highlands are transporting materials downslope to new locations in the lowlands. People are contributing changes to physiography faster than nature, as evidenced by sand and gravel pits, road cuts, and landfills.
Our perception of physiography depends upon our proximity to the landscape. A geographer looking at the eastern United States would say that New England is comprised of the Appalachian Highland, but a Vermonter would describe his state as divided into at least five regions: the Champlain lowland, the Taconic Mountains, the Vermont Piedmont, the Northeastern Highlands, and the Green Mountains. And still, to say that Wilmington and Dover are within the Green Mountains, does not describe their physiography sufficiently. Actually, four subregional expressions of the Green Mountains can be identified in the towns: the Ordovician Highlands, the Pleistocene Highlands, the Pleistocene Lowlands, and the Pleistocene Valley. While their origins are speculative, their geographical identity is accurate.
The degree of slope constitutes savings or costs to building construction. Slopes exacerbate hazards of overlying soils. Physiography is the essential visual component of the landscape.
# Soils
In New England, soils are derived from a soil-building process called podzolization. Decomposing organic matter on the ground surface produces acids which leach through the organic horizons downward through the mineral layers below, dissolving and removing carbonates of all kinds. This process produces soil horizons which from top to bottom are: a partially decayed organic zone where litter is decomposed into humus; a leached acidic siliceous zone (called the A horizon); and an accumulated zone containing carbonates and other soluble salts leached from above (called the B horizon). When fully developed, this is the profile of a true podzol soil.
Less developed variations of the true podzol, known as podzolic soils, characterize southern New England. Their organic, leached, and accumulated horizons are smaller. The towns are comprised of both podzol and podzolic soils. Soils data for Wilmington and Dover are limited to soils associations, constituting large groupings of soils related to one another. However, many are comprised of different combinations of the same soil types. Moreover, many soil types in the same association may be very different from one another with respect to their intrinsic suitability for different land uses. This means that association data for Wilmington and Dover cannot be interpreted for planning purposes. Other than identifying locations of alluvial, muck, and peat soils, the towns must ask the Soil Conservation Service to undertake a modern detailed soil survey, from which the data can be used in town planning.
# Hydrology
Groundwater occurs in rock interstices. Consequently, thick deposits of glacial sands and gravels have the most consistent potential for groundwater storage. Few interstices were created during the metamorphic period creating the area's bedrock. Most of them are derived from secondary joints and fractures created after the rocks were formed. Mapping of surficial geology has just been initiated, and knowledge of bedrock fracture zones is very limited. However, a search for groundwater would initially seek deep sands and gravels in the Deerfield Valley in the vicinity of recharge sources such as streams, ponds, and bogs. The next choice would be the Wilmington gneiss (the most weathered rock with the most fractures), particularly in locations overlain by surficial deposits and near recharge sources. The amphibolites seem to be the least-probable groundwater sources. Little information is available on groundwater in the other bedrock formations, although they apparently are not as good as Wilmington gneiss or as bad as the amphibolites.
Of 53 inches of average annual precipitation, 25 to 30 inches become natural runoff in the streams. Highest flows occur with snowmelt, and thereafter decline continuously, reaching annual lows in August or September. During the fall and early winter streamflow gradually increases to another peak and remains steady or declines slightly until spring again. Harriman Reservoir receives all of the Deerfield's drainage, and is a closure to an otherwise open stream system. Conversely, Lake Raponda (in Wilmington) is located at the head of a drainage area, making it less vulnerable to pollution from entering streams.
# Plants
Other than a generalized forest map and some field observations, little is actually known about the towns' plant communities. Present vegetation originates from the close of the Pleistocene over 11,000 years ago. Following the retreat of the last glacier, Wilmington and Dover were probably initially occupied by tundra, later by boreal trees, and then predominately by the hardwood forest. Early people, following not long after the last glaciation, are thought to have caused extensive burning. Their disturbances and, later, eighteenth- and nineteenth-century agriculture and twentieth-century logging, have continually rearranged the composition of the forest. Moreover, plant succession itself is a dynamic process, responsible for steady change in plant associations without human intervention.
The towns are comprised of both the northern hardwood and boreal forests. The hardwood forest is comprised predominately of American beech, yellow birch, and sugar maple, in association with eastern hemlock, sweet birch, red maple, basswood, white ash, and northern red oak. Pioneer species after cutting or fire include aspen, birch, spruce, or fir, depending upon site conditions. The spruce-fir forest is comprised of red, white, and black spruces and balsam fir. Pioneer associations after fire or cutting may include those same species or hardwoods, depending upon site conditions.
Stressful climates and thin, infertile soils above approximately 2,500 feet elevation are extremely unfavorable for all plant forms. Yet, plant life is necessary if this fragile zone is to contribute significantly to the water-holding capacity of the land.
# Animals
Diverse and abundant populations of birds, mammals, and fish reside in the towns (figure 13). White-tailed deer, skunks, woodchucks, chipmunks, squirrels, red fox, rabbits, hare, porcupines, and grouse may be found practically anywhere. Bear, bobcat, and possibly coyotes live in the highlands. Ducks, woodcock, beaver, mink, muskrats, and probably otter live along streams and other wet areas. Streams and lakes contain brown trout, brook trout, smallmouth bass, perch, and many other species. Mappable data are limited primarily to white-tailed deer, beaver, and trout.
White-tailed deer are ubiquitous forest dwellers, but they are also found along forest openings, orchards, and farmlands where they forage. With few natural predators such as the bobcat, populations are controlled only by hunters, starvation, and disease. The major diet is twigs, although herbaceous plants and fruits are eaten in the summer. The towns have numerous colonies of American beaver, who build dams of sticks, mud, stones, and tree trunks to impound water to a sufficient depth to prevent its freezing to the bottom. Beavers build lodges in their impoundments with entrance chambers opening under water, thus protecting themselves from their enemies.
Most of the towns' streams are good trout waters, despite seasonal low flows, warm temperatures, and soft waters with low buffering capacities. Particularly good reaches are steep gradient streams mixed with riffles and pools having wooded streamsides protected from the sun. Harriman Reservoir is one of Vermont's largest lakes and is potentially a very important fishery resource.
Figure 13. Fish and wildlife habitats in Wilmington and Dover, Vermont.
# Climate
Adiabatic cooling of air masses rising across the Green Mountains causes higher precipitation on mountain slopes and valleys. Local observations suggest the occurrence of a very heavy snowfall zone at elevations above approximately 2,200 feet, whereas heavy snows occur throughout the region. Prevailing winds are from the northwest in winter and west or southwest in summer. Consequently in winter, exposure is maximum on west-facing slopes and minimum on east-facing slopes. Tops of ridges and hills are always windy. Local winds tend to be channeled down or up the Deerfield Valley.
Because of their high elevations, the towns' temperatures are as low as those of northern Vermont. Frost-free periods average fewer than 60 days annually. Local temperatures vary as much as 10 to 20 degrees, such extremes occurring between steep slopes facing the sun (south) and steep slopes oriented away from the sun (north). Fog and frost pockets tend to occur in low areas adjacent to the Deerfield River. Climatic categories identified are:
1. Heavy snow Leeward wind conditions
South slopes, over 3%
West slopes, over 8%
2. Heavy snow Moderately windy
South slopes, over 3%
West slopes, over 8%
3. Heavy snow Leeward wind conditions
All slopes, 0–3%
West slopes, 3–8%
East slopes, over 3%
North slopes, over 3%
4. Heavy snow Moderate wind conditions
North slopes, over 3%
East slopes, over 15%
5. Very heavy snow Leeward or moderately windy
South slopes, over 8%
6. Very heavy snow Leeward or moderately windy
All slopes but south, over 8%
7. Heavy snow
Windward conditions
All slopes, all orientations
8. Very heavy snow Windward conditions
All slopes, all orientations
9. Fog areas All wind conditions
All slopes, all orientations
# Life Zones
As a sum of interacting natural processes, the place can be expressed in terms of the diverse environments it imposes upon people and other life forms. Employing available data on microclimate, surficial geology, physiography, hydrology, and soils, twenty terrestrial life zones were identified and ranked in a gradient of winter stress. Aquatic life zones were also identified. While the data are limited and imperfect, the life zones have considerable value as a first approximation of locations unfavorable or propitious for human habitation. The life zone data have been reviewed by a number of town residents, several of whom corroborated its general accuracy by agreeing upon its representation of winter stress conditions in areas with which they were familiar. Also seen is a coincidence of winter deeryards and locations of least winter stress.
# The Place Is the Sum of Natural Processes
Natural processes and phenomena have been described thus far as discrete components: mesoclimate, microclimate, bedrock geology, surficial geology, groundwaters, surface waters, soils, plants, and wildlife. But the place must be seen synoptically as a single expression of all its parts. The science of ecology seeks this perspective, being derived from the Greek word, oikos, meaning house. Ecological planning in turn seeks to understand the place before it prescribes alternative futures.
The place can be described at various levels of sophistication. It can be mathematically modeled, either manually or with computers. Many hydrologic models, for example, have been developed to describe relationships among physical, chemical, and biological characteristics of streams and lakes. The place may also be seen through graphic techniques, using overlays indicating the coincidence, for example, of microclimate, geology, soils, plants, and animals. It can also be identified descriptively.
Most simply, the place can be described as a "layer cake." Its bottom layer is the metamorphosed rocks, the oldest of which are derived from Pre-Cambrian times dating back over one billion years ago. The Taconic mountain building sequences occurring at the close of the Ordovician period some 350 million years ago produced the Green Mountain anticlinorium.
The place's surficial geology is its next layer, derived from the Pleistocene period which occurred between one million and 10,000 years ago. During that time the place experienced two glaciations, separated and followed by times of post-glacial erosion and deposition, which plastered the pre-existing bedrock with unconsolidated sediments.
The third layer is the place's physiography, a product of geologic history, expressed as four subregions within the Green Mountain physiographic province: the Ordovician Highlands, the Pleistocene Highlands, the Pleistocene Lowlands, and the Pleistocene Valley.
Groundwater is interbedded within the geologic and physiographic layers of this conceptual layer cake. Only small quantities are found in the original interstices of metamorphic rocks. Most of it is in joints and fractures which have formed subsequent to the period of metamorphism and in the unconsolidated overburden deposited during the Pleistocene.
Surface waters are the next layer. The place's dendritic streams have steep gradients and variable flows. They are soft and slightly acid, making them especially vulnerable to change. Ecologically, Harriman Reservoir has created a closed system for nutrient movement.
Soils come next, derived from geologic parent material, physiographic characteristics and other layers above, i.e., climate, plants, and animals. The place's soils develop from podzolization, a natural acid-leaching process forming soil horizons of humus, silicates, and carbonates and other soluble salts.
Plants are the next layer, influenced by all of the physical characteristics of the place which produce a variety of sites appropriate for a diversity of species from two distinct forest associations. The northern hardwood forest is the predominant climax association. However, the northern boreal forest is also seen because of the place's high elevation physiography.
Animals are the next layer, depending upon all of the place's attributes for their habitats and survival. In addition to people, we see diverse and abundant populations of birds, fish, and other mammals, which include white-tailed deer, skunks, woodchucks, chipmunks, squirrels, red fox, rabbits, hare, porcupines, grouse, bear, bobcats, ducks, beaver, mink, muskrats, brown trout, brook trout, bass, possibly coyotes and otter, and many other species.
Micro- and mesoclimate comprise the top layers of the cake. Microclimate, i.e., the climate near the ground, varies with elevation, landform, slope, gradient and orientation, and vegetative cover. Wind is important, reaching its maximum on west-facing slopes, and the tops of hills and ridges. Insolation is equally so, reaching its minimum on steep, north-facing slopes. The place's mesoclimate is controlled principally by prevailing sub-arctic westerly winds and the frequent passage of warm, moist air coming up from the Gulf Coast.
This description of the place would be inadequate without emphasizing its dynamism. It is comprised of many systems involving energy movement through precipitation, erosion, deposition, transport, heating-cooling, freezing-thawing, evaporation-condensation, weathering, insolation, and countless other physical-chemical processes. These physical-chemical processes involve and affect all life forms, their birth, development, movement, reproduction, and death.
# The Land and the People
A review of the historical development of the towns of Wilmington and Dover offers little insight into the future. The most probable form of development will have little relation to and will place small value on the history of these communities. Artifacts and structures which recall an earlier day abound in the area and are of interest to at least a portion of the present residents. Their value in the future, however, may be more in evoking a general American myth of a comfortable rural past rather than their specific relationship to the histories of the towns.
While the specifics of the past are of little interest here since they instruct little, the more general trends which have set the stage for the future deserve some attention. The towns were both first settled in the eighteenth century and grew predominately as farming and herding communities with such attendant industries as these activities would comfortably support. By the latter part of the nineteenth century, the farms were exhausted by questionable agricultural practices and were no longer able to support even a marginal level of production. At about the turn of the century, both communities turned to tourism to replace declining agricultural industries. The Village of Wilmington, the Handle Road area, and Lake Raponda became popular summer resorts. But the decline of the resident population, begun by the depletion of the meager soils, continued to the middle of the present century. These patterns meant that neither town ever went through a period of intensive urbanization. They have been, and they still remain essentially rural. This pattern is now threatened by recent development.
# The Immediate Past
It is in the immediate past that the key to the future may be found. Following World War II, an increasingly affluent American population sought escape from the boredom and tension of routine daily life by increasing expenditures in a broad range of leisure activities. The entrepreneurial response was quick and is still continuing.
In Wilmington and Dover this response took the form of developing ski areas. Mt. Snow, Haystack, and Carinthia provide one of the most popular ski complexes in the State of Vermont. As a result, the ski industry, including the ski areas themselves and the supporting commercial activity, is the primary economic activity in the towns. In the last five years, the growth has assumed logarithmic proportions. During this period, Mt. Snow doubled its lift capacity, and plans are now afoot for even further expansion.
Significant to the future of the towns has been a shift in preferences of this tourist population. The original day and weekend skier has been joined by a vacationer of a less transient nature. This person, often not primarily a skier, is seeking a fixed vacation spot for both summer and winter in a pleasant rural environment. Satisfying this desire often requires the purchase of a second vacation home. The trend of second home ownership has gradually increased in Wilmington and Dover over the past ten years, culminating in Chimney Hills, a subdivision of about 800 lots. As of 1971, there were 539 second homes in Wilmington and 411 in Dover. An additional 2,000 acres have been sold throughout the two towns for second home development. The pattern of the future has been established (figure 14).
Figure 14. Facilities and development patterns in Wilmington and Dover, Vermont.
# The Structure for Change
Presently the Village of Wilmington has a sewage treatment and a water supply system. A small private treatment plant serves Snow Lake Lodge at Mt. Snow, and a few of the subdivisions provide private central water supplies. The remaining developed areas rely on individual wells and septic systems. These services are inadequate to meet the needs of the anticipated growth of the area.
The provision of large-scale central sewer facilities in Fire Districts 1 and 2 (new sewer service areas in the Town of Dover) will dramatically alter the pace and the pattern of future development. Both of these sewer plans are in the proposal stage, and it remains to be seen whether or not either can satisfy the provisions of Acts 250 and 252. Without discussing the relative merits of the proposals, it is sufficient to say that, given the present market situation the two systems, if constructed, will attract development from elsewhere in the area in spite of other considerations of accessibility, land use, and environmental suitability.
The new population in Wilmington and Dover will be largely seasonal in nature. Employment and major purchases (cars, durable goods, etc.) will take place elsewhere. From the local communities these residents will require police and fire protection, road maintenance, trash and garbage disposal, occasional (probably emergency) health service, and general government services such as code enforcement. Sharp increases in town expenditures can be expected in these areas. These same residents will add little to the public school enrollment. They will not vote. They will pay property taxes. They will not support local churches or other institutions. In general they will establish a social life separate from that of the permanent residents. Finally, the seasonal population will add little to the human resources of the towns in terms of leadership or technically skilled personnel.
Regional access provided by the interstate system through Brattleboro is excellent. The decrease in travel time to major centers of population occasioned by this system is largely responsible for the development pressure upon Wilmington and Dover. It is doubtful whether further major improvements in travel time can be produced by additional construction in Vermont.
Travel between the interstate system and the destinations in the two towns can be made easier and thus more attractive by improvements to both Route 9 and Route 100. To maintain the desirability and the development demand of the area, Route 9 at least should be improved to ease traffic movement from Brattleboro to Wilmington.
# Agents of Change
Wilmington and Dover are well located to exploit the second home market. The largest concentration of population in the United States lies within four hours' automobile travel of the area. This includes Boston, Providence, Hartford, New Haven, New York, and the northern part of New Jersey. Within six hours of travel lie Allentown, Trenton, and Philadelphia. While general population growth in this megalopolis is not high by national standards, the proportion of people moving into income brackets of $15,000 or more per year exceeds the general population growth rate. It is this higher income level that constitutes the market for second homes. As far as location is concerned, Wilmington and Dover can tap the largest potential market in the country.
The area is already well endowed with the kinds of large-scale recreation facilities for both summer and winter which provide the attraction for second home users. The three ski areas already mentioned provide a total of 19 cable lifts, and plans are underway to expand this total. Excellent snow conditions and good trails particularly at Mt. Snow, combine to ensure that Wilmington and Dover will retain or even increase their present share of the ski industry.
Summer facilities are less spectacular. Nevertheless, either actually or potentially, the elements for year-round recreation are well established. Two golf courses are already constructed. Lake Raponda offers limited swimming and boating. Lake Harriman is presently underutilized for summer water sports, but has an enormous potential. Molly Stark State Park in Wilmington and the Green Mountain National Forest in both Wilmington and Dover provide trails, picnic spots, and camp grounds. Both towns are laced with streams which offer good fishing in season.
As has been stated, Wilmington and Dover have been and are now essentially rural. The villages, the only population concentrations in the area, retain the patterns of public and publicly related uses which serve the communities. Suburban development, though increasing, is still too sparse to alter the general rural character of the towns.
This pattern means that the vast majority of land remains in relatively large holdings unencumbered by intrusion of public or institutional uses which tend to resist change. Land held in this pattern is relatively easy to assemble into even larger parcels if there is economic incentive to do so. This is precisely what has been happening in the two towns.
A number of landowners, corporate and private, in Wilmington and Dover have an avowed intent, either through direct development or sales to entrepreneurs, to subdivide the land for building lots. This pattern is already well underway.
As these owners continue to realize their intentions, more rural land will be converted to suburban residential patterns, and occupied mainly by second home purchasers. In addition to the 2,000 acres sold or developed since 1967, another 1,500 acres are in subdivisions which have been platted and approved but not yet sold. About 2,400 dwelling units will eventually be constructed in these areas. An estimated 7,500-10,000 acres are being held in lands whose owners intend to develop in the next ten years. Construction of this magnitude will change the existing environment and the established pattern of uses and functions.
# The Spectre of Future Growth
From now through the next ten years, the full fury of the long developing storm will break over the two towns. The forces of market demand, land ownership, established recreation attractions, and accessibility will converge and mature to produce a holocaust of construction and change. Compared with the immediate future, the past, i.e., the period of preparation, will seem tranquil and benign.
It is possible to outline the dimensions of this change, given no new major effort on the part of the towns to control or direct growth to ensure the greatest local benefit. For clarity, new growth can be considered in two categories: the permanent population and the second home population. Totals can be expressed in terms of dwelling units because it is extremely difficult to assign population to seasonal homes. The actual number of occupants in a second home can vary widely from zero to a dozen or more during ski season.
Table 12 projects the permanent growth in terms of both population and population converted to households (so that it can be compared with the second homes).
Table 12. Resident Population and Households, 1970-1980
The following set of assumptions were made to determine the total future growth:
1. Currently platted subdivisions will fill.
2. Reasonably certain proposals for new subdivisions will be platted, approved, and filled.
3. The national economy will remain expansive over the long term and incomes in the eastern megapolitan area will continue to rise at or near the present rate.
4. At least one other major and one minor all-season recreation complex will be developed by the middle of the decade.
5. The expansion of the Mt. Snow ski trails will take place as planned.
6. Sewer plants will be constructed for Fire Districts 1 and 2.
7. Route 100 will be realigned and Route 9 will be substantially improved.
8. Major development corporations with well-trained and aggressive sales forces will continue to invest in the area.
9. National publicity will remain favorable for the southern Vermont area and particularly for the two towns.
It was assumed that certain other events would not come to pass.
1. No new major ski area will be built in or adjacent to the towns during this decade.
2. Somerset Reservoir and the Town of Somerset will not be opened up by improved or new roads.
3. Harriman Reservoir will continue to be used for limited recreation but no major residential development will be built on utility holdings.
4. No major new land use controls will be adopted nor existing ones rigorously enforced.
With these assumptions the total number of dwelling units in the towns can be estimated. Table 13 lists total dwelling unit count by 1980, existing and to be built, and permanent and second homes.
Table 13. Total Dwelling Units by 1980, by Town
Table 14 identifies those units which are constructed to meet the demands of the second home market in Wilmington and Dover.
Table 14. Second Homes by 1980, by Town
Under the assumptions set forth above, the estimated growth by 1980 will be between 671 and 967 permanent households and between 9,791 and 12,551 vacation homes. Such is the likely face of the future for Wilmington and Dover, a 1,000 percent increase in dwellings. The process is visible, it has clear implications for the future. The prospect is a spectre of sprawl. How can it be resolved? What perceptions and powers proffer alternatives?
# Powers to Preserve Natural and Social Values
The threat is imminent. It has catastrophic proportions. It is vitally necessary to devise a strategy. The first element is an ecological basis for a plan based upon substantial information and intelligent interpretations. This locates areas intrinsically suitable for all land uses. Next we must consider state regulatory powers. But they are not enough. Town ecological laws must be promulgated and passed. Nor is that enough. These will merely prohibit or refrain. It is necessary to develop positive inducements to ensure that development will occur on the most propitious locations. But first, let us examine social values and the powers of Vermont Acts 250 and 252.
Natural processes and phenomena have intrinsic values to the natural systems of which they are a part. Plants are used by animals. Flood plains accommodate floods. Natural processes also constitute social values to people. Geologic processes produce areas of scenic splendor. Skiers use the snow-covered slopes of Mt. Pisgah. Favorable soils and slopes are savings for construction. People have created their own values as seen in their villages, schools, roads, churches, and recreation areas. Once accepted that the place is a sum of social values, inferences can be drawn regarding its utilization to ensure optimum use and enhancement. These values can be identified in Wilmington and Dover, and they can be protected by the performance standards of Act 250.
... a development or subdivision:
1. Will not result in undue water or air pollution...
2. Does have sufficient water available for reasonably foreseeable needs...
3. Will not cause an unreasonable burden on an existing water supply...
4. Will not cause unreasonable soil erosion or reduction in the land to hold water . . .
5. Will not cause unreasonable highway congestion or unsafe conditions...
6. Will not cause an unreasonable burden on municipal educational services...
7. Will not place an unreasonable burden on local governments to provide services...
8. Will not have an undue adverse effect on the scenic beauty ... aesthetics, historic sites, or rare and irreplaceable natural areas.
9. Is in conformance with a duly adopted land-use plan...
10. Is in conformance with any adopted plan under Chapter 91 of Title 24.
These standards are used by district commissions in reviewing individual applications for subdivisions and developments. Clearly, these same standards must be seen in terms of their formal implications for entire towns and regions. Consequently, the evidence assembled for Wilmington and Dover must now be reconstituted as opportunities and constraints, employing standards 1 through 8 above. Thereafter, a synthesis of those values reveals intrinsic environmental and community structure, which can then serve as the basis for plans envisaged in standards 9 and 10 above.
# Protection of Water Quality
Most tributaries of the towns are healthy, whereas the main stem of the Deerfield is becoming unhealthy as a result of treatment plants below Mt. Snow and the Village of Wilmington, septic tanks, and urban construction practices. An essential insight to any water quality management program for the towns is that Harriman Reservoir and the entire Deerfield drainage area above it represent an aquatic ecosystem and a single management unit. There is an immediate need to study the long-term effects of wastes entering this system. Fire Districts 1 and 2 are totally inadequate instruments for planning sewage disposal and should be so recognized. Vermont's water quality objectives have been stated:
... the waters be protected and used to promote the general public welfare and interest... [and] maximum beneficial use and enjoyment of all the waters of the state; and... that all the waters of the state be of a quality conforming with or exceeding the classification standards for Class B water...
No waste may be discharged into waters of the state without a permit from the Department of Water Resources. Wastes are prohibited in certain streams.
No new discharge of wastes, which will degrade in any respect the quality of the receiving waters, will be permitted above elevation 1,500 feet or ... at a rate of flow of less than 1.5 cubic feet per second at any elevation... which waters are hereby described as pristine streams and tributaries.
Except for the northeastern edge of Dover (below elevation 1500 feet), the towns' tributaries are pristine streams, meaning that no discharge permits will be authorized other than at locations already having permits, i.e., Mt. Snow and the Village of Wilmington. Discharges at those locations cannot exceed a dilution ratio of 30 to one at any time. The capacity of Wilmington's treatment plant can be enlarged modestly only if a 30 to one dilution ratio is to be maintained at low flow conditions. Interbasin transfer of wastes (outside of the towns) will probably not be permitted by the Department of Water Resources. Consequently, little if any increase in sewered population appears possible in the towns, using conventional treatment and stream effluent disposal methods. New sewered developments must employ advanced technologies for treatment and disposal which do not use streams. Recycling should be considered as a serious alternative. Disposal methods through lagooning, spray irrigation, and subsurface recharge and other devices may be feasible, some of which are being explored by the Water Resources Department.
However, the burden of proving that a proposed waste solution will not degrade water resources must be borne by the developer.
State laws also protect groundwaters and surface waters from pollution by septic tanks. No landowner may subdivide prior to obtaining a permit from the Department of Health. A permit is granted only after an applicant demonstrates that site conditions are suitable, accounting for soil percolation rates, lot size, proximity to water supplies, ground slope, elevation above flood plains, the groundwater table, and bedrock conditions.
Both towns have severe limitations for septic tank systems because of seasonal high water tables, presence of an impervious fragipan (24-30 inches below the soil surface), shallowness to bedrock, and steep slopes. All of these conditions cause sewage to appear on the ground surface from flooding or downslope flows, thus contaminating surface waters and constituting a serious health threat. Without adequate soils data, only some of the alluvial and muck soils can be actually identified as locations where septic systems should be prohibited, despite general knowledge that large areas of the towns are unsuitable for such use. The towns must obtain a detailed Soil Conservation Service survey. Local experience suggests that slopes approaching 15 percent and higher should be avoided in the interest of public health.
# Provision of Water Supplies
In the absence of detailed groundwater survey data, a water budget was developed to quantify the towns' water resources. A water budget simply is an accounting method to indicate amounts of water arriving, stored, and leaving an area over a given time period. It is derived from the following equation of hydrologic equilibrium:
Surface + subsurface inflow + precipitation + imported water + decrease in ground and surface storage
Equals
Surface and subsurface outflow + consumptive use + exported water + increase in surface and groundwater storage
Safe yield or the annual quantity of water which can be withdrawn on a sustained basis was chosen as 25 percent of base flow or that portion of stream flow coming from groundwater. Base flow estimates were made for different times of the year, and expressed in safe yields per acre.
Safe Yield Season | Persons/Acre at Gal/Day/Acre | 100 gpd/Person
---|---|---
July, August, September | 50.6 | 0.5
October, November, December | 286.0 | 2.9
January, February, March | 233.0 | 2.3
April, May, June | 410.0 | 4.1
Water availability during the low-flow periods of August and September suggests that an all-season home with a domestic well should be located on a six-acre lot. Conversely, a ski-season chalet used only in winter would require a 2.5-acre lot. These calculations are guides and are not reliable for small areas where hydrologic conditions will vary. Nor are they substitutes for detailed hydrologic surveys. Yet, they suggest that development cannot be accommodated forever, solely using groundwater or free-flowing streams.
Harriman and Somerset Reservoirs are potential solutions to all future water supply problems in the towns, although both projects are currently operated for hydroelectric power generation. Some gross calculations show that Harriman could serve both purposes. Assuming a tenfold population increase in the next 15 years, about 2 percent of Harriman's storage would be required for water supply. Aside from the fact that it already exists, Harriman Reservoir is inherently better for large water supplies than those studied by the Soil Conservation Service for the Wilmington-Dover Area. Such small upland reservoirs are less reliable and floods fill them with cobbles. Today's most distressing water supply problem is Wilmington's public system, declared unfit to drink for reasons still unknown. Future demands will rely upon new systems utilizing Wilmington gneiss, Deerfield sands and gravels, and Harriman or Somerset Reservoirs as the most promising sources.
# Erosion and WaterRetention
Erosion sediments shorten reservoir life spans, destroy fish habitats, increase floodstage levels, and clog drainage ways. Eroded soils have little utility for agriculture. Water infiltration, percolation rates, and storage capacity of eroded soils are diminished and, therefore, so is groundwater recharge. Erosion is exacerbated by surface runoff from rainfall, which loosens or picks up material by turbulence or eddy currents, aided by sediments already in solution which provide a scouring action. Slope gradients, as they increase, accelerate flow velocity, increasing the erosional force of running water. Surface runoff traveling over exposed soils, dislodges particles more easily than over vegetated surfaces where velocities are reduced and the soil surface is protected.
Most soils of Wilmington and Dover are inherently erodible because of their simple structures and fine grain textures, particularly on steep slopes. Inadequate soils data prohibit identifying locations of the towns'most hazardous soils. Removal of vegetation and prolonged exposure of soil surfaces on slopes should be accompanied by a management plan to restore vegetation and trap sediments. Soil disturbance should be avoided especially during nongrowing seasons of the year when immediate re-establishment of vegetation is impossible. Removal of forest vegetation in heavy snowfall zones (e.g., above elevation 2,200 feet) should be minimized to maintain snowpack conditions as long as possible for sustained groundwater and stream recharge. Excavation of sand and gravel aquifers should be prohibited to protect their storage capacities. Erosion hazard districts should be established from detailed soil and slope data.
Particularly fragile areas which are important sources of water are the upper Green Mountain zones, typically above elevations of 2,400-2,500 feet, where the sugar maple or beech hardwood forest is replaced by a transitional forest dominated by yellow birch, white birch, or red spruce. These areas have been described by Vogelman et al.
Mountain soils with their high organic content hold large quantities of water which come from the high rainfall and fog moisture collections of forest trees. The water filters through the soils and eventually adds to stream flows, springs, and groundwater supplies in the valleys... the severe climatological environment of the upper mountain slopes imposes great physiological stress on plants growing in these areas. Removal or even disturbance of these fragile plant communities opens the soil to severe erosion and irreparable damage. The vegetation returns to these disturbed areas very slowly and with great difficulty.
Vogelman has recommended that ecological disturbances in any form should be kept to an absolute minimum. Roads, structures, and septic tanks should be prohibited. Activities should be restricted to hiking and skiing. Vegetation openings and trails must be designed and maintained not to impede natural drainage, organic matter cycling or cause erosion. Vegetation on ski trails and lift line areas must be intensively managed to avoid soil loss.
# Highway Congestion and Safety
The spirit of Section 12(5) of Act 250 is that no use will be permitted which will cause increase in traffic volume or interference with traffic flow which may result in highways becoming inadequate or unsafe, unless an applicant shares proportionate costs for improvement, expansion, or construction of highway facilities required insofar as such costs exceed the amount budgeted or planned by responsible agencies for such work in and about the area of the proposed use. This principle becomes complicated in Wilmington and Dover, where Highway Routes 9 and 100 are already heavily congested and unsafe at times during ski months and occasionally during the summer and fall tourist seasons. Consequently, any new developments exacerbating those existing problems should be prohibited prior to improving present conditions. Yet, new Route 100 proposed by the Highway Department (Line FP) could be a disaster for both towns. The highway, proposed solely as a device to solve the traffic problem, presumes that the towns wish to encourage population growth. The proposed highway will attract vehicles which otherwise would not be in the towns, and could ultimately increase traffic congestion on town roads. Moreover, such an alignment will be a blight on the towns' scenic character and stimulate new developments in locations unsuited to such uses. Prior to accepting the proposed alignment, other options must be examined: improving existing roads, and facilitating traffic flow by removing certain obstructions, creating new turning lanes, and making roads one way at peak hours.
# Schools
The ease by which the towns can provide education services as population increases is determined largely by capacities of present facilities, numbers of vacant seats, and the rates that those seats will be filled. The rates at which the schools will be filled in the years to come are hard to assess because of the difficulty of projecting increases in permanent population in a second home housing market.
The spirit of Section 12(6) of Act 250 is that no development or subdivision should be permitted which will increase the number of students attending public schools beyond their capacities in terms of space, facilities, instructional, and administrative staff, transportation, and other school-related services, unless the applicant can ensure that such uses will return their proportionate share of expected costs.This will require a developer to prepare an impact statement indicating numbers of dwelling units and bedrooms, price ranges of units, age distribution of children, percentage of seasonal and permanent populations, and other information necessary to determine the tax return to the community from units he is producing, and the number of children he will be adding to the school system. The towns, in turn, must have a reasonable program for expanding their school facilities. Should the proposed development put more students into the school system than it can take, the towns can insist on adjustments such as delayed construction, rearrangement of housing mix, or monetary contributions to the school system.
# Governmental Services
Section 12(7) of Act 250 states that a proposed subdivision or development shall not place an unreasonable burden on the ability of a town to provide services such as police and fire protection, highway maintenance, storm drainage, refuse and garbage collection and disposal, and sewage and water systems. Otherwise, an applicant must ensure that his proposed use will return to the town its proportionate share of the costs for expanded services, or provide the services himself.
A 1964 study of Vermont towns examined relationships between size and efficiency of operation, and concluded that larger communities (6,000 persons average) did achieve economies of scale and provided more and higher quality services than smaller towns. However, that study also found numerous exceptions to its general conclusions. Moreover, a 1965 study suggests that Dover's property tax burden per capita is much lower than the average Vermont town of similar size, whereas Wilmington's is much higher. These phenomena are determined by the pattern of urban development, although their relationships are poorly understood. For example, whether the towns' economy of skiing and second homes really pay back in tax revenues what they cost is not actually known. However, there is little doubt that forecasted sprawled growth will exacerbate the problem of public services, and that much better land-use options exist for the towns.
# Aesthetics, Natural Areas, Historic Sites
Section 12(8) of Act 250 requires a developer to consider his project's impact on scenic and natural beauty, aesthetics, historical sites, and rare and irreplaceable natural areas. Wilmington's and Dover's scenic values are comprised of many geophysical forms as described below in six categories:
1. The mountain range and its base. Mt. Pisgah and Haystack comprise the range, with gentle sloping lands at the base.
2. Ridge tops, steep slope highlands, highland plateaus, and terraces. Rice Hill, Whites Hill, Cooper Hill, Johnson Hill, and others are part of a high ridge system of ridge tops and steep slope highlands. Interspersed are highland plateaus and terraces.
3. Hillock-knoll-terrace, prominent hills. Wilmington's southeastern corner has a complicated low-relief topography comprised of hillocks, knolls, and terraces. Just north of this area, prominent hills such as Castle Hill accent major valley walls and lowland plateaus.
4. Lowland plateaus and terraces, gentle regular slopes. Lowland plateaus and terraces are found above major valley walls and below the mountain range. These are bounded or interlaced by gentle regular slopes isolated from other stronger geophysical forms.
5. Valleys. Tongues of small, V-shaped valleys are in the northeastern corner of Dover. Small but broader, U-shaped valleys are best developed in a band across the southeastern corner of Wilmington. The Deerfield Valley with its walls and rims is a major valley in both towns.
6. Water. Harriman Reservoir, Lake Raponda, Haystack Pond, smaller ponds, and the Deerfield River comprise this category.
Landscapes vary both in eminence and continuity, e.g., the mountain range and ridge tops are strong, whereas gentle regular slopes and terraces are weak. But, the strongest landscape is not always the most enduring. It may be vulnerable to human presence, as along ridge tops where even scattered residential subdivisions create broken teeth along the ridge line. To maintain the towns' natural landscapes, their tolerances to urbanization must be recognized.
* Mountain range, major valley walls, rims. Intolerant of urbanization. Any development would be broadly visible from miles away. The valley's continuity is highly vulnerable.
* Ridge tops, steep slope highlands. Visible from distant locations. All but very low-density development would destroy their continuity.
* Gentle, regular slopes. Not identified with other strong landscapes. Often, the immediate landscape near which many residents live. Very low-density development allowed as a conditional use if their identity can be maintained.
* Prominent hills, V-shaped valleys, Deerfield Valley. Modestly tolerant of low-density development, although their utilization must be designed to retain eminence and continuity.
* Highland plateaus and terraces, U-shaped valleys. Tolerant of urban development if sited carefully to be unobstrusive.
* Hillock-knoll-terrace. Tolerant because if properly utilized, they can conceal developments from one another.
* Lowland plateaus and terraces. Ideal locations for urban settlement. Their identity cannot be easily lost, and their relationship to adjacent landscapes offers diverse visual experiences.
The towns' spruce-balsam-fir forests, taken for granted by many Vermonters, are amenities cherished especially by non-Vermonters for their northern evergreen qualities. Existing stands deserve protection. Additionally, they are valuable winter wildlife cover, and are frequently underlain by wet soils. Consequently, these forests are poorly suited for urban development.
White-tailed deer are an amenity. In the winter they yard, i.e., they congregate in relatively protected areas with a food supply and where their movements are not hindered seriously by deep snow. These yards must be protected if white-tailed deer are to survive the winters. Similarly, other wildlife amenities are beaver colonies, streamside furbearers such as mink and otter, and sport fish, particularly trout.
Although Nell M. Kull has provided a beautiful account of Dover's history, neither town has a documentary study identifying the degree to which history is reflected in existing buildings and places. An architect-historian should be enlisted to produce a documented inventory. Yet, considerable anecdotal information on the history of individual dwellings resides in town residents, which should be recorded before it is lost permanently. The villages of both towns appear strongly reminiscent of the way they looked over 100 years ago, and they should be considered as special historical preservation districts.
# The Towns' Carrying Capacities
An approximation of the towns' carrying capacities can be expressed according to the populations supportable under stated conditions of sewage treatment and water supply (figure 15). Had adequate data been available, similar estimates would have been made for highways, education, and other social services.
Sewered Populations. Stream flow conditions in major tributaries were estimated for different times of the year. Of particular concern is late summer-early fall when stream flow is lowest and the allowable quantities of sewage in streams is most restricted by Act 252. During that period the towns can accommodate approximately 4,400 persons (at 100 gallons per day per capita) if a 30-to-1 dilution ratio is maintained. Stream flow is so low during that period that a relaxation of the dilution ratio to 15 to 1 still could not accommodate the forecasted 1975 population. This means that recycling and other innovative conservation measures are absolutely necessary to sustain a future permanent population with centralized sewage treatment systems. Yet, during the ski season when stream flows are near peak, perhaps 30,000 persons could be accommodated with a conventional sewage system without violating the 30-to-1 standard. Consequently, the concept of carrying capacity using a minimum dilution ratio becomes complicated by variable flows, indicating an immediate need for further research, particularly of the Deerfield and Harriman Reservoir, to determine the long-term effects of human wastes in the area's aquatic ecosystems.
Figure 15. Town carrying capacities compared to population forecasts.
Septic Tank Populations. Town carrying capacities based upon exclusive use of septic tanks and leach fields depend upon dwelling unit lot sizes which, in turn, are determined by soil/slope characteristics and system design. In the absence of adequate soils data, calculations must be based upon the extent of vacant private land with slopes under 15 percent, which is approximately 13,000 acres. Assuming one half of that acreage is used for residences (the remainder needed for dedicated open space, streets, utilities, etc.), a net density of one-acre lots (at 3.4 persons per dwelling unit) gives a carrying capacity of 22,100 persons. However, local experience suggests that large, one-acre lot subdivisions would contaminate ground and surface waters, and that two acres should be a minimum lot size, thus indicating a population of 11,050. Obviously, all estimates are speculative in the absence of detailed soils data.
Water Supply Populations. The largest potential single sources of water are Harriman and Somerset Reservoirs. Harriman could possibly support 500,000 people if it were used solely for water supply. Water-budget calculations show how groundwater recharge rates vary throughout the year, determined mainly by temperature and evapotranspiration, since monthly precipitation remains about constant. Consequently, the carrying capacity of groundwater supplies also varies, being lowest in late summer and early fall (about 24,700 persons at 100 gallons per capita) and much higher during the January through March ski season (about 113,000 persons). Spring groundwater supplies are highest (over 200,000 persons) while fall-early winter conditions are slightly higher than ski season. These figures are approximate, indicating magnitude of carrying capacity and not absolute limits. Both towns need a detailed hydrologic survey of groundwater, and should request immediate assistance from the Vermont Department of Water Resources.
# Intrinsic Suitability
It is now possible to assemble all of the spatial values previously identified with performance standards 12(1) through 12(8) of Act 250, to invoke a single expression of the towns' intrinsic suitabilities for prospective land uses. This final step constitutes what in effect is a local land-capability plan cited in paragraph 12(9) of Act 250, with which Act 250 applications must conform to receive a permit. Moreover, such a synthesis will represent an essential component of a local or regional plan cited in paragraph 12(10) of Act 250, the final requirement of an Act 250 application.
A summation or synthesis of spatial values reveals two sets of potential structures for Wilmington and Dover. One represents a structure for the natural or given environment, comprised of locations where nature is hazardous, stressful, or performing work for people. These locations should be managed in the interest of the public health, safety, and welfare of present and future generations. The other is a community structure, comprised of locations providing the maximum concurrence of propitious factors for human habitation. Future population growth should be induced into these locations by appropriate incentives and land-use controls. Finally, the towns' committed structure must be acknowledged, comprising both natural and community elements such as the Green Mountain National Forest and the Village of Wilmington. That structure must be considered as permanent and irrevocable. If the town residents, in fact, concur with the spatial values portrayed in the synthesis, they may then seek those measures by which the two intertwining environments of community and natural process can actually be realized.
Based upon available data, a potential structure for the natural environment is derived from the geographic distribution of primarily physiographic phenomena and scenic values. Ten districts are identified below in order of their need for preservation.
* Region 1. The fragile ecosystem above 2,500 feet and alluvial or muck soils, all located in landscapes (cited in the other regions below) intolerant to urban uses in varying degrees.
* Region 2. The fragile ecosystem above 2,500 feet, alluvial and muck soils, wherever they occur outside of the intolerant landscapes cited in Region 1.
* Region 3. Slopes over 15 percent located in landscapes very intolerant to urban uses, i.e., the Mountain Range and the walls and rims of the Deerfield Valley.
* Region 4. Landscapes very intolerant to urban uses, i.e., the Mountain Range, and walls and rims to the Deerfield Valley (excluding slopes over 15 percent of Region 3).
* Region 5. All slopes over 25 percent except those above elevation 2,500 feet on the Mountain Range, and the walls and rims of the Deerfield Valley, which are accounted for in Regions 1 through 4.
* Region 6.15 to 25 percent slopes in landscapes intolerant to urban uses, i.e., ridge tops, steep slope highlands, and gentle regular slopes.
* Region 7. Landscapes intolerant of urban uses, i.e., those given in Region 6 (excluding 15 to 25 percent slopes of Region 6 and slopes over 25 percent of Region 5).
* Region 8. 15 to 25 percent slopes in landscapes moderately intolerant of urban uses, i.e., the Deerfield Valley floor, minor V-shaped valleys, and prominent hills.
* Region 9. 15 to 25 percent slopes in landscapes tolerant to urban uses.
* Region 10. Landscapes moderately intolerant of urban uses (cited in Region 8) without any other known restrictions.
Some of the phenomena given in the regions above are already committed to preservation when located in relatively permanent undeveloped areas such as the Green Mountain National Forest and other public open spaces, golf courses, and ski slopes. Finally, it must be recognized that the ten regions are based upon scant data and can be expanded enormously with better information on the towns' natural phenomena and processes.
Given a choice, new or expanding communities would seek those locations with plentiful groundwater, favorable microclimates, good soils and slopes opportunity, and within convenient reach of available governmental services such as fire protection and health facilities. Available data suggest six suitable urban districts, described below in order of their desirability.
* District 1 (most desirable). Locations which are very accessible (i.e., within one-half mile of a village center), with slopes under 15 percent, favorable life zones, the highest probability for finding groundwater, and near an existing recreation area.
* District 2. Locations generally very accessible, or within one-half mile of a major or secondary road, with slopes under 15 percent, and adjacent to, or within one-half mile from an existing recreation area. Either groundwater or life zone conditions may be restrictive.
* District 3. Conditions same as District 2, but with added restrictions. Includes locations adjacent to a recreation area but with groundwater or life zone conditions are restrictive. Accessible locations, not adjacent to a recreation area, which have good groundwater and life zone conditions.
* District 4. Conditions similar to District 3, with added restrictions. Very accessible locations but having no other attributes except slopes under 15 percent. Accessible locations with variable site conditions and distances from existing recreation areas. Poor access locations with other conditions as good as District 1.
* District 5. Locations which mayor may not be accessible or adjacent to an existing recreation area, with slopes under 15 percent. Groundwater and/or life zone conditions are restrictive.
* District 6 (least desirable). Locations accessible or adjacent to an existing recreation area, without any other attributes except slopes under 15 percent. Locations with poor accessibility and variable site conditions.
These urban districts are based upon scant data and should be developed further with better information on natural processes and phenomena. Yet, they are sufficient to constitute a new urban structure for the towns whose benefits will exceed those of uncontrolled growth. Recent subdivisions have not chosen urban suitable locations, nor is it likely that future development will be more responsive. Therefore, a strategy must be devised to guide future growth.
# Concepts for Growth
The towns' economic health will continue to depend upon the second home and recreation market mainly because alternatives such as extractive industries, farming, logging, and most forms of manufacturing are not good prospects, at least during the next ten years. Specialized and finished product manufacturing aimed at markets on the East Coast may look favorably at Wilmington or Dover. However, these industries are also aggressively sought by other communities having larger labor forces and closer to metropolitan areas. Therefore, both towns must depend upon a recreation economy which, in turn, requires a beautiful unspoiled environment. This means that all future growth must conform to the towns' intrinsic suitabilities and that the pattern of trend growth as identified in this study must be avoided.
The next concern is how much growth should occur in Wilmington and Dover? It was Verne Howe, a member of the steering committee for this study, who finally gave focus to discussions between the committee and the consultant on the kinds of futures Wilmington and Dover residents really wanted. The issue, she said, was population growth, on which three conflicting opinions prevailed. One viewpoint is to minimize growth and keep Vermont rural. Another is to accommodate growth, since it seems inevitable. The third viewpoint is to encourage growth to improve the local economy.
Although it was beyond the study's scope, the consultant examined two of these viewpoints, i.e., to minimize or accommodate population growth as two different concept plans. An encouraged growth plan was not considered in that the population forecasts in both towns are so high that the differences between accommodating and encouraging growth is only a matter of degree.
# A Concept to Minimize Growth
Population growth can be minimized by restricting town services, by preventing increases in the towns' carrying capacities, and by discouraging the creation of new recreation opportunities which attract people. This means limited snow-removal services and road maintenance, no new highways, no new sewage systems, no enlargement of existing sewage treatment plants, no new public water-supply systems, no new ski areas, very restrictive town ordinances, and acquisition of development rights on lands imminently subject to urbanization. Rigorous enforcement of Act 252 is essential because it can stop high-density developments unable to use septic tanks or discharge wastes into surface waters. However, Act 252 will also indirectly stimulate low density sprawl if developers build septic tanks to avoid the high costs of sophisticated sewage disposal or recycling systems. Consequently, actions to minimize growth must also consider impacts upon settlement patterns. Settlement can take the form of sprawl, where development is discouraged everywhere but may happen almost anywhere, or be nucleated, where development is discouraged almost everywhere save for a few selected prime locations.
A sprawl settlement pattern resembles trend growth, previously identified, although it would not be quite so disastrous. The nucleated settlement pattern is far more preferable even for a minimize growth concept. It would bring the least disturbance to the landscape and would be much less of a burden on town services. However, it would require enormous public control or private self-restraint and cooperation to ensure that all lands conform with a nucleated plan. This concept is most feasible when only a few large landowners are involved. Wilmington and Dover have large landowners, but they also have many small landholdings. This plan also presents the problem that the few fortunate landowners holding lands identified as prime urban suitable can profit from development, whereas all other owners of unsuitable lands are deprived of such gains. A private real estate syndicate comprised of any number of landowners is one device whereby all participants can share in the profits from development. Landowners benefiting from the plan would share their profits with those who give up their rights to development.
To demonstrate the minimize-nucleated concept, two prime urban-suitable locations were selected, one north and the other south of the Village of Wilmington. These sites, totaling about 350 acres, are appropriate locations for new village centers accommodating some 6,500 persons, assuming the availability of waste recycling, or advanced techniques for sewage disposal. Other good locations having the same acreage could also be selected in a minimize growth plan. However, the two sites chosen appear to have more attributes than any other locations in the towns. They have extensive edges along adjacent areas proposed to remain in a natural state, high sewering efficiency, superior scenic value, and favorable life zones.
# A Concept to Accommodate Growth
This plan must accommodate a forecasted growth of about 9,000 to 12,000 new dwellings during the next ten years. The arguments for nucleating a major portion of that growth are even stronger than for the minimize growth plan. Moreover, if new construction relied exclusively on septic tanks, contamination of groundwaters and surface waters would be almost certain. Therefore, the density of a major part of this new growth must be sufficiently high to justify a new centralized sewage system. Expanded highway capacity, new public water supplies, and more town services are also implied.
Locations in the two highest categories of urban suitability were selected as centers for new growth to demonstrate an accommodate-nucleated growth concept. These locations reveal a new western corridor for future growth. It traverses both towns, situated between the Mountain Range and the Deerfield Valley, and extends from Mt. Snow to Harriman Reservoir. This corridor seems propitious for a year-long recreation economy based upon winter skiing and summer activities related to Harriman Reservoir. The developable area identified in the corridor includes 2,150 acres capable of supporting some 40,000 persons, assuming the availability of waste recycling or advanced techniques for sewage disposal.
Any plan to regulate and distribute population growth can only be so described if it contains the powers to realize its stated objectives. Therefore, an enumeration of those powers constitutes the essential ingredient of the planning process, i.e., a strategy for success.
# A Strategy for the Future
Recent urban development has been increasingly destructive to the towns' natural resources and ignorant of their intrinsic suitabilities. In the absence of new powers which can regulate growth, both towns will experience large increases in population which, in spite of existing state and town regulations, will select locations mainly in response to the real estate market. Positive values to human location, such as good slopes and soils, plentiful groundwater, favorable climates, scenic beauty, and proximity to recreation opportunity and municipal services may be ignored, while locations unsuitable for development may be utilized. This can only worsen town problems of polluted streams and reservoirs, inadequate water supplies, soil erosion, despoliation of scenic beauty and historic village centers, degradation of natural plant and animal communities, congested highways, rising costs of municipal services, and higher taxes.
But other options exist. Future growth need not be destructive. Once understanding the towns as composed of natural systems constituting values either favorable or unfavorable to human location, two intertwining fabrics emerge. One is a natural structure where nature is hazardous, stressful, or performing work for people. The other is a community structure where nature beckons people and offers in varying degrees those environmental attributes from which they can benefit. This being the case, Wilmington and Dover with assistance from the State of Vermont and the Windham Regional Planning and Development Commission, must discourage future development from occurring in those locations which should remain in a natural or near-natural state, while inducing it to select those locations suited for such purposes. That can be achieved only through a strategy employing every device available in both the public and private sectors.
# An Explicit, Replicable Public Planning Process
The first essential component of such a strategy is an explicit, replicable public planning process, as has been initiated by this study. It rests upon the simple proposition that good judgment requires good evidence. That is, given adequate information on the climate, geology, physiography, groundwaters and surface waters, soils, plants and animals, and equivalent socioeconomic data, it becomes possible to discern the place's opportunities and constraints and its intrinsic suitabilities for all prospective uses. This study was produced with a paucity of such evidence. Much better data are urgently needed. The four highest information priorities are: a new U.S. Geological Survey map (at 1:24,000 scale) to be used for plotting slopes and existing land uses; a modern Soil Conservation Service soil survey to identify opportunities and constraints of all soil types; a survey of urban tolerances of the aquatic ecosystem comprised of Harriman Reservoir and its drainage area; and a detailed groundwater investigation to identify the towns' groundwater resources and their management needs.
However, this study has produced sufficient evidence to make important judgments about the towns' futures. It reposes in maps and accompanying text which must be declared as public documents which, over time, can be enriched and corrected as better information becomes available. They can be recognized as elements of town plans as prescribed by the Vermont Planning and Development Act or by any other device that assures their prominent recognition in planning decisions.
All future planning must be the product of an explicit and replicable procedure existing at all levels of Vermont government which honestly appraises the social costs and benefits of all land-use proposals. Such a procedure has already been initiated by the federal and state governments through the National Environmental Policy Act and Vermont Act 250, both of which require environmental impact statements. Clearly, an equivalent system must be established at the town level. These decision-making procedures must be integrated at all government levels and employed to take a fresh look at highway and sewage problems and other land use issues requiring immediate solutions in the towns.
A new Highway Route 100 as proposed by the Vermont Highway Department should not be accepted until it is demonstrated that local improvements to the existing road system cannot satisfactorily increase carrying capacities. If the new highway can be justified, its alignment must conform to the intrinsic suitabilities of the towns, as identified in this study.
Fire districts should be abandoned as sewage planning areas, and a larger district encompassing the ecosystem of Harriman Reservoir should be established. In the absence of better information on the urban tolerances of that ecosystem, existing sewage discharges should be phased out in favor of recycling.
# Regulation by the State
The State of Vermont administers a large group of environmental regulations to protect the health, safety, and welfare of the public. Those regulations set certain performance standards as a condition for obtaining a permit. Waste discharge, land-under-water, and stream alteration permits are the responsibility of the Department of Water Resources. A storage permit for flammable liquids must be obtained from the Department of Public Safety. The Department of Highways gives permits for junkyards and highway access. Dam permits are obtained from the Water Resources Board. Operation of restaurants, hotels, and sanitary landfills are licensed by the Division of Environmental protection. Subdivision permits are administered by the Board of Health, while public water supply permits are provided by the Division of Health.
Vermont Acts 250 and 252 affect Wilmington and Dover more than any other state regulations. In the absence of town statutes which consider ecological realities, those acts should be employed as the major instruments for environmental protection. Act 252 and Section 12a(1) of Act 250 can be employed to prevent sewage effluents from being discharged into locations other than those which already have discharge permits. Those same acts can be employed to prevent increases in discharge volumes at locations having such permits, prior to a detailed investigation of the urban tolerances of the Harriman Reservoir ecosystem. Quality of sewage treatment should be upgraded immediately to meet state standards.
Under Section 12a(2) of Act 250, no developer using groundwater for his supply of water should be issued a permit prior to providing a record of pumping tests indicating yields per foot of drawdown of the groundwater table. This is necessary because of the inherent limitations of the region's groundwater resources to provide sustained yields, as well as the quality limitations of Wilmington's public supplies. Flood plains, peat and muck soils, sand and gravel aquifers, steep slopes over 15 percent, and ecologically fragile areas above 2,500 feet elevation, as identified in this study, should be prohibited from most developments, under Section 12a(4) of Act 250. This is necessary to prevent soil erosion and maintain the towns' water-holding capacity.
Areas scenically intolerant to urban uses should be exempted wherever possible from future development, employing Section 12a(8) of Act 250. These include the mountain range comprised of Mt. Pisgah and Haystack Mountain; the major valley walls and rims of the Deerfield Valley; the high ridge system comprised of Rich Hill, Whites Hill, Cooper Hill, Johnson Hill, and others; gentle regular slopes separated from other strong landscapes, prominent hills; and V-shaped valleys, Section 12a(8) of Act 250 should also be used to prevent future developments from destroying winter deeryards and beaver colonies, and places of historical significance such as the old village centers of Wilmington, West Dover, Dover, and East Dover.
The intrinsic suitability and synthesis maps produced by this study complement and enrich the interim capability plan prepared by the State Planning Office. They should be adopted as local capability plans by Wilmington and Dover, and, thereafter, adopted by the Windham Regional Planning and Development Commission. They may then be used by the District Environmental Commission to require conformance by all applicants, in accordance with Act 250, Section 12a(9).
# Regulation by Wilmington and Dover
An inherent conflict must be recognized between sophisticated state regulations, such as Act 250, and the strong desires of towns to determine their own futures. Given the findings of this study, it is now possible for Wilmington and Dover to make their own laws to prevent material, irreparable, and permanent damage to their natural resources. These laws should be prepared by attorneys experienced in the field of environmental legislation and litigation.
A proposal to draft town statutes has been prepared by Victor J. Yannacone, Jr., Esq., of Patchogue, New York, and Arthur Palmer, Esq., of Cold Spring Harbor, New York, in collaboration with the consultant. Their proposal, already submitted to Wilmington and Dover, recommends the enactment of a new group of town laws derived from this study, which would preserve the ecologically fragile elements of both towns and those natural resources of regional significance. Initially, it is suggested that the towns enact an environmental policy resolution, followed by an environmental protection ordinance designating those town officials responsible for carrying out the policies enumerated in the policy resolution. The ordinance would require certain future developments to be authorized on the basis of a statement of environmental impact, to be submitted to the town. Additional town statutes would then be drafted and enacted individually to cover specific environmental subjects such as water quality, water supply, fragile areas, soils and erosion, plants, and animals.
These protection statutes would, in turn, be followed by a building code incorporating ecological factors and subsequent ordinances to direct future growth. They may include a system of penalties and credits wherein development responsive to natural processes is permitted multiple land uses and higher density, whereas development which contravenes ecological values be required to build remedial works and be constrained as to land use and density. Similarly, existing zoning ordinances prescribing one-acre and commercial districts bearing no resemblance to the towns' natural geometry should be superseded by new land-use districts. Regulations for the new districts should provide for flexibility in design, building coverage, and combinations of building types to reflect intrinsic suitability.
# Tax Policies
Gerald Witherspoon, former Vermont Commissioner of Taxes, has offered several recommendations in his 1969 report on Vermont Tax Policy, which should be adopted.
Every permanent housing unit employed as a principal place of residence should be entitled to a $4,000 homestead exemption, as a means of reducing property taxes. Newly constructed housing units would be qualified for this exemption only if they conform to a voter-approved land-use plan. Agriculture should be encouraged by allowing double household exemptions to farmers.
Rural landowners forced to sell their lands because they are in an area of changing land use and rising taxes (e.g., on the fringe of an expanding ski area) should have an option of deferring, with interest, the excess tax attributable to the changing conditions until such time as their properties are sold. A special capital gains tax should be applied to profits from the sale of land, in which the tax rate decreases with the number of years the property is held. Proceeds from the tax would be earmarked for land-use planning and conservation purposes.
Individual property owners should be eligible for a preferential assessment if they restrict their lands from development by entering into restrictive use convenants with the towns.
# Public Land Acquisition and Disposition
The Green Mountain National Forest should be invited to acquire lands within its jurisdiction, identified as requiring environmental protection. Both the state and Wilmington and Dover are empowered to acquire lands in fee simple, less-than-fee, and by lease. This method should be employed to preserve prime resources such as streamside areas and lakeshores, and other features requiring environmental protection where the use of noncompensatory regulations is not feasible. Towards that end, the towns should accept and manage gifts of land and historic buildings. The proposed Vermont Land Acquisition and Development Agency should be pursued by the towns as a vehicle to induce urban growth into urban suitable locations. Should that be unfruitful, the towns may create their own public corporations to acquire and improve lands, conveying these to private enterprise for appropriate uses.
# Private Action
The final component of the strategy is concerted action by the private sector. Private foundations, trusts, and other civic organizations must be encouraged to acquire and maintain lands and buildings for public purposes. In their own interests landowners should undertake multi-lateral voluntary agreements to save their properties from despoliation. Landowners should consider establishing real estate syndicates to control the timing and location of future development and to secure an equitable distribution of profits from development, while concurrently protecting the towns from the ravages of uncontrolled growth. This private organization would purchase and develop lands especially suited for urban purposes. Part of the profits from this enterprise would be devoted to purchasing development rights of those properties which should remain in a natural state as identified in this study. Developers whose subdivisions are already platted must be urged by the towns and citizens groups to revise their plans wherever possible in conformance with the intrinsic suitabilities of their properties.
In conclusion, the future of Wilmington and Dover must not rest upon traditional zoning and subdivision regulations. These controls are insufficient. Both towns must undertake a new strategy, utilizing a new group of public and private devices to direct urban growth.
# Epilogue
Wilmington and Dover are on the edge of disaster. The second home invasion threatens to engulf them, destroying as it goes. A few will benefit at the expense of many. But you did not need a study to discover this, although it may have given more authority and precision to the dimensions of the threat. Perhaps a thousand new permanent households and ten thousand new vacation homes!
What can we do about it? The first step was a good one. Make a plan. Moreover, the planning process includes a large number of townspeople who learned much of the region in developing data for the study. There is now an informed electorate. It has participated in a study which now emerges as a document. What does it say? Where now? First, it has revealed the towns as a natural system with supervening values both in nature and in human activities. It has revealed areas where there is a hazard to life and health, intolerant areas and those of high social value. The powers reposing in Acts 250 and 252 have been invoked to ensure that development be so regulated as to protect people and land.
Next, it assembles all of the factors, which together constitute the best places for development, and locates these in the towns. If development will occur here these are the locations where it can cause at once the least despoliation of natural resources, provide the best place to live and play, and add the greatest enhancement to the towns. But law operates best to constrain. Can development be deflected from sprawl, from poor areas into nucleated communities in the most propitious situations? Acts 250 and 252 can be used to exclude development from unsuitable locations. Town ordinances can give these areas greater precision and power. Moreover, positive actions based upon public expenditures on highways, water, sewers, and treatment plants can be expressly designed to induce development into preferred areas. The plan designating areas intrinsically suitable for development can also have a positive effect. Those who wish to develop, can identify locations where such development will receive public support, where it will not be destructive and may be enhancing. This too should act as a positive inducement.
Where should development be located? The sites are clearly revealed. They provide the maximum edge to recreational opportunity and scenic value, the best climate areas, the most propitious factors of slope, soils, water, and accessibility. Here people can build new complementary communities, employing the best sites in the towns, enhancing them with buildings, places, and spaces consonant with the land, the people, and their history.
It takes no high intelligence to accomplish this, no great art to locate this marriage of people and the land. Modest architecture, landscape architecture, and planning should be able to achieve this end. Much more difficult is the task of halting the wave of despoliation, stopping "hit and run" development, initiating ecological ordinances, and enforcing them, indeed, changing the image of the towns as tolerant to inchoate, shabby building, to that of an effective community which insists upon the best practices of development.
The approval of this plan in public meetings and the invocation of Acts 250 and 252 to realize it are a splendid beginning. The formulation of town ordinances regulating development and prohibiting it from unpropitious areas is the right step, one of enormous significance. The continuous enrichment of data and their interpretation is a further objective. Reorganization of sewer districts, expansion of the National Forests, intelligent planning for highways and sewers, can all provide beneficial results. But the most important task is the gradual transformation of the image of the towns as a vulnerable place for casual second home sites, to a highly organized community where only excellence is permitted. This requires greater vigilance, dedication, and persistence in the planning process than has been required in the decision to initiate a study or to conduct it. The benefits are great—the maintenance of a beautiful landscape and stable communities. The threats are greater—a rich, diverse, and beautiful landscape quickly transformed into Sprawl City, U.S.A. It is unthinkable, yet it is a most likely future. The postwar decades are testimony to the numberless cases of good people on beautiful landscapes who have been engulfed and destroyed by growth. What reason is there for Dover and Wilmington to succeed where failure is normal fate? There is reason, an informed electorate, a plan, regulating powers, and a powerful instinct to establish a landmark for the nation to see. Good new communities placed on beautiful landscape—people and nature. We have been proud to share in this adventure, we wish it well. America needs an example of intelligent planning, responsive to nature's values, deferential to her constraints. Dover and Wilmington may indeed be a landmark. We hope so.
# Credits
Principal Project Advisors: Benjamin Huffman (State Planning Office), William Schmidt (Executive Director, Windham Regional Planning and Development Commission)
Project Coordinator: Ellen Reiss (Windham Regional Planning and Development Commission)
Project Steering Committee: Laura Heller, Verne Howe, Robert McCafferty (Chairman), Henry Meyer, Elva Turner, Wesley Ward, Albert White, Rodney Williams, Peter Zilliacus
Wilmington Planning Commission: Denise Allen, Peter Barton, Philip Davis, Verne Howe, Michael Kimack, Lynd Kimball (former position), Henri Logher (former position), Robert McCafferty (former position), John McLeod, Henry Meyer, William Palumbo, Lee Rich (former position)
Dover Planning Commission: Donald Albano, Dwight Blakeslee, John Christie, Robert Jalbert, Richard Joyce, John Nisbet, Ellen Reiss, Elva Turner
Wilmington Selectmen: Ervine Bishop (former position), Kingsley Dumont, Robert Grinold, Merril Haynes, Earle Howe (former position)
Dover Selectmen: Rexford Bartlett (former position), Stephen Chontos, Richard Joyce, Henry Sweeney
General Advisors: Robert Davidson (Extension Service), James Edgerton (Extension Service), Schuylar Jackson (Agency of Environmental Conservation), Bernard Johnson (State Planning Office), Forest Orr (Interagency Council), John Plonski (Town Manager, Wilmington), Alden Rollins (former Town Manager, Dover), William Stone (Extension Service), Joseph Teller (Windham Regional Planning and Development Commission), Charles Trebbe (former Town Manager, Wilmington), Robert Williams (Agency of Environmental Conservation)
Office and Administration: Janet Anderson (Secretary, Windham Regional Planning and Development Commission), Karen Davis (Summer Intern), Riley Robinson (Student Aide), Ellen Skelton (Secretary, Windham Regional Planning and Development Commission), Lucy Walkonen (Secretarial Assistant)
Climatic Data and Analysis: Frances Bond, John Christie, Guido Cimonetti, Ernest Gay, Robert Lautenheizer, John McArthur, Phyllis Shippee, Morey Storm, Oscar Tenenbaum, Philip Wagner, Marshall Witten, Esq.
Soils Data and Analysis: Frances Bond, Robert Brigham, David Gilman, David Grimwood, Larry Hamel, Earle Howe, Jasper Howe, Marshall Howe, Ray Pestle, Al Tallerico, Charles Turner, Elva Turner, Bruce Watson
Geology Data and Analysis: Charles Doll, Grant Moyer, Charles Ratte
Hydrology Data and Analysis: Robert Aiken, William Albert, Chris Allen, Joseph Anderson, Thaddeus Betts, Klaus Boettcher, David Coburn, Barbara Cole, Bruce Cole, Dean Davis, Bob Dufresne, Helen Dumont, David Gavett, George Gordon, Ralph Haslund, Mary Horan, John Howe, Verne Howe, Wesley Howe, Jim Hudson, Rachael Hurlbutt, Gerry Jenkins, Gerald James, Martin Johnson, Richard Joyce, Joseph King, Karen Larson, Joseph Lassiter, John McCloud, Ted Nelson, Charlotte Peck, Pat Potter, Richard Rader, Charles Ratte, Edward Rice, Tonia Rolf, Winifred Sargent, Tim Sockett, Charles Towle, Robert Wernecke
Wildlife Habitats: Richard Ackerman, Arthur Ball, Gilbert Cameron, Kenneth Davis, Helen duMont, Russell Hanson, Halsey Hicks, Alonzo Hodgedon, Roy Hood, Al Johnson, Peter Johnson, Eugene Keenan, Edward Kehoe, Richard Kenyon, Floyd Marieta, James McMartin, Charlotte Peck, John Poor, Wayne Rowell, John Stevens, Hubert Vogelman
Socioeconomic Data and Analysis: Stephen Anderson, Robert Augenstern, Emerson Baker, Justin Brande, Elizabeth Brown, Lensey Cole, W. Thompson Cullen, Gale Day, Jefferson Eaton, Thomas Farmer, Esq., Porter Farwell, Margaret Garland, Arthur Goss, John Gray, Margaret Greene, Russell Hanson, Hugh Henry, Rufus Horton, Ralph Howe, Wayne Kivi, Arthur Knight, Nell Kull, David Lyons, John Marshall, Robert Matteson, Howard Mitchell, James Monahan, Elbert Moulton, Richard Nicholls, Gerald Osier, William Palumbo, Ivor Pelsue, Peter Plastridge, Arthur Ristau, David Rockwell, Fred Sargent, Olin Stephens III, Robert Steward, Richard Thomas, Clarence Truesdell, Elva Turner, Jack Ware, John Williams, Robert Wilson, Betty Wolf, Lawrence Wright, Peter Zilliacus
Report Photographs: Alan Seymour, West Dover
# Notes
A development is the construction or improvements on a tract of land for any purpose other than residential farming, logging, or forestry. It applies to properties one acre or over in a municipality not having permanent zoning and subdivision regulations, and ten acres or more for municipalities having those permanent regulations.
Rule 1: Policy. Regulations covering Water Classification and Control of Quality, State of Vermont Agency of Environmental Conservation and Water Resources Board, May 1971
10 V.S.A., Chapter 33, paragraph 910(a).
Ibid. Rule 10(a).
Ibid. Rule 10(3).
WMRT examined a stream disposal waste credit system, assuming that every landowner had an equal right to put wastes in the streams at a 30 to 1 dilution ratio. To exceed that amount, a landowner would have to buy pollution rights (or credits) from other landowners in his sewage disposal district.
18 V.S.A., paragraphs 102 and 1203, and Vermont Health Regulations, Chapter 5, Sanitary Engineering.
Other conditions must also be met. See Vermont Health Regulations, Chapter 5, Subchapter 10.
Under the supervision of Dr. Charles Ratte of Windham College, groundwater studies were initiated during the summer of 1971. However, important data such as well yields/unit time/feet drawdown, were not available for this study.
Twenty-five percent is a judgmental figure. The consequences of slightly higher or lower percentages are not known.
Assuming 100 gallons per day per person, and three persons per house: 300 gallons per day divided by 50.6 gallons per day per acre equals approximately six acres.
Assuming 100 gallons per day per person, and six persons per house, 600 gallons per day divided by 233 gallons per day equals about 2.5 acres.
National Community Water Supply Study for the State of Vermont, by the Public Health Service, 1970, p. 21.
Vogelman, H., J. Marvin, and M. McCormack. 1969. "Ecology of the Higher Elevations in the Green Mountains of Vermont." A report to the Governor's Commission on Environmental Control, Burlington, Vermont (7 pages, mimeo).
Agency of Environmental Conservation Memorandum, "Development of Criteria," Section 12, Act 250, October 30,1970.
Ibid.
LeSourd, David A. and Sinclair, Robert O. 1964. State and Local Taxation and Finance in Vermont. Burlington: Agricultural Experiment Station, University of Vermont.
Sinclair, R. O. 1965. Procedure for Comparing Vermont Towns in Terms of Local Tax Base, Taxes Paid, and Effort. Burlington: Agricultural Experiment Station, University of Vermont.
Kull, Nell M. 1961. History of Dover. Brattleboro, Vt.: Book Cellar.
# 22
# Ecological Plumbing for the Texas Coastal Plain (1975)
with Jonathan Sutton
The Woodlands is one of McHarg's most influential projects. It is the best example of ecologically based new town planning in the United States during the 1970s. Today the approach might be called "sustainable development." As with current efforts toward sustainability, The Woodlands' plan sought not only environmental sustainability, but also economic and social balance. In fact, the two latter objectives may have been achieved better overall in The Woodlands. Ecological values have faired somewhat less well, outside the initial development with the especially strong McHarg imprint. The Woodlands remains an important model for new community planning.
In 28 square miles of forest, north of Houston, a new city—The Woodlands, Texas—is taking shape (figure 16). It is a unique experiment. The plan of the new city is based on a comprehensive ecological study; indeed, it is the first city plan produced by ecological planning. Yet, its uniqueness will not be immediately visible. The housing groups contain pleasant and decorous homes set in the forest. The only exceptional fact is that the trees and shrubs will be so close to homes. Although not immediately apparent, the unique qualities of The Woodlands are pervasive and they derive from its ecological plan.
The planning process for The Woodlands presented a novel experience for Wallace, McHarg, Roberts, and Todd (WMRT) since the initiation of work in 1971 as part of a team of consultants planning an 18,000-acre new town for the Mitchell Energy and Development Corporation. It consisted of a four-year relationship beginning with a study of the region, selection of the new town site, and the development of principles applicable to a single house or lot. The Woodlands job presents a planning method, developed during the project which is overt, explicit, and replicable.
Figure 16. Regional context of The Woodlands, Texas.
Having accumulated and interpreted the biophysical data describing the region and the 18,000-acre site, a method was developed which ensured that anyone who employed the data and the method would reach the same conclusions. Moreover, the data and method were printed in four technical reports, describing the ecological inventory and interpretation as well as the ecological plans and consequential principles to be employed in land and site planning. Thus any engineer, architect, landscape architect, developer, and the client himself were bound by the data and the method.
Because of its comprehensive analysis from the region to the residential setting, The Woodlands project represents a generic study with broad application to the entire physiographic region of the coastal plain from Long Island, New York, to Florida and Texas. In this region there is a wide commonalty of geophysical and biological phenomena.
# Challenges of The Woodlands Site
The Woodlands site presented many problems. First, it is almost entirely forest—at once attractive, but also a difficult environment in which to build. Secondly, it is flat, indeed, so flat that local wisdom holds that you cannot predict where water will run unless you know which way the wind is blowing. Next, a high percentage of the soils are poorly drained. And finally, the site's streams are characterized by very low base flow and very high peak flows causing excessively broad and shallow flood plains in the flat topography. Nearly one-third of the site is within the 100-year flood plains of Panther, Bear, and Spring Creeks.
A crucial ecological concern was how to preserve the woodland environment while draining the land for the new community. Orthodox drainage would have required destruction of the forest in order to lay drain tiles.
As most landscape architects and planners know, engineering principles related to storm drainage throughout the United States were largely developed in the northeastern cities and were appropriate to conditions of the crystalline piedmont on which these cities were located. These principles emphasize accelerating runoff and disposal of this runoff in piped systems. An elaborate vocabulary of equipment and techniques composes this drainage technology.
The coastal plain, a great groundwater resource, requires the opposite approach, often inviting retardation of runoff to maximize recharge. Ecological planning for the coastal plain suggests solutions contrary to the orthodoxy of engineering. This was dramatically true of The Woodlands. It therefore became necessary not only to create an ecological planning method appropriate to this unique and difficult region but also to demonstrate its efficacy and economy to dubious engineers. Its effectiveness has now been clearly demonstrated. The elimination of a piped storm-water drainage system as anti-ecological and excessively expensive was an important factor in the engineers' conversion to ecological planning. The development of an inexpensive natural drainage system was a necessary corollary (figure 17).
Figure 17. The Woodlands hydrological cycle.
The economic benefits of this approach were clearly demonstrated in the first engineering cost estimates for the storm drainage system of the new city. The costs of a natural drainage system were compared by the engineers with those of the conventional method of site drainage (Turner, Collie and Braden, Houston). The construction cost calculations for the storm sewers required for the conventional system were $18,679,300 while the natural drainage system costs were $4,200,400, an impressive saving of $14,478,900. Such figures accelerate conversion to ecological principles. There is no better union than virtue and profit.
# The First General Plan
In 1970 WMRT was retained to undertake an ecological study for The Woodlands New Town. Gladstone Associates were retained to prepare market analyses. Richard Brown provided engineering studies and William L. Periera was responsible for land-use planning. Robert Hartsfield was the coordinating planner for the client. It is fair to say that the ecological studies were contracted less with a profound conviction of their necessity than as a concession to public environmental consciousness. The immediate objectives were three-fold: to ascertain market feasibility, to design a plan consonant with the market and environmental opportunities of the site, and to apply for a U.S. Department of Housing and Urban Development (HUD) Title VII loan guarantee of $50 million.
The solution to these problems became so critical to the feasibility of the project that the ecological study moved from inconsequence to dominance. It determined the form of The Woodlands. Moreover, the HUD review process concentrated upon environmental factors. The ecological study was conducted simultaneously as a planning and environmental impact analysis and this contributed largely to the guarantee of $50 million in 1971.
All program allocation evolved from the ecological imperative that required maximizing recharge, protection of permeable soils, maintenance of water table, diminution of runoff, retardation of erosion and siltation, increase in base flow of streams, and the protection of natural vegetation and wildlife habitats. The satisfaction of these requirements provided the major structure of the plan and became influential at the smallest scale of development.
In the general plan for The Woodlands, arterial and collector roads were sited on ridge lines away from drainage areas. Development density was generally most intense near major roads and intersections and decreased with distance from these locations. Intensive development was located on areas of impermeable soils (for example, the Metro Core). Minor residential streets were used as berms perpendicular to the slope of the site to impede flow over excessively permeable soils. Design solutions such as the installation of permeable paving were recommended to increase the storage of storm water. (An experimental parking lot with porous paving has been constructed at the Commercial Leisure Center—the village focus for The Woodlands' First Phase—and is part of a monitoring program established by Rice University with grants from the U.S. Environmental Protection Agency and The Woodlands Development Corporation.)
# Phase I Planning
The next engagement of WMRT in The Woodlands involved the first phase of development—an area of approximately 2,000 acres which was called Grogan's Mill Village. Detailed soil and vegetation surveys were computerized, from both field surveys and false infra-red photography, and one-foot contour interval topographic maps were prepared.
At this point three members of WMRT refined the concept of natural drainage, their objective to develop a method utilizing the attributes of the existing drainage system—streams, swales, ponds, natural vegetation (with its litter layer), the storage capabilities of soils—and to enhance this network by designing all infrastructure to satisfy water management objectives. Impoundments, settlement ponds and basins, berms to encourage recharge, golf course construction, highways, roads, and streets were all considered as adaptive strategies to meet demands of a natural drainage system.
Having determined the structure's essential concept, it became necessary to quantify it. Elaborate studies were conducted to ascertain the contribution of each soil/vegetation type to the proposed water regimen. This investigation developed into the description of coverage/clearing ratios.
# Determination of Landscape Tolerance as Established by Coverage/Clearing Ratios
Landscape tolerance is an index based on the requirements of the natural drainage system and the quality and adaptability of vegetation types. The measure of landscape tolerance is the percentage of a given soil and vegetation type which may be covered by impervious surfaces or cleared.
Permissible coverage is that fraction of an area which can be rendered impervious without affecting the ability of the remaining soil to absorb local runoff from high frequency storms (one inch in six hours). The amount of permissible coverage was derived from the excess storage capacity of each soil (down slope), which in turn was based on the depth of the upper pervious layer, its percolation rate and the height of the seasonal high water table and direction of slope (table 15).
Table 15. Coverage Disturbance Percentages
Soil groups A and B were termed "Recharge Soils" and depending on their location were recommended for protection. Permissible clearing for each vegetation type reflects its quality and landscape value, its importance as a wildlife food source, and its tolerance to compaction and development activities. Permissible clearing is expressed as a percentage of area which may be cleared. Since both permissible coverage and permissible clearing are expressed as percentages, they were readily combined on one chart which served as an index of landscape tolerance.
# Determination of Land-Use Impacts as Measured by Coverage/Clearing
Commercial uses, residential types and densities, golf courses, and roads were examined for their landscape impacts. Each use was evaluated in terms of the amount of impervious surface and clearing involved. Expressed as a percentage of a given unit of land, this coverage-to-clearing ratio measured development impact.
A study of the areas of impervious coverage and minimum clearing of 5, 10, or 15 feet around all building types (depending upon number of stories and construction technique for different densities) shows that coverage and clearing generally rise per unit area with higher densities. In order to check this conclusion against a wider sampling of developments, coverage and clearing as a percentage of total development area were taken from 22 representative site plans. The plans were published in various forms, and densities varied from 1 to 20 dwelling units per acre. All schemes analyzed relied on at-grade parking.
# Design Synthesis
In order to match landscape tolerance with development impact a design synthesis base was created to illustrate the distribution of landscape tolerance. In addition, this base kept the land-planning process up-to-date and made it site-specific. A base map was developed which included a composite of all the salient ecological data. The synthesis involved a method of adding ecological factors in stages so that the contribution of each stage could be properly understood. The data were added to the map in sequence:
1. Field topography (1-foot contours)
2. Drainage pattern (watershed boundaries)
3. Soil types
4. Vegetation types
5. Adaptive strategies
Adaptive strategies include minimum open-space corridors associated with the 25-year flood plain or the minimum of 300-feet-wide drainage easements and storm water storage areas, which for the most part take the form of Waller ponds (shallow ponds, predominantly open, occasionally forested, which are situated on the impermeable Waller soils). Degrees of permissible coverage/clearing were established, based on the coincidence of soil types, vegetation types, and drainage patterns. These ranged from preservation to nearly total modification. The four steps leading to delineation of tolerance ranges in the design synthesis are illustrated:
1. Natural drainage system
2. Recharge soils
3. Prime vegetation
4. Impermeable soils, flat land less than 1 percent, and permeable soils and sloping land greater than 1 percent (to be developed according to the percentages indicated on the Coverage/Clearing Chart, Table 15)
Step 4 makes a distinction between land which is relatively well drained (because it has either permeable soils or slopes greater than one percent) and flat, poorly drained, impermeable areas. One percent slope was adopted as the limit for positive drainage in a forest with a little layer. Therefore, impermeable soils on slopes less than one percent are considered poorly drained. The soil's ability to drain affects directly the permissible coverage/clearing percentages. The Design Synthesis Map shows the essential structure of Phase I on which the various program elements were subsequently tested.
# Matching Development Impacts with Landscape Tolerance
The combination of landscape tolerance (permitted coverage/clearing) of given soil and vegetation types with land-use requirements (required coverage/clearing) yielded an allocation of development impacts in a gradient from complete protection to extensive modification. Development types having extensive coverage/clearing impacts were matched with soils of low permeability and vegetation of low quality, and development types having slight coverage/clearing impacts were matched with soils of excess storage capacity for surface water and/or vegetation of high quality.
Each level of constraint required some additional development cost such as for increased drainage improvements, fill, or landscape treatment. An opportunity implied a decreased or normal development cost. The costs were then compared with the anticipated value created by both natural and human-made amenities. The final synthesis revealed the least cost/greatest benefit solution on the basis of the existing landscape characteristics.
There were persuasive social or economic factors that modified the design synthesis—altering locations of roads, golf courses, and shopping areas. Detailed vegetation studies caused us further to modify the plan. We calculated the acreage of each soil and vegetation type within each watershed to find its "development capacity"—i.e., the maximum area that should be covered and cleared. For example, the Grogan's Mill Village center was located directly upon impermeable soils—rather than over one of the absorptive recharge areas. We put all golf courses to a similar test—balancing development opportunities against costs to the environment.
Such were the steps leading to the land-use plan, which was derived from these structural elements: (1) drainage ways; (2) road net; (3) golf courses; (4) community facilities; and (5) pedestrian paths.
This plan showed us those areas suitable to become development parcels. To each of these parcels we assigned a "program module" depending on their aggregated landscape tolerance, surrounding amenities, physical configuration, marketing potential, and staging strategy. The exercise of matching "program modules" or density types with specific sites in Phase I (1,947 acre) suggested 4,350 dwelling units on 1,255 developable acres, including community facilities, a business park, and the commercial leisure center (note this reserved 692 acres as open space). In order to ensure a sufficient pace and market response in Phase I much of the program called for single-family lots.
# Plan for the Village of Grogan's Mill
The residential areas of The Village of Grogan's Mill are organized around the Community Leisure Center (CLC), which includes commercial and office facilities, a community conference center, and sites for recreational activities. This large package of leisure facilities will give the CLC a catchment area well outside The Woodlands. It will supply the community focus and marketing impetus for Phase I.
The neighborhood in Zone I relates to the championship golf course, country club, tennis courts, and conference center. The neighborhood in Zone 3 is focused on the Sawmill Community Center at the corner of Grogan's Mill and Sawmill Roads. The public golf course, the existing Lamar Elementary School, with adjacent Tamina Mill Community Center and proposed park, provide neighborhood facilities and identity in Zone 4.
The three neighborhoods will be linked to the CLC by six miles of pathways for pedestrians and bicycles. The rights-of-way of the pedestrian system will provide at some future point the option of an alternative mode of vehicular transportation to the automobile. Grade separated crossings of Grogan's Mill Road are planned at the CLC and at the Sawmill Community Center to reduce conflict between the pathways and the arterial road.
# Guidelines for Site Planning
In order to carry out the actual "fitting" of development types to specific site conditions, a decision tree was devised. Using summary sheets for hydrology, limnology, soils, vegetation, wildlife, and climate, the method reviewed the critical site conditions revealed during the ecological inventory. It also suggested the environmental objectives and physical adaptations related to each natural phenomenon. The main concern of the manual, though, is with the natural drainage system and the preservation of the woodland environment. This is to be achieved through careful organization and communication of site planning information for each development parcel.
The key method was developed to identify specific site conditions and relate them to design strategies. Ecological data related to soils and vegetation have been organized in a form which can be readily utilized by planners and designers. A manual outlines the step-by-step process of applying selected criteria to a particular site. Given the criteria and the procedure, all consultants who employ the method can produce a variety of designs, all of which will satisfy the stated environmental objectives. The outcome of the process provides only the framework for the plan or design. The quality and character of the final product will still depend on the ingenuity of the individual site planner.
# Site Design
The execution of a site plan for two parcels in Grogan's Mill Village illustrates how the key method of the guidelines is employed (figure 18). The parcels in question contain a total of 48 acres, separated by a residential collector road and bounded by golf fairways and a primary drainage easement.
Figure 18. After guidelines were applied to two parcels of 48 acres, the above site plan resulted. The drainage system has been designed so that the roads perpendicular to the slope of each parcel intercept and direct runoff to points easily drained.
Taken together the sites were to form a sub-community of distinctive high- and medium-priced homes. Of the total acreage all except the 50 feet vegetation easements along Millrun Drive (to be assigned to the community association) and the land required for internal streets was to be converted to deeded home-sites. A preliminary scheme called for sites for a total of 113 units consisting of 43 single-family detached high (SFD-H) and 70 single-family detached medium high (SFD-MH) (figure 19). Up to 20 percent of this total could be planned as clustered patio-type units. The patio option was exercised to achieve the necessary match between program impact and permissible coverage-to-clearing ratios.
Figure 19. Typical new town houses in 1975.
# Dwelling Unit Prototypes
Several housing prototypes were recommended as a response to specific site conditions identified in the "Guidelines for Site Planning." Each site condition suggested a different design approach with respect to foundation type, setback requirements, access, parking, yardspace, and housing configuration. A representative group of unit prototypes indicates the range of creative responses to specific site conditions.
# General Plan Revisions
Since 1972 changes in the first general plan have been made as more detailed ecological, marketing, transportation, and design studies have been completed and the 1,947-acre first construction phase begun. In 1973, the following consultant team was assembled to prepare a revised general plan. In addition to WMRT, Land Design Research was responsible for land-use planning; LWFW for marketing, Turner Collie and Braden for engineering, Green Associates for transportation; and Espey and Winslow for hydrology.
The methods that had been developed in Phase I were used in revising the general plan. A design synthesis base was established for the whole new city, indicating degrees of permissible coverage/clearing, ranging from protection to nearly total modification. Levels of landscape tolerance are shown on the map by a range of colors. The map also gives an indication of the broad, open-space framework and the distribution landscape thresholds which structure the revised general plan. Three major zones, each with a range of appropriate land uses, are defined by the map.
# Primary Open Space—At the Townwide Scale
Primary natural open space for the whole town of The Woodlands is conceived of as a conservation zone incorporating the ecologically valuable areas with respect to hydrology, wildlife, and vegetation. These areas would include the 25-, 50-, and portions of the 100-year flood plains of existing streams as well as preserves of highly diverse vegetation suited for conservation, passive recreation, and wildlife food and cover.
# Secondary Open Space—At the Development Zone Scale
Secondary open space includes components of the natural drainage system not included in primary open space: secondary swales, storm-water impoundment sites, and storm-water recharge areas. The remaining wet-weather ponds, some of which may be impounded, are included as parts of the storm-water control system and for their natural and aesthetic value. Wooded fringes outside of the primary open space system are part of this category as are greenways, large stands of trees selected for preservation, and uncleared areas between development parcels.
# Tertiary Open Space—At the Development Parcel Scale
Tertiary open space is important at the scale of the individual development parcel. Based on detailed vegetation mapping and analysis, this open-space system is comprised of those areas of vegetation not cleared for development purposes: small natural green spaces among detached houses or in open areas in higher density development (vegetation buffers along roads or in traffic islands). The on-lot drainage system would also be incorporated into the uncleared areas which make up tertiary open-space.
The hierarchy of open spaces just described will provide a multi-purpose network integral to the maintenance of natural drainage, of the forested environment and of certain species of indigenous wildlife. High-intensity recreation areas such as golf courses and playgrounds are situated to complement the basic structure of the natural open-space system but do not infringe upon it.
# Test of Land Availability
In order to test how the development program could be adapted to the open-space system, three alternatives of land availability were examined. Alternative A respected all elements of the design synthesis in terms of primary natural open space, and recreational open space. The acreage outside of the open-space zone was divided into areas which could accept low, medium, or high coverage in terms of development intensity, based on the coverage/clearing guidelines of the design synthesis.
However, the available buildable acreage and the no-proceeds open space of alternative A did not balance with the projections of the economic model. To achieve a larger buildable area, Alternative B developed certain areas of the vegetation preserve and 100-year flood plain. Alternative C increased the open-space area over Alternative B in order to restore the balance between open-space priorities and the need for an adequate economic return, while permitting selected recreational and low-density residential uses within the secondary open-space system.
Alternative C formed the basis for the general plan revision, suggesting the overall capacity of the site for development, and establishing guidelines for land-use allocation. When an evaluation of coverage/clearing was applied to this alternative, the analysis suggested a "carrying capacity" of 33,000 dwelling units and 9,972 developable acres, given the assumptions of the economic model.
The primary objective of the revised general plan was to provide the direction for up-to-date development decisions that would be realistic as well as innovative. The plan will remain flexible in order to respond to changing conditions brought on by such concerns as energy crisis or new market preferences. However, the site-specific basis of the plan will ensure the attainment of certain fundamental environmental goals.
# Community Structure
Once the open-space network was established for the new city, the areas with highest priority for development were defined. The activity focus of these development zones and the business hub of the new community was the Metro Center. Residential villages of approximately 2,000 acres each were organized around the Metro Center. The Village of Grogan's Mill was the first of these to be developed, and University Village will follow in the next phase.
The revised general plan intermixed employment, residential and community facilities at all scales. The higher density residential areas were related to the major roadways, to areas of maximum coverage/clearing, and to convenience shopping at the village center and Metro Center. The medium-density residential areas (town houses, quads, and patio units) were generally located near the major roads for easy access. These units allow for more flexibility in site planning, screening, and orientation than do single-family lots. The single-family dwellings were generally related to the smaller-scale streets, neighborhood collectors, loops, and cul-de-sacs. Since about 60 percent of the proposed residential land is for single-family detached housing, the edges of major roads will be protected by vegetation easements, landscape improvements, and special fencing.
# Lessons Still to Be Learned from The Woodlands
Much of The Woodlands story remains to be told. The opening day for the new city was October 17, 1974. However, the first three years of its development may be critical to the success of the new community. The ecological, social, and economic objectives of the new city will be greatly influenced by initial decisions and commitments. The objectives of The Woodlands plan over the next 19 years are to minimize ecological and social costs while achieving economic goals. As each new area is considered for the next phase of development, the current benefits and costs as well as feasible alternatives will be considered.
Underlying such an evaluation is the continuing process of evolution in ecological planning initiated over the last three years. It has established a set of analytical methods and vocabulary of adaptive form responsive to the hydrology, limnology, soils, vegetation, wildlife, and climate of the coastal plain.
However, it is the quantitative capabilities of the method which deserve the greatest attention and refinement. While the data and the hypothesis employed in formulating the conclusions await testing, they represent a dimension of causality and quantification not heretofore accomplished in any projects by WMRT.
As the building of the new city is carefully monitored by the environmental planning staff of The Woodlands Development Corporation and several outside research groups, an in-depth understanding of this physiographic region will be gained. During the course of this research, the ecological planners' understanding of man's impact on the natural system of this area and of the manner in which these impacts can be mitigated will be greatly enhanced.
In fact, the construction of the first phases has already given testimony of the effectiveness of the natural drainage system. James Veltman, director of environmental planning for The Woodlands, reports "that despite 13 inches of rain in three days, and four inches of rain in one hour, there was no surface water within six hours and that during this period there was effective operation of detention ponds which filled when it rained and reverted to their normal level within six hours."
The results of this planning and such observations should be immediately communicated to The Woodlands residents. They should understand the drainage function of wet-weather ponds and temporarily wet lawns. They should realize that the presence of water in their yards is critical to the survival of The Woodlands themselves. They should be aware of the special characteristics of their environment, their community, and the exciting experiment of which they are a part.
If such a level of environmental awareness was reached in a new community, it would be unique. During the period when Reston and Columbia were being planned, environmental issues had hardly been voiced and ecological planning was at a rudimentary stage. It would be fair to say that no ecological planning, as it is now understood, was employed in the initial design of either of these new towns. The planning of The Woodlands occurred in a much different social climate during a peak of environmental sensitivity and in a particularly intolerant landscape. Moreover, the environmental impact analysis composed a major requirement for the HUD Title VII application.
The effort to achieve increased public (not just professional) awareness of the generic qualities and resources of the coastal plain landscape is one of many new directions in the ecological planning work for The Woodlands not touched on in this chapter. These new directions are the logical extensions of what has already been achieved. The next steps in the planning process are as follows: monitoring the development of a new city; understanding social processes in The Woodlands; energy conservation as a community design determinant; transportation as a design determinant; and adaptive architecture in The Woodlands. All these new directions, if vigorously and carefully pursued, will lead to an increasingly successful adaptation of the human-made to the natural environment.
# Epilogue
The Woodlands site is a mute record of ancient seas and the deposition of clays and sands which underlie the forest. The seasons of the year, the hydrologic cycle, and the recycling of vital nutrients continue. Hurricanes sweep over from the Gulf of Mexico and produce intense storms. Incident precipitation enters the ground and in time replenishes the stream flow. Vegetation holds the sandy erodible soil in place and provides food and cover for wildlife. It is critical to recognize the dynamism of these physical and biological processes, for they affect man and are affected by his intervention.
As projects such as The Woodlands are planned, built, and monitored throughout the country, many lessons will be learned from each, thereby increasing the collective knowledge of the landscape architecture profession. Such knowledge of each physiographic region should be thoroughly documented and understood. This resource, if exploited with creativity and imagination, will usher in a new era of environmental design.
# 23
# A Strategy for a National Ecological Inventory (1992)
with John Radke, Jonathan Berger,
and Kathleen Wallace
We are surrounded by information. A few clicks on the Internet can produce a flood of data about a place. This situation poses both great opportunity and peril. Can we convert this information into knowledge to help us to create a more sustainable future? Or, will we be drawn into the sea of data, unable to make connections to a shore?
McHarg has steadfastedly advocated means for connection. First in the early 1970s, then in the early 1990s, he proposed to the U.S. Environmental Protection Agency a comprehensive national ecological inventory. The latter proposal resulted from the 1990 request from EPA Administrator William Reilly for advice from McHarg about the agency's computing system. With his Penn colleagues John Radke, Jon Berger, and Kathleen Wallace, McHarg suggested a three-scale inventory in 1992 that would incorporate the "entire spectrum of environmental science." They noted that the inclusion of appropriate social scientists was "essential" and "necessary to understand human environments." Their suggestion emphasized the centrality of the need to integrate and synthesize data. They recommended a shift from a focus on data collection to the synthesis of information. They proposed that the work be undertaken by teams of regional scientists rather than a centralized bureaucracy. Finally, they noted that "the establishment of a national ecological inventory be seen as the most important act protecting the American environment in history, indispensable for intelligent regulation and management, but moreover, capable of transforming both environmental research and education."
# The National Ecological Inventory
Some years ago a realization emerged within the U.S. Environmental Protection Agency (EPA) that the agency was unable to provide an appraisal of the quality of the environments which comprise the nation for the President, Congress, the American people, or for its own use in regulating the environment.
In 1972 Russell Train, then administrator of EPA, asked Ian McHarg to describe a process for undertaking a national ecological inventory. His proposal (Wallace, McHarg, Roberts, and Todd 1973) recommended that an environmental laboratory be established in each of the nation's 40 physiographic regions, plus one national laboratory in Washington, D.C. Each would be constituted with the appropriate scientists and charged with undertaking inventories and performing monitoring. While the proposal was warmly received, it transpired that existing computer capabilities were inadequate to the task. Dramatic advances in computation have since developed and an increasing body of digital data is now available for the scientific community and is denoted as geographic information systems (GIS).
The proposal for a national ecological inventory persisted, and in 1988 the EPA Science Advisory Board again recommended a program to inventory and model the environment to ensure that agency policies were maintaining or improving environmental quality. The program mandate was to initiate a national inventory of ecosystems along with their status and trends. This required assessing the health of ecosystems and the effectiveness of existing regulation in protecting or enhancing those systems. The method selected consists of applying a hexagonal grid, each of 40 km2, over the United States from which sample sites will be identified and studied. This effort came to be entitled the Environmental Monitoring and Assessment Program (EMAP).
The governmental decision to undertake a national ecological inventory may well be the single most important act in the field of the environment. Since World War II, when the greatest environmental transformations have ensued, it is accurate to state that the vast majority of these occurred without reference to the environmental consequences of contemplated actions. However, it should be observed that the culprit is not the absence of data. These are abundant, although of variable quality. The major deficiency derives from the fact that data are generated by distinct agencies for their internal purposes—EPA, NOAA, the U.S. Geological Survey (USGS), the Soil Conservation Service [now the Natural Resources Conservation Service] the U.S. Army Corps of Engineers, DMA, and CIA—and they have not been integrated as a result of which their utility is diminished.
Environmental research has been fragmented, partial, uncoordinated, and underfunded. An elaborate indictment is contained in the documents produced by the National Institutes of the Environment and need not be elaborated here. Undoubtedly, the generation of new data will play an important role in the national ecological inventory, but it is reasonable to assert that the integration, synthesis, interpretation, and evaluation of existing data may be one of its most valuable contributions. Clearly additional data must be developed, if for no reason other than to ensure that data are current, but the largest energies should be devoted to integrating available data. It cannot be overstated that we are swamped by available data but they are not being utilized. The major commitment to data should be integration of existing resources.
Ignorance of environmental data, partial information, or the inability to predict the consequences of modifications, all culminate in bad judgments. The contemplated inventory should dispel this ignorance and permit more knowledgeable and sensitive environmental adaptations. The product should lead to preservation, conservation, intelligent planning, management, regulation, and, not least, restoration.
We must learn how the world works. The national ecological inventory is a direct response to this resolve and an unequaled instrument for applying this understanding to the face of the nation.
However, it is important to recognize that the current EPA program, EMAP, does not occupy this important role, neither in the scientific community, the conservationist fraternity, nor the public. Indeed it has received little attention from scientists who at best view the enterprise with little interest, at worst with skepticism and opposition.
It is intolerable that such a vital objective should be so lowly regarded. The remedy for this situation constitutes our primary recommendation.
# Executive Committee
It is recommended that the most distinguished scientists of the environment be constituted into an executive committee to advise the EPA administrator on the conduct of the enterprise. This act alone should provide legitimacy and authority to the inventory. Members should be selected from officers in all of the relevant scientific societies. By this decision, not only the scientist officers in these societies, but also the membership will be co-opted into the national inventory.
The creation of such a committee will assure the scientific community that the enterprise will benefit from the most august minds in environmental science. It will also provide the leadership which this monumental task deserves.
The composition of the EPA Science Advisory Board contains over forty persons with the largest component derived from medicine, epidemiology, and toxicology. Next in size is environmental science and engineering, with the smallest contingent representing environmental science. This composition may well be appropriate for the agency's regulatory function but it is not suitable for designing the national ecological inventory.
A useful comparison can be found in the composition of the Advisory Council for the National Institutes of Health. This body of forty-five members includes such luminaries as Carl Sagan, Kenneth Boulding, Paul Ehrlich, George Woodwell, Edward Wilson, George Wald, and the presidents of Harvard, Texas, and Notre Dame. The composition of the recommended executive committee for the national ecological inventory should aspire to a comparable caliber but it must be distinguished by the representation of all of the environmental sciences.
The organization of the EPA long preceded the determination to undertake a national ecological inventory. The question is whether it requires reorganization to fulfill this role. The current elements engaged in the EMAP process are invaluable but, it is suggested, insufficient. The small staff in Washington, D.C., the bio-statisticians at Corvallis, competence in remote sensed imagery in Las Vegas, air photography, and landscape ecology at Warrenton, together with the diffuse groups engaged in research, represent rich resources but, in sum, do not constitute the essential elements for the task. However, these small, scattered groups lack the resources and the leadership capable of comprehensive integration and synthesis of environmental data, which is essential for the achievement of the objective.
It is recommended that the executive committee identify the constituent roles in the inventory and monitoring processes and advise on the identification and selection of the staff. This contemplated staff should include representation of all of the environmental sciences. Physical science should include meteorology, physical oceanography, geology, hydrology, and soil science. Plant and animal ecologists, limnologists, marine biologists would be included in the biological sciences. Ethnography, anthropology, resource economics, geography are conspicuous among the social sciences. Biostatistics, remotely sensed imagery, and computation should be represented. The staff should be selected, not alone for their competence within their discipline, but for their demonstrated experience in interdisciplinary and multi-disciplinary research.
The current organization includes research on agroecosystems, arid lands, forests, estuaries, surface water, wetlands, Great Lakes, air deposition, and more. We observe that while arid lands are an appropriately ecological descriptor amenable to research, the same cannot be said of the phenomena of surface water, wetlands, and estuaries, all of which are interdependent elements in continuous processes. Forests as a title include so many types as to diminish their validity as a research subject. We recommend for consideration the replacement of the current reductionist phenomena-based research by a regional/ecological approach employing physiographic regions, Kuchler's ecoregions and, perhaps including metropolitan regions among others (Kuchler 1964). As a consequence we recommend that the executive committee review and advise on existing and contemplated research.
# A Title
We are of the opinion that one of the reasons why EMAP is so cavalierly dismissed or ignored is its title. EMAP is just another bureaucratic acronym of which there are far too many. Certainly this title will not evoke passions and commitment yet, given the signal importance of enterprise, arguably the most important single act in the history of the American environment, it does deserve a better title, capable of attracting supporters, advocates, enthusiasts, participants from the public at large, conservationists, governmental committees, and the scientific community.
In order to stimulate discussion on the matter we offer examples for consideration:
"The Face of the Nation"
"The State of the Nation"
"The American Landscape"
"The National Environment"
"The Health of the Land"
At that time when the executive committee is formed and it produces a report on the plan for the national ecological inventory and identifies operational structure, staff, plans, objectives, time table, budget, and anticipated benefits, there should be a concerted public relations effort. This should involve title, logo, a published report, and these should be presented to the American people with appropriate fanfare. There should be a continuing effort to keep the objectives and accomplishments of the venture in conspicuous public and scientific view and to engender widespread debate.
# Basis for Recommendations
The recommendations for a national ecological inventory are based in part on the authors' experience with colleagues and students at the University of Pennsylvania since the early 1960s. This group has engaged in ecological planning studies for the Potomac and Delaware River Basins, their physiographic regions and selected sample sites within each of these. For the past four decades, sample sites in the Philadelphia metropolitan region have been examined. McHarg in his partnership (Wallace, McHarg, Roberts, and Todd) undertook elaborate planning studies in Iran and Nigeria and for many of the metropolitan regions of the United States. This association of physical, biological, and social scientists was described by the magazine Science as the most interdisciplinary and multi-disciplinary academic department in the United States (Holden 1977). Thus these observations and recommendations derive from extensive experience (see American Institute of Architects 1965-1966; Development Research Associates and Wallace, McHarg, Roberts, and Todd 1972; International Planning Associates 1979; The Mandala Collaborative and Wallace, McHarg, Roberts, and Todd 1975; University of Pennsylvania 1963; and Wallace, McHarg, Roberts, and Todd 1969, 1972, 1974, 1976, 1978).
In these exercises, notably of river basin studies, a three-part analysis was performed. The first study addressed the entire river basin, the second level concentrated upon the constituent physiographic regions, the final step involved examination of specific sites. Scales ranged from 1:250,000, 1:100,000, and 1:24,000. In this process sites became comprehensible within physiographic regions (Fenneman 1931, 1948; Hunt 1974), river basins, and in certain cases, metropolitan regions.
So, while we subscribe to the sample method of inventory, we suggest that while necessary, it is insufficient. A medical analogue identifies blood, urine, sputum, and tissue samples. Each can be analyzed within the characteristic attributes but as samples of an organism. So too with inventory samples; they should be selected from meaningful contexts.
For example, samples from Alaska and Florida, California and Maine, Minnesota and Texas, even adjacent Pennsylvania and New Jersey, will reveal dramatic differences. Are these normal, or abnormal? In the last example the explanation for profound distinctions lies in the observation that one sample reposes in the Crystalline Piedmont, the other in the Inner Coastal Plain. As all of the environmental sciences have developed taxonomies, it would be cavalier not to employ them. There are climate zones, geological and physiographic regions, river systems and groundwater basins, soils regions, bio-climatic zones, ecosystems, habitats, metropolitan regions, and more.
A six percent sample would be unable to ascertain boundaries of regions, yet the identification of samples, within the appropriate regions, will vastly facilitate analysis and evaluation.
# Three-Part Inventory
So, we recommend that the contemplated national ecological inventory be performed at three scales: national (at 1:2,000,000), regional (at 1:250,000), and site (at 1:24,000).
The national inventory should extend to include contiguous regions which affect the environment—adjacent areas in Canada, Mexico, the Caribbean, oceanic and atmospheric processes. At this scale all regions should be delineated—climatic zones, geological and physiographic regions, rivers and groundwater basins, soil regions, big-climatic regions, biomes, ecosystems, and wildlife habitats. These should also include metropolitan, agricultural, energy regions among other social descriptors.
The regional inventory at the scale of 1:250,000 should include all environmental data nationwide which can be utilized for all regional analyses and provide the basis for sample selection in a stratified sampling process.
The final component, sites at 1:24,000 is provided by the hexagonal lattice. Thus, the national inventory would present data in its most aggregated form, appropriate for examination and resolution of large-scale problems. Regional data would amplify resolution and be amenable to analysis of homogenous areas. The sample at site scale would provide the richest data, highest resolution, and permit the most discriminating analyses.
It would take sixteen years of 6 percent samples to complete the inventory, an intolerable delay. As more digital data are currently available for both regional and national scales these inventories could be completed within a few years. Moreover, the role of samples should be limited to areas of crucial importance. They would not comprise the building blocks for the national inventory. However, employment of this method gives great importance to the criteria employed in selecting samples.
# An Ecological Inventory
The questions to be addressed in landscape characterization are: What are the environments which comprise the nation; how did they come to be; what processes characterize them; what tendencies do they exhibit; what has been the effect of human use; what is their current status; and, most of all, what should we do?
The first prerequisite to an adequate characterization is an inventory of environments and their status. This step has long been needed and has been long awaited. It is commendable that the initiative has begun, but it is important to recognize that it is still in the process of design.
Given the mandate to undertake a national ecological inventory, as we have seen, the EPA concluded in favor of the sample method using a national hexagonal lattice, each of 40 km2, and sampling within those hexagons. This permits resolution at a scale of 1:24,000 and is perfectly suitable for landscape characterization. However, adoption of such a general method does not resolve the issue of what data are to be employed nor does it inform which samples should be selected.
If the environment must be inventoried, there is no problem in identifying data sources. The environment has been dissected by science. Distinct subjects are proprietary to discrete fields and this relationship has been formalized in the sciences, university departments, and government agencies. Each major discipline—meteorology, geology, oceanography, hydrology, pedology, plant and animal ecology, limnology, marine biology, and others— has its own academic and bureaucratic constituency. Science remains obdurately reductionist and shows no sign of change, yet the process of characterization demands multi-disciplinary and interdisciplinary exercises. Above all, it demands the kind of integration and synthesis which are inseparable from the ecological method of landscape characterization.
We note that the current process of the characterization in EPA has emphasized landscape ecology and placed large reliance on remotely sensed data. We consider this necessary but insufficient and recommend a comprehensive data set to include:
Meteorology
Geology
Surficial geology
Groundwater hydrology
Physiography
Surficial hydrology
Soils
Plant ecology
Animal ecology
Limnology
Physical oceanography
Marine biology
Ethnography
Cultural anthropology
Cultural geography
Archeology
Planning
# Chronology as the Structuring Device
The task of characterization requires that landscapes be understood as the product of processes, laws, and time. The exercise should identify meaningful processes and phenomena. To accomplish this, a structure of data organization should be employed which assimilates incremental data and which culminates in understanding and in meaningful representation.
The process of characterization needs to incorporate data from many disciplines. Given the fact that the normal circumstance will reveal, not a paucity of data, but a surfeit, there is a large premium in employing an organizing structure which absorbs data and expands meaning to culminate in a rich expression of the region under study. Chronology is just such an organizing structure.
The chronological method can be employed as the structural basis of characterization, used to reveal cause and effect as described by process and pattern. This suggests that characterization begin with the history of geological evolution, involving 500 million years of evidence in the case of the Pre-Cambrian. This geologic history begins to describe and explain the regions under study and reveal their dynamism. Meteorology follows with some 50 million years of climatic evidence, culminating with the major events of the Pleistocene. These can be employed to describe the glacial processes and events of this period. These data, comprising geology, meteorology, and geomorphology, provide the major bases for the phenomena of groundwater and for physiography. The addition of physiography provides the essential data to explain surficial hydrology. By this point, the cumulative nature of the evidence becomes manifest—no layer is comprehensible without reference to the antecedent layers, each building upon the previous one's stratum. Data assembled can now be reinterpreted to explain soils with their variable characters and distribution—derived from bedrock, affected by climatic conditions, transported by glaciation, alluvial or aeolian processes. All of the foregoing—geology, physiography, hydrology, and soils—can be reinterpreted as vegetative environments described as bio-climatic zones, biomes, or ecosystems. This addition of vegetation provides the final component to explain the region's wildlife.
This method, which we call the layer-cake approach, employs data in chronological sequence to progressively provide meaning to a landscape and to ultimately reveal the historic processes which explain contemporary phenomena characterizing regions and sites. Mountains and oceanic abysses, valleys and coastal plains, rivers, streams, wetlands and deltas, soils with variable characteristics, boreal and deciduous forests, prairies and deserts, sundry niches and habitats, all become comprehensible when viewed through the lens of chronology. Each place is because; natural history explains natural systems, and ecology is employed as the unifying science. Processes, laws, and time reveal the present. Therefore, characterization must not only represent spatial patterns as they exist today, but also identify the critical processes which have determined prior landscapes and which are inexorably converting the landscapes of the present into those of the future. The ecological inventory method of landscape characterization is ideally suited to this task.
This permits examining the nation, regions, and sites to disclose probability, causality, and meaning. It is an exemplary method. However, there are certain qualifications: data must be extensive, including all relevant environmental data, and must be of a consonant degree of resolution and time scale. If these characteristics are met, the method is a device for accomplishing integration and synthesis. As no data layer is comprehensible without reference to underlying layers, the accumulation of layers itself constitutes a synthesizing process. This ecological inventory method will explain climate, rocks, water, soils, plants, and animals; alone, however, it will be incapable of addressing human interventions and the realities of existing human environments.
This introduction of the subject of human intervention requires a further step, a more recent field which is less well developed than environmental science but which is nonetheless crucial to the process of landscape characterization. It can be described as human ecology (see, for example, Bennett 1983, Berger and Sinton 1985, Meinig 1979, Wiken et al. 1981). This begins with the primeval environment and its aboriginal population. Like ecological inventories, it employs chronology as its structuring device and it requires reference to a variety of disciplines, but here they include cultural and political geography, regional planning, history, ethnography, and anthropology to supplement physical and biological sciences.
Starting with the primeval landscape and its aboriginal inhabitants, human ecologists identify resources, technology, values, and resultant settlement patterns and land uses. The arrival of Europeans to America brought new values and technologies, thereby changing the definition of important resources and yielding new settlement patterns and land uses. Massive social events such as the Revolutionary War, the Civil War, the great migrations, the industrial revolution, and the World Wars were all influential. As important to the changing landscape was the introduction of technology: water transportation, water power, iron, coal, steam power, roads, canals, railroads, and the automobile with its associated interstate highways and suburbanization.
As history proceeds the environment is continuously modified, reflecting fundamental changes in the values of the inhabitants and the resources and technology available to them. The process is inexorable, with some landscapes being transformed, others neglected, and still others returning as forest succession; the changing landscape reveals the perceptions and values of past and present, its patterns gravid with meaning.
As with the biophysical layer-cake analysis, historic land-use maps reveal the changing human environment so that an informed viewer can observe the marks and changes resulting from earlier activities, fit them into the contemporary scene, and make predictions of the future landscape. The product is a synthetic view of a dynamic landscape continuously shaped by biophysical and social processes.
# Regions and Stratified Sampling
Regional structures are indispensable for selecting samples. The first decision must be to select controls—pristine or near pristine environments representing the full range of regions, ecosystems, and varied habitats (lakes, forests, prairies, deserts) located presumably in national parks, national forests, wilderness areas, refuges, and conservancies. Pristine or near pristine environments constitute the upper limits of quality; similarly, samples should also be selected in the most debased environments. Among these would be U.S. Department of Energy installations—Hanford, Savannah River, Oak Ridge, Brookhaven, Rocky Flats, and nuclear test sites in Nevada; the most notorious toxic sites—such as, Love Canal and Times Beach; and also the most depraved urban sites revealing dereliction, abandoned buildings, brickbat soils, water in gutters and catch basins, virtually no vegetation, and wildlife of rats, mice, lice, fleas, cockroaches, sparrows, and starlings.
A subsequent structure for selecting further samples is afforded by the major physiographic regions as subdivided into aquifers, river basins, soil regions, bio-climatic zones, and faunal regions. In urbanized regions, an additional level of structure must be applied based upon both demographic and socioeconomic data.
Geographers have identified urban regions—central business district, zones of transition, inner city residential neighborhoods; inner suburbs; suburbs; outer suburbs; and the urban-rural fringe. Each of these may be associated with specific data on air quality, urban heat sink, degree of permeability, runoff, water quality, vegetative cover, and wildlife. Employment of socio-economic data suggests that other regions be identified. While resources originate from natural history which explains their location, exploitation has specific consequences. Thus coal regions are associated with environmental degradation and acid mine drainage; cereal production relates to ancient prairies and is now associated with groundwater pollution, notably of nitrates. Economic minerals (iron ore, bauxite, limestone, sand and gravel, etc.) have regional locations and attendant environmental problems, as do other extractive land uses such as forestry, fishing, and grazing. There is a systematic relationship between regions, resources, exploitation, and environmental effects.
Data collected for hexagons within the context of larger physiographic regions and river basins would permit an analysis of the effects of cumulative environmental and social processes. For example, the Delaware River transects the Allegheny Plateau, the Ridge and Valley Province, the Great Valley, the Reading Prong, the Crystalline Piedmont (including the Triassic Basin), and the coastal plain. The changing water quality of the Delaware River reflects the regions transected. Headwater streams in the Allegheny Plateau are of high quality. Tributaries in the Ridge and Valley Province, a long-term locus of coal mining, contribute a massive burden of acidity from mine drainage. These tributaries and the main stem of the Delaware are effectively neutralized as they pass through the limestone Great Valley, but receive a large assault of sediments and fertilizers from agricultural activities, among others. When the river reaches the Crystalline Piedmont and concentrated urbanization, it receives an engorgement of urban sewage and industrial wastes. Thereafter, the river encounters the coastal plain where it impacts extensive aquifers and experiences estuarine processes.
How many samples would be required to evaluate the Delaware River; how many for the Schuylkill River, its largest tributary? Could the contemplated resolution permit comment on Wissahickon Creek, a small tributary which is nonetheless a major feature in the landscape of Philadelphia?
The physiographic regions enumerated above are transected by the Delaware, Susquehanna, Potomac, and James Rivers. A useful comparison could be undertaken between each of these rivers and their samples within each physiographic region. This method could be applied to comparative evaluations not only for physiographic regions and river basins, but also for climate zones, aquifers, soil regions, forest types, wildlife habitats, and, of course, for metropolitan areas.
The major purpose of this discussion on the basis for the selection of samples is, first, to emphasize the taxonomic regional structures which are employed by the environmental sciences and should be incorporated. Secondly, the opportunity of comparative, stratified samples, as for instance, for between river and groundwater regions in similar physiographic regions, or as between urban regions ranging from central city to the rural fringe, in their regional contexts.
# Regional Expertise
Much knowledge exists about the ecology of the nation, not only by the distinguished scientists of the environment, but also of regional groups wherein repose the major perceptions and data. This is an important observation, that the largest body of knowledge required for inventories reposes in the knowledge, data, and repositions of existing scientists and institutions. As important, they are likely to possess not only current but also historic data. As a result we recommend that the conduct of both regional and hexagonal samples be devolved to regional groups. However, we recommend that EPA assume the responsibility for the national characterization, possibly in concert with the USGS and NASA.
This proposal would invest the effort in persons and institutions familiar with the relevant sciences and regions and avoid the necessity of creating a large bureaucracy.
A centralized computing facility should be installed in EPA to support the regional groups generating data. This would ensure quality assurance and quality control of data and would be the most cost-effective way to process specialty data sets. Furthermore, it is not necessary to know the study regions in order to process the data. Centralization of data processing allows the assembly of a highly skilled group, the development and updating of software to solve problems, and an economy of scale in setting up the facility. This economy of scale is quite significant because of the very large data sets involved, so large that even high-end workstations cannot efficiently process the data.
# Synthesis
The nature of science is reductionist. This is reflected in the separate disciplines, journals, university and government departments devoted to individual specialties. Occasionally the investigation of ad hoc problems produces interdisciplinary and multi-disciplinary research, but such efforts remain an aberration. Yet, it is incontrovertible that a valid landscape characterization requires that all of the environmental sciences be involved and engaged in integration and synthesis.
We are not fundamentally committed to clouds, rocks, rivers, soils, plants, or animals; instead, we are interested in the quality of the human environment which partakes of all of these and more. The particular attribute which is crucial to the characterization process is synthesis.
Synthesis involves both science and art, two views thought to be dramatically opposed in traditional philosophy. When each is viewed with its larger pretensions, the conflict between rationality and intuition appears irreconcilable. However, when both nouns are reduced to their root meanings, reconciliation seems possible. Science means knowledge, art means skill. Knowledge without skill has limited utility, while skill without knowledge is incomplete. At this level it would appear that each is essential to the other. This is certainly valid in the synthesis required for characterization.
The scientist searches for irrefutable knowledge and refrains from making statements when such evidence is incomplete. Yet management of the environment will always include imperfect knowledge, and so government must employ the best available knowledge and act prudently with respect to it.
In spite of these difficulties it is possible to undertake landscape characterization. The first problem to be resolved results from the fact that no person exists who is coequally informed in all of the environmental sciences, particularly at threshold levels. It thus becomes clear that the exercise must be multi-disciplinary From the physical sciences of meteorology, geology, physical oceanography, hydrology, and soils it is possible to find scientists who, while identified with one discipline, are conversant with the other physical sciences of the environment. But more important is the agreement that while physical processes have been dominated by plate tectonics, these and other processes are manifested in geomorphology. Physiography provides the initial synthesis for landscape characterization.
It transpires that biological systems synthesize both physical and biological conditions so that the organisms in an ecosystem represent a second level of synthesis, certainly prior to human intervention. This investigation is simplified by agreement on ecological principles espoused by plant and animal ecologists, limnologists, and marine biologists. It has also been enhanced by the perceptions of the emergent discipline of landscape ecology (Forman and Godron 1986).
The last function of producing a physical, biological, and social model is the least understood process and faces the serious problem that social scientists are generally uninformed in natural science. The reverse is equally true. Yet the natural environment is intertwined with human society. Values and attitudes toward the environment, combined with available resources, capital, and technology, constitute the basis for landscape history. If the combined socio-natural ecological inventory proceeds carefully, the current situation of the environment will become comprehensible, as will the population, its economy, the values they espouse, the institutions they select to fulfill their objectives, and their settlement patterns and land uses. There are a few precious people who possess the rare combination of skills to complete this integrative process of landscape characterization (Hills 1974, for example). From their work come the frames of reference for synthesis.
Ecological planning provides a means for synthesis (McHarg 1969). Ecological planning can be augmented by an immersion in cultural anthropology (Berger and Sinton 1985).
The key questions which must be addressed for each inventory and characterization are: What is the place? How did it come to be? What processes characterize it? What is its status? Where is it going? There is an equally important sequence of questions: Who populated the region, historically? Who today? How did they utilize its resources? How do they evaluate the environment? What issues do they perceive? What are their responses?
This interface between nature and people deserves extended treatment. It is indispensable for understanding human environments. Moreover this concern has not been conspicuous in the history of EPA which has tended to concentrate on biophysical environments. It cannot be too strongly stated that the EMAP process must address human environments.
The world is a single interacting biophysical-social system. There are no physical systems unaffected by life, no biological systems impervious to matter, no biophysical systems uninfluenced by people. Yet the world is subdivided by science and language, specific realms are jealously guarded as proprietary by the discrete sciences.
Traditional ecological inventories do not include ethnographic histories or historic analyses of economic and social processes. From our extensive and evolving experience, we assert that such historic analyses are absolutely crucial to landscape characterization. Jonathan Berger and John Sinton (1985) provide an example of such an ethnography in their book about the New Jersey Pinelands.
# Conclusions and Recommendations
1. Create an executive committee drawn from the most distinguished leaders in the fields of environmental science.
2. Charge the executive committee to design the entire national ecological inventory process, determine the structure necessary to realize the objective, identity staff, computer operation, inventory, monitoring, and characterization procedure.
3. Design and implement a public relations campaign.
4. Investigate and appropriate scales of inventory, including national, regional, and site specific.
5. Investigate and resolve requirements for each scale of investigation.
6. Develop criteria to be employed in selecting sites.
7. Investigate the subject of an appropriate title for the enterprise.
8. Consider chronology as a suitable rubric for the characterization process.
9. Consider the utility of the layer-cake mode of representation.
10. Determine the obligatory contents of the inventory.
11. Discuss and resolve the matter of historical and ethnographic data to be incorporated into the inventory.
12. Investigate regional groups with the appropriate range of competence in physical, biological, and social science, competent in integration and synthesis.
13. Review the current research operation and determine whether or not this reductionist, phenomenon-based approach should be continued or replaced by a more integrated structure.
# References
American Institute of Architects Task Force on the Potomac; Wallace, McHarg, Roberts, and Todd; and the University of Pennsylvania. 1965–1966. The Potomac. Washington, D.C.: U.S. Government Printing Office.
Bennett, John. 1983. "The Micro-Macro Nexus: Typology, Process, and Systems." 1981 Annual Meeting of the American Anthropological Association.
Berger, Jonathan and John Walter Sinton. 1985. Water, Earth, and Fire: Land Use and Environmental Planning in the New Jersey Pine Barrens. Baltimore, MD: Johns Hopkins University Press.
Development Research Associates and Wallace, McHarg, Roberts and Todd. 1971. Ecological Studies of the Regional Transportation District, Denver, Colorado. Interim Technical Memorandum, Task 5 of the Joint Venture. Denver: Regional Transportation District.
————. 1972. Ecology-NaturalSuitabilities for Regional Growth. Denver: Regional Transportation District.
Fenneman, N. M. 1931. Physiography of Western United States. New York: McGraw-Hill.
————. 1948. Physiography of Eastern United States. New York: McGraw-Hill.
Forman, Richard T. T., and Michel Godron. 1986. Landscape Ecology. New York: Wiley.
Hills, G. Angus. 1974. "A Philosophical Approach to Landscape Planning." Landscape Planning 1 (4):339-81.
Holden, Constance. 1977. "Ian McHarg: Champion for Design with Nature." Science 195 (4276,28 January):379-82.
Hunt, Charles B. 1974. Natural Regions of the United States and Canada. San Francisco: W. H. Freeman.
International Planning Associates. 1979. The Master Plan for Abuja, The New Federal Capital of Nigeria. Lagos, Nigeria: The Federal Capital Development Authority.
Kuchler, A. W. 1964. Potential Natural Vegetation of the Conterminous United States. Special Publication 36. American Geographical Society.
The Mandala Collaborative/Wallace, McHarg, Roberts, and Todd. 1975. Pardisan, Plan for an Environmental Park in Tehran. Prepared for the Department of Environment, Imperial Government of Iran. Philadelphia, Penn.: Winchell Press.
McHarg, Ian L. 1969. Design with Nature. Garden City, N.Y.: Doubleday, The Natural History Press. 1992 Second edition, New York: Wiley.
Meinig, D. W. 1979. The Interpretation of Ordinary Landscapes. New York: Oxford University Press.
University of Pennsylvania. 1963. Metropolitan Open Space from Natural Processes. Philadelphia: Urban Renewal Administration and the States of Pennsylvania and New Jersey.
Wallace, McHarg, Roberts, and Todd. 1969. Ecological Study for Twin Cities Metropolitan Region, Minnesota. Prepared for Metropolitan Council of the Twin Cities Area. Philadelphia: U.S. Department of Commerce, National Technical Information Series.
————. 1972. An Ecological Planning Study for Wilmington and Dover, Vermont. Brattleboro, Vt.: Windham Regional Planning and Development Commission and the Vermont State Planning Office.
————. 1973. Towards a Comprehensive Plan for Environmental Quality. Washington, D.C.: American Institute of Planners for the U.S. Environmental Protection Agency.
————. 1974. San Francisco Metropolitan Regional Environmental Impact Procedure Study, California. San Francisco: Bay Area Council of Governments.
————. 1976. Lake Austin Growth Management Plan. Austin, Texas: City of Austin, Department of Planning.
————. 1978. Ecological Study for Northwestern Colorado Council of Governments. Frisco, Colorado: Northwestern Colorado Council of Governments.
Wiken, E. B., J. P. Semyk, and E. Oswald. 1981. "Inventorying Land Resources through an Ecologically Based Approach." In In-Place Resource Inventories: Principles and Practices. Orono: University of Maine Press.
# Prospectus (1998)
My battle began almost fifty years ago with a tentative presumption in favor of nature that became explicit in the 1960s. I employed this view wherever opportunity was afforded—in small sites and in metropolitan regions, in urban areas and rural landscapes, and across North America and abroad. By applying increased knowledge and developing a powerful model and method, I engaged a large army of practitioners: first, students, next, readers of Design with Nature, then, viewers of Multiply and Subdue the Earth, and finally, audieMultiply and Subdue the Earth, and finally, audiencesnces throughout the country and the world.
This commitment recognized the enemy to be those devoted to anthropocentrism and anthropomorphism, those who believed that man was the justification of evolution, that he exclusively shared divinity and was given dominion over the earth, that he was licensed to subjugate nature and empowered to treat the planet as a storehouse awaiting plunder.
So there was a war between those who would treat the world and its creatures with, at least, deference, and perhaps better, with reverence, and those who would destroy without care or remorse. I resolved to mobilize all of my energies on behalf of the stewards.
For over a quarter of a century, I have been a strong advocate of ecological inventories at all scales, from the site to the whole earth. If we are to undertake a global ecological inventory and engage in continuous monitoring, the questions will quickly arise: what, where, and how should we restore? In brief, do we select areas massively destroyed or, in contrast, select the pristine which merely needs protection? Whatever course we follow will require a massive rediscovery of the earth in times of good health. For the West, that would be before the industrial revolution. For most of the developing world the time span would be much shorter, possibly only half a century. But we must describe the primeval world and consider how to restore it, how to green it, how to heal it.
Of course the greatest agency involved in this process is natural succession, regeneration, that process which is most dramatically portrayed on the slopes of recently active volcanoes but occurs worldwide. We must do more than sustain the planet, we must design regenerative communities and landscapes. Or we must get out of the way and let the earth regenerate itself.
A dramatic example of this is the demilitarized zone in Korea, which is 250 kilometers long, 4 kilometers wide, and is bordered by a high fence that has succeeded to forest, where many species of plants and animals once thought to be extinct are now abundant. Where are the healthy tissues of the world life body, the virgin areas—Arctic and Antarctic ice caps, miles of deep ocean trenches, mountains as high as they are deep, deserts, forests, bare rock, snow and ice, but, above all, water?
We know that succession is a real phenomenon. The example of regeneration after volcanic eruptions alone is persuasive. How fortunate we are to have such an ally. But let us consider the possibility that there exists a process, which might be called "infectious health ," whereby pristine areas contribute to the regeneration of disturbed neighbors.
Could infectious health enable healthy systems to cause adjacent lands to recover? Then we should not only protect the pristine, but also their recovering neighbors. We would see pristine colonies expanding whereby species, microorganisms, and seeds initiate regeneration. We would see pristine realms coalesce, wounds and welts shrink. We would see the descent of health from elevated islands spreading symmetrically. We would see carbon being fixed on coral reefs, benthic limestone, and old-growth forests, and gas made palpable, changed from threat to inheritance.
And so I've advocated using ecology, employing all the environmental sciences, to understand our world and guide design and planning decisions. The determination to undertake national, digital, ecological inventions awaits us. It would be beneficial if this were performed successfully by a large country. Recent experience has been limited to small nations and regions, including New Zealand, Western Australia, and Taiwan. There is, of course, the most successful inventory performed anywhere—the environmental atlas of Alaska, led by Lidia Selkregg, but it is not yet in digital form. Selkregg's marvelous work consists of maps, diagrams, photographs, and tables, all of which should be digitized to constitute an exemplar for this nation. Because Alaska is perhaps the most pristine environment in North America, it is the most appropriate candidate for the first comprehensive, digital statewide ecological inventory.
If the primary objective of healing the earth were to begin with inventories to identify the pristine and near pristine environments which should be preserved to contribute to infectious healing and regeneration, there emerges the problem of cities. Conventional economic thought places cities at the apex of human achievement, followed by agriculture, while wild lands and native ecosystems are held at the lowest value.
An ecological perspective would reverse this hierarchy. Cities, rather than an ecological apex, would be considered to be a catastrophe. As the major sources of pollutions and toxins, particularly greenhouse gases, they exclude the vegetative contribution of fixing carbon. Agriculture makes a contribution to deterrence of the greenhouse effect but also contributes to excessive water use, soil erosion, and climatic extremes. Its carbon fixing falls far below that achieved by rain forests. Wild lands, lakes, oceans, and coral reefs, least valued in conventional wisdom, are, in contrast, the primary agents for fixing carbon and for climatic amelioration.
Present urbanization constitutes a major global ecological challenge with the maximum costs to the environment and none or few benefits. This suggests that cities, particularly megacities that can number up to thirty million in population, should be greened to diminish environmental costs.
In South America and Southeast Asia vegetation is profuse and can grow in unlikely places. In Honolulu parking garage roofs and walls are covered with ficus. In Singapore overpass structures are covered in vegetation. But this remedy should not be limited to tropical regions; temperate rain forests are admirable fixers of carbon and produce rich and beautiful environments and high biodiversity. Green roofs and walls should be investigated to reduce urban impacts on the environment.
New knowledge about our environments and new theories will continue the advance of ecological design and planning. There is nothing so practical as a good theory, or so the saying goes. Landscape architects and ecological planners should become the physicians of the land. We should be healers, but we should also be practitioners of preventive medicine. Medical doctors need to know human biology to practice their art. Similarly, we should be well-versed enough in landscape ecology to be able to make diagnoses and describe interventions.
The goal of earth medicine should be the support and the creation of healthy places. Health is the ability to recover from injury or insult. Our planet and its landscapes have suffered many injurious and insulting actions—the work to be done is cut out for'us. The process of making landscapes healthy involves diagnosis, prescription, and intervention. As a medical doctor uses knowledge of anatomy during the checkup, so too must an ecological designer or planner understand natural and cultural processes in order to identify the nature of the illness.
Design and planning should be arts based on a body of knowledge built from case studies of practice. We should undertake actual projects and then assess our work through writing. This is the path to reflective practice and theory-building. Physicians learn their art from a solid foundation of literature, and they then build upon that foundation in the form of new research and papers. Surgery on the brain or heart or any other vital organ takes courage. So too does intervention in an urban neighborhood, a degraded wetlands, or any of the other tissues of the living landscape. To heal places, landscape architects and ecological planners must not only have the knowledge and skill to make reasonable diagnoses and to prescribe sound actions, but also the courage to take on difficult challenges.
# Acknowledgment of Sources
Most essays and articles in this collection have been revised and reprinted with the kind permission of the original publisher. The following papers were published in these original sources:
"Man and Environment." In Leonard J. Duhl and John Powell, eds. The Urban Condition. New York: Basic Books, pp. 44-58 (1963).
"The Place of Nature in the City of Man." The Annals of the American Academy of Political Science (Urban Revival: Goals and Standards) 325 (March):1-12 (1964).
"Ecological Determinism." In F. Fraser Darling and John P. Milton, eds. Future Environments of North America. Garden City, N.Y.: The Natural History Press, pp. 526-38 (1966). From Future Environments of North America by Fraser Darling and John P. Milton. Copyright © 1966 by the Conservation Foundation. Used by permission of Doubleday, a division of Bantam Doubleday Dell Publishing Group, Inc.
"Values, Process and Form." In The Smithsonian Institution. The Fitness of Man's Environment. New York: Harper & Row, pp. 207-27 (1968). Used by permission of the publisher.
"Natural Factors in Planning." Journal of Soil & Water Conservation 52 (1, January-February): 13-17 (1997).
"Regional Landscape Planning." In A. J. W. Scheffey, ed. Resources, the Metropolis, and the Land-Grant University. Proceedings of the Conference on Natural Resources, No. 410. Amherst: University of Massachusetts, pp. 31-35 (1963). (Comments by others on McHarg's remarks continue through p. 37.)
"Open Space from Natural Processes." In David A. Wallace, ed. Metropolitan Open Space and Natural Process. Philadelphia: University of Pennsylvania Press, pp. 10-52 (1970). From Metropolitan Open Space and Natural Process, edited by David A. Wallace. Copyright © 1970 by the University of Pennsylvania Press. Reprinted with permission of the publisher.
"Must We Sacrifice the West?" In Terrell J. Minger and Sherry D. Oaks, eds. Growth Alternatives for the Rocky Mountain West. Boulder, Colo.: Westview Press, pp. 203-11 (1975).
"Human Ecological Planning at Pennsylvania." Landscape Planning 8:109-120 (1981).
"Ecological Planning: The Planner as Catalyst." In Robert W. Burchell and George Sternlieb, eds. Planning Theory in the 1980s (2d edition, 1982). New Brunswick, N.J.: The Center for Urban Policy Research, Rutgers University, pp. 13-15 (1978). Reprinted with permission of Rutgers University, Center for Urban Policy Research, from Planning Theory in the 1980s: A Search for Future Directions, edited by Robert W. Burchell and George Sternlieb. Copyright © 1978 Rutgers University.
"The Court House Concept." Architectural Record 122 (September):193-200 (1957). Reprinted with permission from Architectural Record, Vol. 122 (1957).
"Architecture in an Ecological View of the World." AIA Journal 54(5):47-51 (1970).
"Nature Is More Than a Garden." In Mark Francis and Randolph T. Hester, Jr., eds. The Meaning of Gardens: Idea, Place, and Action. Cambridge, Mass.: The MIT Press, pp. 34-37 (1990).
"Ecology and Design." In George F. Thompson and Frederick R. Steiner, eds. Ecological Design and Planning. New York: Wiley, pp. 321-32 (1996). Copyright © 1996. Reprinted by permission of John Wiley & Sons, Inc.
"An Ecological Method for Landscape Architecture." Landscape Architecture 57(2):105-7 (1967).
"A Comprehensive Highway Route Selection Method." Highway Research Record 246:1-12 (1968). In Transportation Research Record 246, Transportation Research Board, National Research Council, Washington, D.C. (1968).
"Biological Alternatives to Water Pollution." In Joachim Tourbier and Robert W. Pierson, Jr., eds., Biological Control of Water Pollution. Philadelphia: Center for Ecological Design and Planning, University of Pennsylvania, pp. 7-12 (1976).
"A Case Study in Ecological Planning: The Woodlands, Texas." In Marvin T. Beatty, Gary W. Petersen, and Lester D. Swindale, eds. Planning the Uses and Management of Land. Madison, Wisc.: American Society of Agronomy, Crop Science Society of America, and Soil Science Society of America, pp. 935-55, chap. 38 (1979).
"Plan for the Valleys vs. Spectre of Uncontrolled Growth." Landscape Architecture 55 (4,April):179-81 (1965).
"An Ecological Planning Study for Wilmington and Dover, Vermont." The study was prepared by Wallace, McHarg, Roberts, and Todd for Wilmington and Dover, Vermont, the Windham Regional Planning and Development Commission, and the Vermont State Planning Office, April 1972.
"Ecological Plumbing for the Texas Coastal Plain." Landscape Architecture 65 (1, January):78-89 (1975).
"A Strategy for a National Ecological Inventory." In A Database Prototype for a National Ecological Inventory. Washington, D.C.: U.S. Environmental Protection Agency, pp. 6-26, chap. 2 (1992).
# Index
Abercrombie, Patrick
Aboriginal peoples
Action, linking knowledge to:
"Ecological Planning Study for Wilmington and Dover, Vermont,"
"Ecological Plumbing for the Texas Coastal Plain,"
overview
"Plan for the Valleys vs. Spectre of Uncontrolled Growth,"
"Strategy for a National Ecological Inventory,"
Adaptation
Addison, Joseph
Administrative structure and water resources
Advocacy, social
Africa, North
Agriculture
coastal plain
Denmark and England
farmer steward of the landscape
highway route selection
natural processes, understanding
New York
physiography of a region
piedmont, the
positive and negative aspects
prime agricultural land
sensitive lands, recommendations for
uplands, the
value attributed to
Agriculture, U.S. Department of (USDA)
Airsheds
Alaska
Alexander, Christopher
Alluvium
American Planning and Civic Association
American planning theory, intellectual influences on
American Society of Landscape Architects
Amphibolites
Andropogon
Animal husbandry
see also Wildlife
Animism
Anthropology
Anthropomorphic–anthropocentric view
Anti-style
Aquifers
Architects/social scientists/landscape architects and communication problems
Architectural press
Architecture separate from biophysical processes
see also Landscape architecture
Aristocrat-landscape architect
Army Corps of Engineers, U.S.
Arsenic
Artificial and natural environments
Art(s):
fitness
prosthetic development
science of ecology tied to planning
and values
Asia, Southeast
Astronomy
Atmosphere
Atomic energy
Atrium house
Audubon Society
Australia, Western
Babbitt, Bruce
Ball, Arthur
Baltimore County (MD)
see also Plan for the Valleys
Barozzi de Vignola, Giacomo
Bartholomew, Harland
Bauhaus philosophy of interdisciplinary interaction
Beatty, Susan
Beauregard, Robert
Beaux Arts
Beaver, American
Belknap, Raymond K.
Benincasa, Eglo
Bennett, John
Berger, Jon
Bhan, Ravindra
Biochemical warfare
Biological Control of Water Pollution
Biology
see also Water pollution, biological alternatives to
Biophysical model
Black American Street Life (Rose)
Boston (MA)
Boulding, Kenneth
Bowden, Laura
Brown, Denise S.
Brown, Lancelot
Brown, Richard
Burchell, Robert
Bureau of Land Management (BLM)
Bush, George
Bye, A. E.
Cadmium
Caldwell, Alfred
California
Calthorpe, Peter
Carrying capacities
Carson, Rachel
Castells, Manuel
Catalyst, planner as
Causality
Center for Urban Ethnography
Central America
Central Intelligence Agency (CIA)
Central Park (NY)
Change, prohibitions and permissiveness to
Change, structure and agents of
Characterization, process of
Charles Eliot, Landscape Architect (Eliot)
Chermayeff, Serge
Chicago Columbian Exposition of 1893,
Children's awareness of environmental issues
Chinese housing
Chlorine
Christianity
Chronological method
Cities:
artificial and natural environments
assault on nature in
challenge, global
court houses
despoliation of resources
ecological method
fleeing the
future, the
growth, unparalleled urban
highway route selection
human ecosystems
modern metropolis
nature in
parks
planning process
pollution
population issues
regression, ecological
tension zone of rural-urban interpenetration
water pollution
Western views: man and nature
see also Open space, metropolitan
Clarke, Michael
Clay, Grady
Climate:
ecological planning
forests and woodlands
history of a landscape
inventory maps
open space, metropolitan
physiography of a region
Vermont, ecological planning study for
Woodlands, The (TX)
Clinton, Bill
Cluster zoning
Coal
Coastal plains
Coe, Jon
Coe Lee Robinson Roesch
Cohen,Yehudi
Collins, Michael G.
Commoner, Barry
Common orientation for planners in education and practice
Communication systems
Communities of land uses
Community structure
Competent people
Composite suitability map
Conquest and exploitation
Conservation Foundation
Constraints/opportunities, ecological planning and
Conventional wisdom
Cook County Park System (IL)
Coral
Le Corbusier
Cornell University
Corporations and water pollution
Cost-benefit analysis and highway route selection
Costruzioni Casabella
Court houses:
Africa, North
challenge to provide new
conclusions
essential quality of
Italy
Mies van der Rohe-Hilberseimer-Johnson
press, architectural
recent
sources, four principle
South and Central America
Coverage/clearing ratios
Creation, defining
Creativity
Creators, transforming from destroyers to
Crowley, Abraham
Cultural imperialism
Cultural memories
Curriculum for students in resource development, core
Cyanide
Dalton, Jim
Dangermond, Jack
Darwin, Charles
Data, incorporating new
Davidoff, Paul
Death
Deer, white-tailed
Defense Mapping Agency
de la Lama, Victor
De La Roziere.J.
Delaware River:
floods
pollution
regional landscape planning
sewage treatment
see also Highway route selection
Denmark
Dependence on natural processes, man's
see also Ecological listings
Descriptive ecology
Design, ecological:
adaptation
"Architecture in and Ecological View of the World,"
art vs. ecology
biophysical model
contemporary examples
"Court House Concept, The,"
Design, ecological (continued)
"Ecology and Design,"
Environmental Monitoring and Assessment Program
fitness
"Landscape Architecture,"
"Nature is More than a Garden,"
opposing camps within landscape architecture
overview
representation drawings
see also English landscape concept; Landscape architecture
Design of Residential Areas, The (Adams)
Design synthesis
Design theory
Design with Nature (McHarg)
compatible ideas
Mumford, Lewis
Plan for the Valleys, The
social factors in planning process
Steiner, Frederick
Despoliation, development without
Destroyers to creators, transforming from
Development
see also Plan for the Valleys; Vermont, ecological planning study for; Woodlands, The (TX)
Development Corporation
Development Research Associates
Dewey, John
Dewey, Margaret
Disasters, national/natural
Discordant Harmonies (Botkin)
Disease, man as a planetary
Domestication of grazing animals
Douglas, John
Dover (VT), see Vermont, ecological planning study for
Drinking water, assumptions about
Drought
Duality of man and nature
Duany, Andres
Dubos, René
Dune formation
Earth, early
Earth Day
Earthquakes
Eastern tradition
Ecological determinism:
conclusions
description of natural processes
Green Spring and Worthington Valley
inventory, ecosystem
see also Inventory, ecosystem
limiting factors, identification of
New Jersey shore
open space, metropolitan
six elements
stability or instability
value, attribution of
Ecological view:
change, prohibitions and permissiveness to
cities
comprehensive plan
data, incorporating new
defining ecology
elements, landscape
English landscape concept
environmental vs. ecological design
flowchart of the planning process
holistic approach to analyzing landscapes
interpretation and synthesis of information
landscape architecture
links between elements and proposed land uses
open space, metropolitan
opportunities/constraints
planning
regression, ecological
Sears, Paul
unity of life covering the earth
see also Design, ecological; Human ecological planning; Methods and techniques for ecological planning; Vermont, ecological planning study for; Woodlands, The (TX)
Economic value system
Ecosystem inventory, see Inventory, ecosystem
Ecosystems, human
Edinger, John
Educating the public
Education, landscape architecture
Edwards, Jonathan
Ehrlich, Paul
Eighteenth–century attitude
Einsweiler, Robert
Eiseley, Loren
Elements, landscape
Eliot, Charles
Eliot, Charles W. (father)
Ellin, Nan
Emerson, Ralph W.
Energy, U.S. Department of
Energy use and production
atomic energy
knowledge of the environment, increasing
nature giving direction
needs met without destroying landscape
photosynthesis
planning
Scotland
social costs
solar and wind
value system of the problem solver
England
English landscape concept:
ecological view
functional landscapes
nature as a garden
pastoral aesthetic, known as the
Renaissance and eighteenth century
transformation by the
Entropic-misfitness-morbidity
Environment:
artificial and natural
decade, environmental
disinterest of scientists/institutions around the
ecological vs. environmental vs. design
impact statement, environment (EIS)
lab, U.S., national
public policy
science
social value of a given
see also Natural processes, understanding; Nature
Environmental Monitoring and Assessment Program (EMAP)
beginnings of
chronology as the structuring device
conclusions and recommendations
ecological inventory
executive committee
expertise, regional
recommendations, basis for
regions and stratified sampling
research, environmental
synthesis
three-part inventory
title
Environmental Protection Agency (EPA)
Eternal questions connected to everyday practice
Ethnographic history
Europe
Evolutionary history of a region
Evolutionary process, nature and man as an
Evolution as reverse to reduction
Ewan, Joe
Executive committee and Environmental Monitoring and Assessment Program
Expertise
Exploitation and conquest
Farmer steward of the landscape
see also Agriculture
Farnsworth House
Fenneman, N. M.
Fill material
Fire
Fish and Wildlife Service, U.S.
Fitness:
creative fitting
defining
design, ecological
form and process
human ecological planning
planning process
Fitness of the Environment, The (Henderson)
Flexner Report
Floods:
alluvium
cities
coastal plain
flood plain parameter
man, consequences of the acts of
Mississippi River
peaks
piedmont, the
Plan for the Valleys
restraints to, natural
state regulations
uplands, the
Woodlands, The (TX)
Flowchart of the planning process
Folklore
Ford, Gerald
Ford Foundation
Forests and woodlands:
climate
coastal plain
destroying
open space from natural processes
piedmont, the
Plan for the Valleys
uses, recommended
Vermont, ecological planning study for
Woodlands, The (TX)
Forman, Richard T.
Form and process
Forshaw, J. H.
Fortune magazine
Fosberg.F. R.
Foundations, building
Fragmentation of knowledge/government
Francis, Mark
Franklin, Carol
Franklin, Colin
Free-standing single-family house
Friedmann, John
Fuller, Buckminster
Furtado, John G.
Future, the
Gaia hypothesis
Gans, Herbert
Gap Analysis Program
Garden City idea
Garden design
see also English landscape concept
Geddes, Patrick
Generalists in resource development and management
Genetic system of communication
Genius of the site
Geographic information systems (GIS)
Geological Survey, U.S.
Geology:
Environmental Monitoring and Assessment Program
history of a landscape
inventory maps
planning process
plate tectonics
Vermont, ecological planning study for
Gingrich, Newt
Giurgola, Ronaldo
Gladstone Associates
Glickson, Arthur
Global positioning systems (GPS)
Glotfelty, Karen
Goddard, David
Godron, Michel
Gordon, Steven I.
Gosberg, F. R.
Gottman, Jean
Government:
fragmentation of
Vermont, ecological planning study for
water pollution
Graham Foundation
Gray, Thomas
Great Britain
Great Valley (Delaware River Watershed)
Greek housing
Green Belt
Green Mountains
Green Spring (MD)
see also Plan for the Valleys
Grigsby, William
Gropius, Walter
Groundwater, see Water-related processes
Growth issues
Halprin, Lawrence
Hamilton, Lawrence
Hamme, David
Harriman Reservoir
Harris, Britton
Hartsfield, Robert
Harvard University
Harvey, David
Haslund, Ralph
Haynes, Merrill
Hazardous resources/occupations
Healing the Earth
Healthy environments
Henderson, Lawrence
Hester, Randy
Highway route selection:
comprehensive method, application of the
criteria for interstate
inadequacy of criteria for
maps, superimposed
parameter analysis
Plan for the Valleys
productive land uses
social values
suitability analyses
Vermont, ecological planning study for
Hilberseimer
Hills, George A.
Hispano-American tradition
Historic analyses of economic/social processes
Historic value and highway route selection
History of a landscape
Holden
Holism and synthesis
Homeostasis
Hopkins, Gerald M.
Housing:
market analysis
multi-story flat
open space, metropolitan
prototypes
second homes
soil, runoff holding capacity of
types
see also Court houses
Housing and Urban Development, U.S.
Department of (HUD)
Howard, Ebenezer
Howe, Verne
Howell, G.
Hubbard, Henry
Hudnut, Dean
Huffman, Benjamin
Human ecological planning
defining terms
fitness
method
model for region under study
national policy
populating region with inhabitants
syntropic-fitness-health
Human ecological planning (continued)
theory which impels
tremendous work/costs involved
values and instruments to attain objectives
Human ecosystems
Hunt, Charles B.
Hurricanes
Hydrology
see also Water-related processes
Illinois
Industry and water pollution
Instabilities/stabilities, identifying
Institute for Urban Studies
Interagency Ecosystem Management Task Force
Interdependence characterizing all relationships
see also Ecological listings
Interior, U.S. Department of the
Intrinsic value
Introversion and court houses
Inventory, ecosystem:
anthropology
data bank for further investigations
descriptive ecology
detail, level of
eighteenth century landscape architects
opportunities/constraints for land uses
overlay technique
unique/scarce/sensitive areas
see also Environmental Monitoring and Assessment Program (EMAP)
Inversion, temperature
Isard, Walter
Island of the Day Before, The (Eco)
Italy
Jackson, J. B.
Japanese housing
Jensen, Jens
Johnson, Arthur
Johnson, Bernard
Johnson, Lady Bird
Johnson, Lyndon
Johnson, Philip
Johnson, T. H.
Jones, Carolyn
Jones, Leslie
Jones & Jones
Journals, scholarly and professional
Joyce, Richard
Judaic-Christian-Humanist tradition
Juneja, Narendra
Kahn, Lou
Kent, William
Kiley, Dan
Kimball, Theodora
Knight, Richard P.
Knowledge, fragmentation of
Korea
Kreditor, Alan
Kristensen, Eske
Kuchler, A. W.
Kull, Nell M.
Lake Erie
Lake Raponda
Land:
abundant land exploited for maximum good
acquisition and disposition, public
availability, test of
highway route selection and productive land use
matrix for possible land uses
prime agricultural
sensitive, recommendations for
use and technical change/social events
values
Landscape and Urban Planning
Landscape architecture:
bridge between science/design/ planning
Carson, Rachel
communication problems
criterion for guiding/evaluating
ecological view
education for
Eliot, Charles
emergence of
English landscape concept
environmental science
Halprin, Lawrence
Le Nôtre, André
Leopold, Aldo
Marx, Roberto B.
Nolen, John
Olmsted, Frederick L.
opposing camps within
science, environmental
sieve-mapping overlays
students influenced by McHarg
training
Western tradition
see also Design, ecological; English landscape concept
Landscape paintings
Landscape tolerance
L'Architettura (Benincasa)
Law
Layer cake model
see also Human ecological planning
Lead
Lee, Gary
Legislation:
Capper Cromptin Act
Clean Water Act
Housing Act of 1961,
National Environmental Policy Act (NEPA)
open space, metropolitan
Vermont Act
Vermont Act
water pollution, biological alternatives to
Water Pollution Control Act
Le Nôtre, André
Leopold, Aldo
Leopold, Luna
Le Play, FrÈdÈric
Lewis, Philip
Lewis Mumford and the Ecological Region (Luccarelli)
Libera, Adalberto
Lieberman, Arthur S.
Life zones
Ligorio, Pirro
Limited development areas and recommended uses
Limiting factors external to the ecosystem
Limnological theory
Local regulations
Location, solving the problem of
Lorrain, Claude
Lovelock, James
Low, Setha
L-shaped court houses
Luccarelli, Mark
Lyle, John
MacDougall, E. B.
MacKaye, Benton
Man and environment:
atomic energy
disease, man as a planetary
duality of
ecological view
natural science and naturalism
power, evolution of
Renaissance and eighteenth century
resources, despoliation of
Western views
see also individual subject headings
Manheim, Marvin
Manning, Warren
Mapping information:
data bank for further investigations
detail, level of
Environmental Monitoring and Assessment Program
highway route selection
interpretation of superimposed maps
inventory, ecological
overlaying mapped information
Mapping information (continued)
Vermont, ecological planning study for
Woodlands, The (TX)
Margulis, Lynn
Marsh, George P.
Marsh, Jonathan
Marshes
Martino, Steve
Marx, Roberto B.
Maryland, see Plan for the Valleys
Massachusetts Institute of Technology
Materialism
Maxman, Susan
McCall, Tom
McCafferty, Robert
McDonough, Bill
McSheffrey, Gerald
Media, the
Medicine
Mediterranean
Meing, D. W.
Mennhenick, Howard
Merchants' creed
Metaphysical symbols
Methods and techniques for ecological planning:
"Biological Alternatives to Water Pollution,"
"Case Study in Ecological Planning: The Woodlands, Texas,"
"Comprehensive Highway Route Selection Method, A,"
"Ecological Method for Landscape Architecture, An,"
overview
Vermont
see also Vermont, ecological planning study for; Woodlands, The (TX)
Metropolis, modern
see also Cities
Microclimate
Microorganisms in water/soil and their symbionts
Miller, Lynn
Minger, Terrell
Minimize-nucleated concept
Minnesota
Mississippi River
Mitchell Energy and Development Corporation
Modernism
Molecular biology
Morbidity
Moslem house, traditional
Movement corridors
Multiply and Subdue the Earth (film)
Multi-story flat
Mumford, Lewis
Nader, Ralph
National Aeronautics and Space Administration (NASA)
National Biological Survey (NBS)
National ecological inventory, see Environmental Monitoring and Assessment Program (EMAP)
National Environmental Laboratory
National Institutes of Health
National Institutes of the Environment
National Oceanic and Atmospheric Administration (NOAA)
National policy
National Science Foundation
Natural and artificial environments
Naturalism, natural science and
Naturalist tradition in the West
Natural processes, understanding:
agriculture
energy use and production
open space, metropolitan
optimism for the future
professional disciplines, need for range of
Vermont, ecological planning study for
water pollution, biological alternatives to
see also individual subject headings
Natural Resources Conservation Service
Natural science and naturalism
Nature:
cities
dependence on, man's
duality of man and
energy needs and looking for directions from
as a process
selected
separateness and dominance of man over
simplification of ecosystems
steward of
transcendental view of man's relation to
value system, intrinsic
Western attitudes toward
see also individual subject headings
Naveh, Zev
Negative value
Negentropy
Neo-Marxists
Netherlands
Neutra, Richard
New Canaan House
New England
New Jersey
see also Highway route selection
New York
New Zealand
Niagara Falls
Nitrates
Nolen, John
Non-renewable resources
Notzold, Richard
Oaks, Sherry
Objectives and human ecological planning
Occupations, hazardous
Odum, Eugene
Odum, Howard T.
Oil
Olmsted, Frederick L.
Olmsted, Frederick L., Sr.
Olsen, Donald
Open space, metropolitan: abundant land exploited for maximum good
airsheds
Baltimore County (MD)
conclusions
creating
criteria for purchasing sites for
Delaware River Basin
developing
ecological approach
exceptions to the general experience
forests and woodlands
Friedmann, John
Hamilton, Lawrence S.
housing, modern
Housing Act of
land values
legislation, analysis of existing/ proposed
mapping and measuring
natural processes, understanding
objective/systematic methods for identifying
paradox/tragedy of metropolitan growth/suburbanization
Philadelphia Standard Metropolitan Statistical Area
professional disciplines, need for range of
reconciling different criteria
recreation demand
values
water and natural processes
Woodlands, The (TX)
Opportunities/constraints, ecological planning and
Optimism, reasons for
Organic approach to architecture
Organismic biology
Orient, the court in the
Orthodox city planning tradition
Orthogonal investigations
Oswald, E.
Outdoor Recreation Resources Review Commission
Overlay mapping process
Painting, landscape
Palmer, Arthur
Paper-oriented investigations
Parameter analysis and highway route selection
Park, James
Parks, urban
Park Service, U.S. National
Partridge, J. A.
Paternoster, Robert
Patrick, Ruth
Paxton, Joseph
Payne, Morse
Penn, William
Pennsylvania
Performance requirements
Periera, William L.
Permissiveness to urban uses
Philadelphia Standard Metropolitan Statistical Area (PSMSA)
see also Open space, metropolitan
Phosphates
Photo electric cells
Photosynthesis
Photovoltaic cells
Physical environment, creating a
Physiography of a region
agriculture
aquifers
climate/geology and interpreting the
coastal plain
Environmental Monitoring and Assessment Program
flood plains
highway route selection
maps, inventory
marshes
physiographic determinism
piedmont, the
steep lands
surface water
uplands, the
Vermont, ecological planning study for
water pollution, biological alternatives to
water regimen, principal roles in the
Physiology of man
Piedmont, the
Pierson, Robert
Plan for the Valleys:
analysis and proposals
development value
features of the valleys, three main
genius of the site
importance of
originality of the study
physical/financial consequences extended into the future
physiographic determinism
powers, accumulation of
prohibition of development
real-estate syndicate
technical report
Planning in the Public Domain (Friedmann)
Planning process:
cities
comprehensive method
conclusions
devolution of planning/management
disasters, natural
ecological see also Vermont, ecological planning study for; Woodlands, The (TX)
energy use and production
environment in public policy
fitness
fragmentation of government/ knowledge
geology
health and safety, natural processes affecting
inadequacies of
public
social factors in
suitability analyses
water pollution, biological alternatives to
see also Human ecological planning; Methods and techniques for ecological planning; Open space, metropolitan; Regional landscape planning; Woodlands, The (TX)
Plants, see Agriculture; Forests and woodlands; Vegetation
Plater-Zyberk, Elizabeth
Plate tectonics
Podzolization
Policy, national
Pollio, Marcus V.
Pollution
see also Water pollution, biological alternatives to
Pope, Alexander
Population issues:
cities
human ecological planning
Plan for the Valleys
threat to modern man
Vermont, ecological planning study for
Postmodernism
Poussin, Nicolas
Power, evolution of
Powers, accumulation of
Precursory ecology
Price, Uvedale
Prigogine, Ilya
Princeton University
Private action
Process, nature as a
Productivity, value as
Professional disciplines, need for range of
Prohibitions and permissiveness to change
Projects, see Action, linking knowledge to
Prospectus (1988)
Prosthetic development
Protection map
Prototype Database for a National
Ecological Inventory, A (Radke,
Berger, & Wallace)
Public awareness
Public education
Public planning
Public policy, emergence of environment in
Public powers
Radke, John
Rapson, Ralph
Raritan River, see Highway route selection
Ratte, Charles
Reading Prong (Delaware River Watershed)
Reagan, Ronald
Real-estate syndicate
Recreational value and highway route selection
Recreation demand
Reductionism
Regenerative communities and landscapes
Regional landscape planning:
"Ecological Planning: The Planner as Catalyst,"
Environmental Monitoring and Assessment Program
"Human Ecological Planning at Pennsylvania,"
"Open Space from Natural Processes,"
overview
"Regional Landscape Planning,"
vocabulary
see also Vermont, ecological planning study for; Woodlands, The (TX)
Regional Planning Association of America (RPAA)
Regulations, state/local
Reilly, William
Reiss, Ellen
Renaissance period
Replacement value
Representation drawings
Repton, Humphry
Research, environmental
Reservoirs
Residential quality and highway route selection
Resources:
curriculum in resource development, core
despoiling
hazardous
inventory maps
natural processes interpreted as
non-renewable
research in developing
Rexroth, Kenneth
Riehle, Ted
Riparian lands
Rockefeller House
Rocky Mountain West, see Energy use and production
Roman housing
Roofs and walls, green
Rosa, Salvator
Rose, Dan
Rushman, Michael
Safety, natural processes affecting
Sagan, Carl
Sampling, regions and stratified
Sand County Almanac, A (Leopold)
San Francisco
Sauer, Rolf
Scenic values
Schmidt, William
Schools
Schoop, E. Jack
Schwindowski, George
Science and landscape architecture
Science vs. art
Scientific knowledge of the environment, see Natural processes, understanding
Scotland
Sears, Paul
Secondary open space
Second homes
Sedway, Paul
Selected nature
Selenium
Selkregg, Lidia
Semyk, J. P.
Sensitive lands, recommended uses for
Septic tanks
Sert, José
Settlement patterns
Sewage treatment
Shaftesbury, earl of
Shenstone, William
Sierra Club
Sieve-mapping overlays
Silent Spring (Carson)
Single-family house, free-standing
Sinton, John W.
Site design/planning
Skiing
Social advocacy
Social Creation of Nature, The (Evernden)
Social epidemiology
Social events and land use/settlement patterns
Social factors/costs:
energy use and production
highway route selection
planning process
Social justice
Social sciences
see also Natural processes, understanding
Social scientists and communication problems
Social value of a given environment
see also Value(s)
Soil Conservation Service
Soil Mechanics Institute at Delft
Soils
coastal plain
erosion
highway route selection
history of a landscape
housing suitability and runoff holding capacity of
maps, inventory
microorganisms and their symbionts
piedmont, the
plant history and nature of
prime agricultural land
scientists
uplands, the
Vermont, ecological planning study for
Woodlands, The (TX)
Solar energy
Soleil, Le Roi
Somerset Reservoir
South America
South Pacific
Space fantasy and man/nature relationship
Stabilities/instabilities, identifying
Staten Island (NY)
State regulations
Steep lands
Stein, Clarence
Steiner, Frederick
Steinitz, Carl
Stengers, Isabelle
Sternlieb, George
Stone, Edward
Stonorov, Oskar
Storm drainage
Strong, Ann
Stutz, Frederick
Style, anti-
Suburban sprawl
see also Cities; Open space, metropolitan
Suitability analyses
see also Woodlands, The (TX)
Surface water
Surficial geology
Sutton, Jonathan
Symbiosis
Synthesis and holism
Syntropic-fitness-health
Szwed, John
Taiwan
Taxes
Technological change and land use/settlement patterns
Temperature inversion
Temple, William
Ten Eyck, Christine
Tertiary open space
Texas, see Woodlands, The (TX)
"The House We Live In" (TV show)
Theology of man-nature-God
Theoretical writings:
"Ecological Determinism,"
"Man and Environment,"
"Natural Factors in Planning,"
overview
"Place of Nature in the City of Man,"
"Values, Process and Form,"
Thomson, James
Thoreau, Henry D.
Togetherness, pathological
Tolerance, landscape
Topography and highway route selection
Tourbier, Joachim
"Towards a Comprehensive Plan for
Environmental Quality" (Wallace,
McHarg, Roberts & Todd)
Town and Country Planning Textbook
Town statutes
Train, Russell
Training
Transcendental view of man's relation to nature
Transformation of natural environments into human habitats
Trout
Trustees of Public Reservation
Turner, Elva
Tyrwhitt, Jacqueline
United Nations
United States
see also individual subject headings
Unity of life covering the earth
University of California-Berkeley
University of Georgia
University of Illinois
University of Massachusetts
University of North Carolina
University of Pennsylvania
see also Human ecological planning
Uplands, the
Urban planning
see also Cities; Open space, metropolitan
Urban Renewal Administration
Utzon, John
Valley and Ridge Province (Delaware River Watershed)
Value(s):
creative process, the world as a
ecological determinism
economic value system
energy use and production
environment, social value of a given
genetic value system
highway route selection
human ecological planning
Judaic-Christian-Humanist tradition
location, solving the problem of
mapping information
nature's intrinsic value system
open space, metropolitan
planning solutions
reassurance and affirmation of
Vermont, ecological planning study for
van der Rohe, Mies
van der Ryn, Sim
Van der Wal, Coennard
Vegetation:
ecological planning
evolution of
maps
roofs and walls, green
soil patterns and history of
Vermont, ecological planning study for
Woodlands, The (TX)
see also Agriculture; Forests and woodlands
Venturi, Robert
Vermont, ecological planning study for
aesthetics/natural areas/historic sites
carrying capacities
change, structure and agents of
climate
epilogue
future, strategy for the
geology
governmental services
growth issues
highway congestion and safety
land acquisition and disposition, public
land and the people
life zones
methods and techniques for ecological planning
natural processes, place as sum of
past, the immediate
physiography of a region
planning process, public
preface
private action
recommendations, summary of
regulations, state and local
schools
soils
suitability, intrinsic
tax policies
values
vegetation
water-related processes
wildlife
Vernacular architecture in the Western tradition
Village of Grogan's Mill (TX)
Voluntary associations
Wald, George
Wallace, David
Wallace, Kathleen
Wallace, McHarg, Roberts, and Todd (WMRT)
Water pollution, biological alternatives to:
administrative structure
biological systems, clean/potable water and
devolution of planning/management
drinking water, assumptions about
government
industry, modern
legislation
planning methods, improving
Water-related processes:
cities
Delaware River, flood capacity of the
Environmental Monitoring and Assessment Program
groundwater
highway route selection
history of a landscape
open space planners
physiographic phenomena, roles of
Plan for the Valleys
state regulations
storm drainage
value attributed to
Vermont, ecological planning study for
Woodlands, The (TX)
Waugh, Frank
West, Rocky Mountain, see Energy use and production
Westbroek, James
Western tradition:
cities
duality of man and nature
landscape architecture
social justice
space fantasy and man's relation to nature
transcendental view of man's relation to nature
White House Task Force
Whitman, Walt
Wiken, E. B.
Wildlife
domestication of grazing animals
ecological planning
highway route selection
husbandry, animal
inventory maps
plant-related
Vermont, ecological planning study for
Woodlands, The (TX)
Williams, Rodney
Wilmington (VT), see Vermont, ecological planning study for
Wilson, Edward
Wilson, James
Wind energy
Wisconsin
Wise, Harold
Witala, S. W.
Witherspoon, Gerald
Woodlands, see Forests and woodlands
Woodlands, The (TX):
availability, test of land
challenges of the site
community structure
coverage/clearing ratios
design synthesis
development, planning for
first general plan
hazards to life and health
housing prototypes
landscape processes and development activities
lessons to be learned from
movement corridors
nature of the site
opportunities/constraints
phase I planning
problems presented by
revisions, general plan
Woodlands, The (TX) (continued)
site planning/design
summary
Village of Grogan's Mill
Wallace, McHarg, Roberts, and Todd (WMRT)
water-related processes
woodland environment, preserving the
Woodwell, George
Work performed, value as
Worthington Valley (MD)
see also Plan for the Valleys
Wright, Frank L.
Wright, Henry
Yannacone, Victor
Yaro, Robert D.
Yosemite State Park
Zilliacus, Peter
Zoning
# Island Press Board of Directors
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Executive Director, Pew Global Stewardship Initiative
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Environmental Lawyer
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a
Since this essay was written, the Clean Air Acts, the Clean Water Acts, and several other laws have established standards for environmental quality and have placed limits on pollution.
| {
"redpajama_set_name": "RedPajamaBook"
} | 2,963 |
Getting to know the City of Trees one event venue at a time.
Boise, Idaho's capital, gets its name from bois, the French word for tree, and the city does not disappoint in that department. During a three-day stay in late September as part of a fam trip hosted by the Boise Convention & Visitors Bureau, I saw hundreds upon hundreds of the more than 25,000 "street trees" maintained by the city's Community Forestry program.
After a short layover in Seattle, I touched down at Boise Airport and hopped a 10-minute shuttle to the Grove Hotel, the only AAA Four-Diamond property in the City of Trees. The 250-room Grove would serve as the home base for our group of journalists and meeting professionals for the duration of our stay. The hotel sits at the intersection of Capitol and Grove streets in downtown Boise, and both the green and wood tones in the lobby and the abstract mural of interwoven aspen branches that covered one wall of my seventh-floor room brought the destination's woodland spirit indoors.
After touring the Grove's 36,000 square feet of event space, including the 6,845-square-foot Grand Ballroom, we peered down at the multipurpose CenturyLink Arena from one of the hotel's large breakout rooms. Next up was Boise Centre, where an expansion last summer nearly doubled the convention facility's meeting and event footprint to 86,000 square feet, which includes 29 meeting rooms and larger spaces like the nearly 25,000-square-foot Eyries Room. After the walkthrough wrapped up, we sat down to a hearty steak-and-potato dinner prepared by Boise Centre's in-house catering team.
The next morning, we headed over to the six-story Holiday Inn Express Boise–University Area for a tour and a quick buffet breakfast; the property offers 159 guest rooms and 2,000 square feet of meeting space. Next on the agenda was JUMP! (Jack's Urban Meeting Place), a nonprofit community venue with a massive lobby for receptions, classrooms, a terrace, and a rooftop deck complete with a giant slide where adults can — and should — test their daring.
We spent the rest of the morning on a quick city tour that included the Anne Frank Human Rights Memorial, with its winding, 108-foot quote wall inscribed with the wisdom of both the well-known and the unknown; and 89.4-acre Julia Davis Park, which encompasses Boise's zoo and several small museums.
Ste. Chappelle, Idaho's oldest winery.
After a tour of the Hilton Garden Inn Boise Spectrum — a 137-room hotel with 4,000 square feet of event space, including the 2,924-square-foot, divisible Garden Room — we visited two vineyards on Idaho's Sunnyslope Wine Trail. Vast, Gothic-inspired Ste. Chapelle is Idaho's largest winery. There, our group sampled four vintages, including a tart, Riesling-grape-based ice wine. Ste. Chapelle's tasting room, mezzanine, and grounds are all available for groups. At nearby Huston Vineyards, which can also host private events, we sampled boutique vintages like the label's Chicken Dinner red and white blends, as well as its full-bodied 2014 Malbec.
After we'd drunk our fill, it was back to Boise for a walkthrough of the 186-room Hampton Inn and Suites Boise Downtown, which offers 4,000 square feet of meeting space. Next we were off to the 303-room Riverside Hotel for a tour of its 21,000-plus square feet of event space, followed by a seated dinner where we were served flank steak topped with mushrooms, a slice of Idaho salmon, and — of course — more potatoes.
The next day, we were up bright and early to check out the 162-room Courtyard Boise Downtown and its 1,000 square feet of meeting and event space. Our next stop was the Old Idaho Territorial Penitentiary — a historical attraction that from 1870 to 1973 served as an actual prison — followed by a drive through the Idaho Botanical Gardens, where visitors can see some 800 species of plant life.
More Basque-Americans live in the greater Boise area than anywhere else in the United States, and so a visit to the city's Basque Museum and Cultural Center was a must. After a tour, we snacked on pinxtos — Basque small plates like fried cod balls and spicy fried peppers — saw a traditional dance performance, and browsed through the handicrafts at an Old World–style market.
After a walkthrough of the 182-room Red Lion Hotel Boise Downtowner, which can host groups of up to 250 in its 8,425 square feet of meeting space, we drove an hour north of the city for a lazy afternoon of rafting under the able guidance of Cascade Raft and Kayak. After we dried off, we dined on grilled flank steak and potatoes au gratin at picnic tables overlooking the banks of the winding Payette River before returning to the Grove to rest up. Because it turns out Boise is pleasantly exhausting. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,860 |
\section{Introduction}
\textsf{Alt-Ergo}\xspace is an SMT solver designed for checking logical formulas generated
by deductive program verification frameworks. For instance, \textsf{Alt-Ergo}\xspace is
used as a back-end in the \textsf{Why3} plateform~\cite{why3}. It is
also used to discharge formulas derived from C programs in
\textsf{Frama-C}~\cite{framac2015}, from Ada programs in
\textsf{SPARK}~\cite{spark} or from B machines in
\textsf{Atelier-B}~\cite{AtelierB, AE-RSSR-2016}.
The \textsf{Alt-Ergo}\xspace input files produced by such tools share the same
structure. They start with a prelude that contains a set of
definitions (datatypes and logical symbols) and axioms for the
encoding of theories specific to program verification (complex data
structures, memory models, etc.). The rest of the file contains a
proof obligation (PO) generated by a weakest precondition calculus.
\textsf{Alt-Ergo}\xspace checks such input files by the use of a combination of decision
procedures (SAT, simplex, congruence closure, etc.) for reasoning
about builtin theories (Booleans, arithmetics, equality, etc.) and a
matching algorithm for instantiating quantified formulas.
Due to undecidability of first-order logic, \textsf{Alt-Ergo}\xspace may not terminate on
some problems. When this happens, one would like to find the reasons
why the solver fails. Most of the time, \textsf{Alt-Ergo}\xspace is either overwhelmed by
a huge number of useless instances of axioms (causing a high activity
in its decision procedures), or it fails to produce the good
instances of lemmas that are mandatory to prove a goal.
A possibility for inspecting the internals of the solver is to output debugging
information. This is however impractical because there is simply too many
things to display and the output rapidly becomes unreadable. To help users (or
developers) find out and understand what is going on, we have designed \textsf{AltGr-Ergo}\xspace, a
graphical user interface for \textsf{Alt-Ergo}\xspace. As shown in Figure~\ref{fig:overview}, our
GUI displays at runtime crucial profiling information about the internal
activities of the solver (time spent in decision procedures, number of
instantiations, etc.). Some interaction features have also been added so that
one can manually help the solver prove a goal (manual instances of lemmas,
selection of hypotheses, etc.).
The main features of \textsf{AltGr-Ergo}\xspace are described in the next sections.
The interface can display the following profiling information:
\begin{itemize}
\item unsat cores (Section~\ref{sec:unsat})
\item number of instances produced by axiom (Section~\ref{sec:instances})
\item time spent in decision procedures (Section~\ref{sec:profiling})
\end{itemize}
Interactive features include the following syntactic manipulations:
\begin{itemize}
\item pruning operations (Sections~\ref{sec:pruning} and
\ref{sec:dep}) for deactivating some part of the prelude;
\item manual instances of lemmas (Section~\ref{sec:manual});
\item selection and modification of triggers
(Section~\ref{sec:triggers}) to change the heuristics used to guide
the matching algorithm.
\end{itemize}
Last but not least, \textsf{AltGr-Ergo}\xspace provides a session mechanism
(Section~\ref{sec:sessions}) which allows a user to save and replay
all his modifications (selections, manual instances, etc.) on a given
problem.
\begin{figure}[htb]
\centering
\includegraphics[width=1.0\textwidth]{agr.png}
\caption{Overview of \textsf{AltGr-Ergo}\xspace's interface}
\label{fig:overview}
\end{figure}
\section{A Short Introduction to Alt-Ergo}
\label{sec:matching}
In order to understand some aspects of the graphical interface, we
briefly present in this section \textsf{Alt-Ergo}\xspace{}'s syntax and a high level overview of
its main components.
The input language of \textsf{Alt-Ergo}\xspace{} is an extension of first-order logic with
builtin theories and prenex polymorphism\footnote{Type variables, if
any, are prenex and implicitly universally quantified.} \emph{\`a
la} ML~\cite{AE_polymorphism}. Figure~\ref{fig:ex} shows a small
proof obligation written in \textsf{Alt-Ergo}\xspace's syntax: first, a type symbol
\texttt{set} parameterized by a type variable $\mathtt{\alpha}$ is
declared. Then, a polymorphic function (\textit{resp.} predicate)
symbol \texttt{add} (\textit{resp.} \texttt{mem}) is
introduced. After that, an axiom \texttt{mem\_add} that gives the
meaning of membership over the \texttt{add} symbol is stated. Finally,
two integer symbols \texttt{a} and \texttt{b}, two sets of integers
symbols \texttt{s1} and \texttt{s2}, and a goal are given.
In addition to the Boolean connective ``$\to$", the ``toy goal'' mixes
symbols from two theories: the free theory of equality (\texttt{mem},
\texttt{add}, \texttt{a}, \texttt{b}, ...), and linear arithmetic
(\texttt{+, -, 1}). It is made of two parts: the hypotheses \texttt{a
= b + 1} and \texttt{s2 = add(b,s1)}, and the conclusion
\texttt{mem(a - 1, add(b,si))} we would like to prove valid. Thanks to
the second hypothesis and a \textit{ground instance} of axiom
\texttt{mem\_add} (where {\tt x} is replaced by {\tt a - 1}, {\tt y}
by {\tt b}, {\tt s} by {\tt s1} and $\alpha$ by {\tt int}), the
conclusion is equivalent to \texttt{(a-1 = b $\lor$ mem(a-1,
s1))}. Moreover, the latter formula always holds because \texttt{a -
1 = b} is equivalent to the first hypothesis modulo linear
arithmetic. We thus conclude that the goal is valid.
\begin{figure}[htb]
\centering
\begin{minipage}[h]{9cm}
\hrule
\begin{tcbfl}[9cm]{altergo}
type 'a set
logic add: 'a, 'a set -> 'a set
logic mem: 'a, 'a set -> prop
axiom mem_add:
forall x, y: 'a. forall s: 'a set.
mem(x, add(y, s)) <-> (x = y or mem(x, s))
logic a, b: int
logic s1, s2: int set
goal g:
a = b + 1 ->
s2 = add(b,s1) ->
mem(a - 1, s2)
\end{tcbfl}
\caption{An example problem in \textsf{Alt-Ergo}\xspace's syntax}
\label{fig:ex}
\hrule
\end{minipage}
\end{figure}
\textsf{Alt-Ergo}\xspace{} handles such proof obligations following the architecture given
in Figure~\ref{fig:archi}. The solver can be called either via its
command-line ``\texttt{alt-ergo}'' or via its graphical user interface
``\texttt{altgr-ergo}''. The front end provides some basic operations
such as parsing, type-checking, triggers inference\footnote{this notion is
crucial to control how axioms are instantiated, and is
explained at the end of this section} and translation of input
formulas to richer data structures manipulated by back end modules.
\begin{figure}[htb]
\centering
\includegraphics[width=9cm]{archi.pdf}
\caption{\textsf{Alt-Ergo}\xspace's simplified architecture}
\label{fig:archi}
\end{figure}
The SAT solver plays a central role in \textsf{Alt-Ergo}\xspace{}. Given a formula, it
tries to build a (partial) Boolean model for the ground part that is
neither contradicted by the decision procedures, nor by the instances
generated from (universally quantified) axioms. Its main operations
are guessing truth values of (immediate) sub-formulas appearing in
disjunctions (\textit{decision}) and propagating unit facts that have
been deduced (\textit{bcp}). Atomic formulas (literals) are sent to
``decision procedures'' to check if they are consistent in the union
of supported theories, and universally quantified formulas are sent to
an ``axioms instantiation'' engine. If an inconsistency that does not
involve any decision is detected, the given goal is valid\footnote{To
prove validity, \textsf{Alt-Ergo}\xspace{} internally assumes the negation of the
conclusion and tries to deduce unsatisfiability.}. Otherwise, when
the SAT reaches a fix-point (\textit{i.e.} succeeds in building a Boolean
model), it asks the ``axioms instantiation'' part for some new ground
instances. These instances are added to the SAT's context and
reasoning continues.
Decision procedures component provides a combination of decision
algorithms for a collection of built-in theories. \textsf{Alt-Ergo}\xspace{} supports some
theories that are useful in the context of program verification, such
as the free theory of equality with uninterpreted symbols, linear
arithmetic over integers and rationals, fragments of non-linear
arithmetic, polymorphic functional arrays with extensionality,
enumerated and record datatypes, and associative and commutative (AC)
symbols. More details of our combination techniques, which are not
necessary to understand the rest of the paper, can be found
here~\cite{ccx, acx, thesisMohamed}.
To reason about axioms, \textsf{Alt-Ergo}\xspace{} uses an instantiation mechanism based
on E-matching~\cite{e-matching} techniques. It generates ground
consequences from assumed axioms based on some heuristics and
information provided by the SAT solver and the decision
procedures. The challenge is to heuristically produce useful instances
that will allow to discard the current SAT's model, thus reducing
the search space, and hopefully derive unsatisfiability (validity).
In addition to axioms, the instantiation engine requires a set of
ground terms, and their partition into equivalence classes (computed
by the decision procedures). In general, considered ground terms are
those that appear in the decision procedures environment when
instantiating. If this does not generate any instance, all the ground
terms that appear in the current Boolean model are considered.
Another key ingredient is the use of the notion of triggers
(\textit{a.k.a} patterns or filters) to guess which instances may be
relevant depending on the SAT and decisions procedures' context.
A trigger for an axiom $\psi \equiv \forall \vec x. \phi(\vec x)$ is a
term (or a set of terms) that usually appears in $\phi(\vec x)$ and
which contains all the (term) variables $\vec x$ and all the type
variables in $\phi(\vec x)$. We use the notation
\[
\forall \vec x\;[\;p \;|\;
p_1, p_2].\; \phi(\vec x)
\]
to indicate that $\psi$ is associated with one mono-trigger $\{p\}$, and one
multi-trigger $\{p_1, p_2\}$. Triggers can either be provided by the user
with the syntax above, or heuristically computed by \textsf{Alt-Ergo}\xspace{}. In the
latter case, \textsf{Alt-Ergo}\xspace{} will choose at most two triggers per axiom by
default. For instance, possible triggers for the axiom {\tt mem\_add}
of Figure~\ref{fig:ex} are:
\begin{center}
\{{\tt mem(x, add(y, s))} \} \qquad
\{{\tt add(y, s), add(x, s)}\} \qquad
$\cdots$ \qquad
\{{\tt x, y, s}\}
\end{center}
The latest multi-trigger is a very bad choice and is never selected by
\textsf{Alt-Ergo}\xspace{}. In fact, it would generate an instance of {\tt mem\_add} for
every (well-typed) combination of terms appearing in the decision
procedures (\textit{resp.} SAT's model). The two first triggers seem
to be good choices. However, only the first one will permit us to
prove the validity of the example in Figure~\ref{fig:ex}. Indeed, the
ground term {\tt mem(a - 1, s2)} \textit{matches} the trigger {\tt
mem(x, add(y, s))} modulo the equality {\tt s2 = add (b, s1)}. The
E-matching process produces the substitution $\{\mathtt{x \mapsto a -
1,\; y \mapsto b,\; s \mapsto s1,\; \alpha \mapsto int}\}$, which
allows us to generate the needed instance.
\bigskip
The rest of the paper describes the features (and their implementation) that
\textsf{AltGr-Ergo}\xspace offers and shows how they can be useful both from an end-user perspective
as well as from a developer's perspective.
\section{Feedback}
\label{sec:feedback}
The first purpose of \textsf{AltGr-Ergo}\xspace is to provide \emph{feedback} which can be useful at
times to understand and evaluate what is happening inside the solver. Feedback
is useful for users as a visual aid to make sense of the solver's progress, but
it is also a precious tools for developers to profile and debug the solver.
\subsection{Unsat Cores and Minimal Context Extraction}
\label{sec:unsat}
An \emph{unsatisfiable core} in SMT, is a subset of the input formulas that
make the problem unsatisfiable. Traditionally, SMT solvers will return sets
where the elements are some of the input, top-level formulas, identified by a
unique name in the source. \textsf{Alt-Ergo}\xspace goes a bit further and identifies sub-formulas
that arise from the CNF (conjunctive normal form) conversion. This allows to
identify more precisely which part of the formula is actually useful in proving
the goal.
Unsat cores production is deactivated by default when running \textsf{Alt-Ergo}\xspace, but the
interface offers a way to change solver options on the fly, even while the
solver is running. In text, mode \textsf{Alt-Ergo}\xspace will spit unsat cores as pretty-printed
formulas on its output. This can become large at times. The interface will
display unsat cores in a more user-friendly way, visually identifying useful
parts of the context, hypotheses, \textit{etc.} by highlighting them in green
(see~Figure~\ref{fig:overview} for instance).
Different shades of green are used to highlight unsat cores in the buffer
window. Top-most declarations and definitions which contain part of the unsat
core will be highlighted in the lightest green and (sub-) formulas that appear
more frequently in the unsat core will be highlighted with a darker shade. In
particular, if an axiom is instantiated several times and the same part of the
resulting instances is actually useful to prove the goal, then the user will be
able to see this information visually. These features make it easy to rapidly
identify which parts of the context are useful and how crucial they are to
prove the goal.
Unsat cores also serve another purpose. By identifying which part of the
context is effectively used to prove the goal by the solver, we can remove any
other information contained in the problem while still having the guarantee
that the goal will be provable by the solver\footnote{This guarantee might be
lost in some particular cases in \textsf{Alt-Ergo}\xspace, namely when an instance of a
useless quantified axiom is used as a \emph{source of terms} to trigger the
instantiation of another useful axiom.}. \textsf{AltGr-Ergo}\xspace offers a button in the toolbar
to quickly remove every top-level declaration or definition that does not
participate in the unsat core. Coupled with the mechanism of sessions (see
Section~\ref{sec:sessions}), this allows to save and replay already proven
goals much more rapidly.
\subsection{Instantiation}
\label{sec:instances}
As remarked in the introduction, quantified formulas are a source of
incompleteness and inefficiencies in most SMT solvers. Providing a way to
accurately and concisely expose information about instantiation is important
for the user experience. \textsf{AltGr-Ergo}\xspace does so by displaying axioms instantiated, in
real time, in a sub-window of the interface as shown in Figure~\ref{fig:inst}.
\begin{figure}[htb]
\centering
\color{black!30}
\frame{\includegraphics[width=.38\textwidth]{inst.png}}
\qquad
\frame{\includegraphics[width=.38\textwidth]{inst_limited.png}}
\caption{Instances and manual limits}
\label{fig:inst}
\end{figure}
Here we report the number of instances produced by each axiom. They are listed
in decreasing order of number of instances and their names\footnote{For
lemmas that are nested in larger formulas, we report the top-level name with
some indication of their position. However, users can access their
corresponding location in the source code by simply double clicking on the
displayed name.} are colored in varying shades of red to denote frequency of
instantiation. Axioms whose name is of a more saturated red denote the ones
which produce more instances with respect to the total number of instances
(regardless of its origin) generated at this point in time by the solver. This
allows to quickly identify potentially problematic axioms which generate too
many ground instances. This feedback gives indication regarding the likely
cause of problems in the instantiation mechanism.
When this happens, we also offer the possibility to \emph{limit} instantiation
of particular quantified lemmas. For example, the left report of
Figure~\ref{fig:inst} tells us that the lemma \textsf{permut\_exists} was
instantiated 1322 times, more than twice as much as any other lemma. This is
thus a good candidate to limit instantiation. On the right screen capture of
Figure~\ref{fig:inst}, we limited the instances of this problematic lemma to
200 and an associated lemma (\textsf{permut}) to 300 (we performed this process
iteratively, by first limiting instances of \textsf{permut\_exists} and looking
at what other lemmas were problematic). Lemmas for which instantiation has
reached its given limit are shown in blue. We can notice that the runtime of
the solver is reduced subsequently by a factor 3 for this particular example.
\subsection{Profiling}
\label{sec:profiling}
Much like in the spirit of the previous section, the bottom-right most
sub-window of the interface (see Figure~\ref{fig:overview}) gives \emph{real
time} profiling information for the different modules and theories of
\textsf{Alt-Ergo}\xspace. These include the time spent in the SAT solver, the matching procedure,
the congruence closure algorithm (\textsf{CC(X)}), the builtin support for
associative and commutative symbols (\textsf{AC(X)})~\cite{acx} and the
theories of arithmetic, arrays, enumerated data-types (\textsf{Sum}) and
records.
\begin{figure}[htb]
\centering
\color{black!30}
\frame{\includegraphics[width=.38\textwidth]{prof_window.png}}
\caption{Real time profiling information}
\label{fig:prof}
\end{figure}
Figure~\ref{fig:prof} shows the state of the profiling information after
running an example. From the information displayed here, we can see that the
problem mostly stresses the theory of arithmetic in the solver. While the
solver is running, users can see which part takes the most time and can follow
the evolution. For instance, if the time reported for the theory of arrays is
too large in proportion and keeps growing, this can indicate that maybe there
are some axioms about arrays which are too permissive. Another use case, for
developers, is the possibility to identify problems where the solver is stuck
in a particular decision procedure (\textit{e.g.}, if only the timer for this
theory increases).
\section{Syntactic Operations}
\label{sec:syn}
A lot of the time when trying to use SMT solvers on real world examples (coming
from program verification tools for instance), the size of the logical context
and the sometimes heavy axiomatizations (that make liberal use of quantifiers
at varying degree of alternation) make the problem hard for purely automated
tools. However, only a fraction of the actual context is usually necessary to
prove the desired goal. Identifying useful information in very large problems
can be challenging. When the information provided by the feedback features
described in Section~\ref{sec:feedback} allow to identify a potential issue,
\textsf{AltGr-Ergo}\xspace offers a number of functionalities to perform syntactic manipulations on
the context that is shown in textual format. This allows for an iterative (and
slightly interactive) approach to SMT solving, where users can experiment and
quantify the effect of different manipulations.
\subsection{Pruning}
\label{sec:pruning}
Pruning can be performed very easily in \textsf{AltGr-Ergo}\xspace by simply double-clicking on the
top-level declaration or sub-formula that one wants to deactivate. Reactivating
previously deactivated items is also possible by performing the same action.
Most of the time pruning is not dangerous from a soundness point of
view. Removing an axiom only under-constrains the original problem which means
that any goal proved valid without some top-level hypothesis is still valid
with in the original context. However we also allow users to prune parts of a
formula. In this case validity can be affected depending on the
polarity of the
removed formula.
Consider the original goal:
\begin{tcb}{altergo}
logic P, Q : int -> prop
goal g: forall x, y: int. Q(x) or P(y) -> Q(x) and P(y)
\end{tcb}
which is trivially invalid. Removing \texttt{P(y)} on the left side of the
implication or on the right side of the implication changes the validity of
this goal. In fact removing both turns this goal into a valid one. \textsf{AltGr-Ergo}\xspace will
allow users to perform these potentially unsafe operations but will notify the
user by showing unsound prunings in red. A session that contains unsound
pruning operations cannot be saved either. This feature is still useful from an
end-user point of view because it allows to attempt proving goals by
strengthening hypotheses or weakening the goal itself. For instance if the goal
is a conjunction, a user can try to prove only part of the conjunction and
gather information from this attempt to help prove the rest.
\subsection{Dependency Analysis}
\label{sec:dep}
\textsf{AltGr-Ergo}\xspace maintains dependency information between declarations, definitions and
their use. It is possible to remove the declaration of a logical symbol and all
top-level declarations that make use of it in a single action. Conversely,
reactivating a previously pruned formula or declaration that uses a symbol will
also reactivate its declaration and/or definition.
A usage scenario for this feature, is to quickly disable a symbol for which we
know the axiomatization is problematic for the solver, then reactivate part of
the axiomatization iteratively in the hope that the solver will not be
overwhelmed anymore.
\subsection{Manual Instances}
\label{sec:manual}
Quantifiers are notoriously difficult for most SMT solvers. Unfortunately some
application domains such as deductive program verification make heavy use of
this feature to encode some domain specific functions. For instance
\textsf{Frama-C} has built-in axiomatizations for various memory models of
C. These are usually very large and complex.
Quantifier instantiation is a heuristic process for SMT solvers in general (although there
exists complete techniques for decidable fragments). \textsf{Alt-Ergo}\xspace uses \emph{matching
modulo equality}. On the other side of the spectrum, interactive theorem
provers like \textsf{Coq} or \textsf{Isabelle} require users to perform
instantiation (\textit{i.e.} application) entirely manually. This is because in
traditional backward reasoning done in theorem provers, only relatively few applications
are necessary and a human can figure out which one to do based on the goal, the
context and knowledge about the current proof attempt.
\textsf{AltGr-Ergo}\xspace gives users the possibility to perform some instances manually. This is
useful for example if one has knowledge that a particular goal cannot be
solved without using specific instances of a lemma. \textsf{AltGr-Ergo}\xspace will ask for terms to
use in the instantiation. These can be constants but also other terms from the
context. Instances can also be \emph{partial}, which means that we allow that
only some of the variables be instantiated. Instances (partial or not) are
finally added to the context as hypotheses (they are shown in the top most
right corner of the window). All of the other presented actions can be
performed on instances.
\subsection{Triggers Selection and Modification}
\label{sec:triggers}
As mentioned in section~\ref{sec:matching}, \textsf{Alt-Ergo}\xspace computes
triggers---\textit{i.e.} patterns or filters used by the matching algorithm to
instantiate axioms---in a \emph{heuristic} way. For certain restricted
categories of axiomatizations, there exists techniques for computing triggers
that make the instantiation mechanism complete~\cite{dross16jar}, however this
is not the case in general and coming up with good triggers is a difficult
problem.
Because they essentially control how instances are generated, triggers play an
important role in proving goals with quantifiers. For example, consider the
following axiom, where \texttt{f} is an uninterpreted function from integers to
integers.
\begin{tcb}{altergo}
axiom idempotent : forall x : int. f(f(x)) = f(x)
\end{tcb}
The trigger \texttt{f(f(x))} is more restrictive than \texttt{f(x)}. This means
that only terms $t$ that appear in larger terms \texttt{f(f(}$t$\texttt{))} will
be used for instantiating this axiom. Having this trigger will thus make the
solver generate \emph{less} instances of this axiom as opposed to the other one
possibility, \texttt{f(x)}.
There is a balance to be found when coming up with triggers, between restrictive
patterns and liberal ones. In some cases, the solver can be overwhelmed by the
instances generated if the triggers are not good, it will likely not terminate
from a user's point of view. If patterns are too restrictive, or if \textsf{Alt-Ergo}\xspace cannot
compute appropriate triggers, the affected axiom will likely not be
instantiated enough, preventing the solver of discovering potential
inconsistencies. In this latter case, \textsf{Alt-Ergo}\xspace will answer ``I don't know'' which
is usually unsatisfactory from an end user perspective.
Triggers can be specified by hand in the source problem for \textsf{Alt-Ergo}\xspace. The interface
\textsf{AltGr-Ergo}\xspace goes a step further by allowing triggers to be modified on the fly in the
source buffer window. The problem displayed to the user will show the triggers
computed by \textsf{Alt-Ergo}\xspace itself (it will also warn the user if the heuristic could not
come up with appropriate triggers for a quantified formula) as annotations in
the source. These can be modified interactively when one figures that the
heuristic did not produce the expected results.
Consider now the following, somewhat degenerate, goal.
\begin{tcb}{altergo}
axiom crazy : forall x, y : int. x + 1 = y
goal indeed: false
\end{tcb}
\textsf{Alt-Ergo}\xspace will not compute any triggers for the axiom. By default, the solver
rejects triggers composed of single variables as they are considered too
permissive (any term of the appropriate type can be used for
instantiation). However in this specific case, we need to instantiate the axiom
with any two integers to discover the inconsistency. Figure~\ref{fig:trigagr}
shows\footnote{Goals in the interface are shown in their negated form, as they
will be seen by the solver.} the functionality, right clicking on the trigger
displays a contextual menu to add triggers in a pop-up window. They can be
entered by the user, and \textsf{AltGr-Ergo}\xspace will then parse and type check the piece of text
corresponding to the trigger using the same internal functions (exposed) as the
ones of \textsf{Alt-Ergo}\xspace. In this example we add the multi-trigger \texttt{[x, y]} which
allows to conclude.
\begin{figure}[htb]
\centering
\color{black!30}
\frame{\includegraphics[width=.8\textwidth]{trigger.png}}
\caption{Adding new triggers by hand}
\label{fig:trigagr}
\end{figure}
Triggers can also be deactivated (\textit{i.e.} removed from the axiom) by
using the same deactivation mechanism as the one for formulas. With these
possibilities, triggers can be modified at will without resorting on complex
solver parameters nor relying on heuristics.
\subsection{Sessions}
\label{sec:sessions}
Because this interface is designed to provide a slight increase in
interactivity for the user, we want all operations to be saved in what we call
\emph{sessions} for later \emph{replay}. In particular, operations that
manipulate the problem and custom tuning can greatly help the solver in its
search. When a user is happy with the state of its modifications---for
instance when they allowed to successfully prove the goal---the interface
offers the possibility to save the information concerning the session on disk.
This mechanism allows to replay sessions \emph{even if the original problem was
modified}. Apart from the list of actions, the only additional information
that is saved in the session information is an association table between symbol
names for declarations and their identifiers (\texttt{id}). Node identifiers in
the AST are sequential integers following a depth-first ordering. When a
session file is read from the disk, the interface computes offsets for
identifiers using this association table in order to figure out the correct
corresponding ones for each action in the stack. Of course this is possible
only if modifications of the file are relatively minor. For instance, this will
work if some axioms were removed or added, or if a modification was performed
locally in a declaration (\textit{e.g.}, a formula was changed inside an
axiom). Nevertheless, sessions will not survive complete refactorings. If the
modifications are too important, the replay will try its best following this
offset strategy but can decide to abort if too many actions cannot be replayed.
Another usage of the session replay mechanism is to reuse sessions between
problems that share a lot of context. This is useful in a scenario where a user
found a satisfying set of modifications and tuning operations on a given
problem, and wants to try the same operations on a similar problem
(\textit{e.g.}, one where only the goal is different but the context is
identical).
\section{Implementation}
\textsf{AltGr-Ergo}\xspace is designed as a new front end for the solver. As such it reuses part of
\textsf{Alt-Ergo}\xspace's front end and \textsf{Alt-Ergo}\xspace's API. The interface is placed at the level of the
typed abstract syntax tree (Typed AST on Figure~\ref{fig:agrarchi}) and can
manipulate this representation at will. The solver itself communicates
various pieces of information to the interface.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{archi_agr.pdf}
\caption{\textsf{AltGr-Ergo}\xspace's architecture}
\label{fig:agrarchi}
\end{figure}
The graphical part of the interface is written in GTK, using the OCaml bindings
LablGTK and runs in a separate thread. When \textsf{Alt-Ergo}\xspace is started in graphical mode,
one thread is created for the interface, and one thread is created for every
instance of the solver (started with the button ``Run''). These threads
communicate asynchronously through shared variables, messages or signals
depending on the functionality. For example, runs of the solver can be aborted
at anytime by clicking the button ``Abort'', a signal is emitted and caught by
the interface and the solver instance.
Most of the work performed by the interface is done on an \emph{annotated}
version of the AST.
\subsection{Annotations}
An annotated AST node is encoded by the record shown below. It contains the
node itself, mutable flags for pruning and polarities, GTK tags, a unique
identifier, the GTK buffer in which it is displayed and the line number at
which it appears in the source.
\begin{tcb}{ocaml}
type 'a annoted = {
mutable c : 'a; (* annotated value *)
mutable pruned : bool; (* pruned mark *)
mutable polarity : bool; (* polarity for sub-terms *)
tag : GText.tag; (* GTK tag associated *)
ptag : GText.tag; (* Another GTK tag for proofs *)
id : int; (* Unique identifier *)
buf : sbuffer; (* Source buffer *)
mutable line : int; (* line numbers can change *)
}
\end{tcb}
Some of these fields are mutable to account for user actions. For instance
pruning a sub-term in the source amounts to changing the flag
\texttt{pruned}. In turn, polarities can be affected by these operations (see
Section~\ref{sec:pruning}).
Tags are used as a way to effectively identify portions of the source code
displayed to the user in the buffer. These special tags of type
\lstinline[language=ocaml]{GText.tag}, allow to efficiently change properties
of the display at any moment. For instance they are used to show the user
sub-terms of the formulas and their type when the mouse is hovered over the
particular location in the buffer area. Tags can be stacked, this is why we
have a second one, \texttt{ptag}, which displays unsat cores and remains
unaffected by any other action (see Section~\ref{sec:unsat}). The operations
described in Section~\ref{sec:syn} are all performed on this intermediate
representation.
When the user clicks on the button \emph{Run}, annotations are stripped form
the AST and the resulting AST is sent to a newly created thread for this
specific instance of the solver \textsf{Alt-Ergo}\xspace.
\subsection{Sessions}
Every action performed in the interface is saved in a stack in the current
environment. This is the stack that is exported when the session is saved.
\begin{tcb}{ocaml}
type action =
| Prune of int
| IncorrectPrune of int
| Unprune of int
| AddInstance of int * string * string list
| AddTrigger of int * bool * string
| LimitLemma of int * string * int
| UnlimitLemma of int * string
type env =
{ ...
saved_actions : action Stack.t;
... }
\end{tcb}
Actions are parameterized by the identifier of the AST node on which they were
performed. Only atomic actions are saved. This is because all other operations
can be expressed in terms of these atomic actions. This simple structure
allows to replay sessions at a later date and obtain a state which is
equivalent to the one that was shown to the user when it was saved on disk.
\section{Conclusion and Future Work}
We've presented \textsf{AltGr-Ergo}\xspace, a graphical front end for the SMT solver \textsf{Alt-Ergo}\xspace. It
provides real time feedback to users and allows to interactively manipulate
problems. We believe such a tool is beneficial both from an end-user and
developer point of view, especially for the kind of goals that arise from
deductive program verification. We have personally (as the developers of \textsf{Alt-Ergo}\xspace)
found the interface precious to tune the solver and tackle larger verification
problems.
Future directions to improve this tool include turning \textsf{AltGr-Ergo}\xspace into a full
featured editor for SMT solvers (at the moment the source buffer window cannot
be edited directly). There are still some inefficiencies in the interface when
one wants to load \emph{very large} files, due to some of the more advanced
features, but also inherent to GTK itself. Another future direction for this
tool, would be to dissociate the interface from the prover completely with the
objective of providing a \emph{web based interface}, which would be more
portable, flexible and easy to use. Also, the session mechanism could be
further improved to allow for more replay scenarios by integrating some more
advanced diff and merge techniques~\cite{autexier}.
This work is a first step towards \emph{semi-interactive} theorem provers. One
more ambitious goal would be to provide a set of \emph{tactics} and an
appropriate language to allow users to do part of their proof manually, or to
finish (also partly manually) proofs that the SMT solver struggle with. This
would involve work both at the level of the proof language as well as the level
of the user interface.
\bibliographystyle{eptcs}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,304 |
Michal Pšenko (ur. 8 lipca 1982) – słowacki skoczek narciarski i kombinator norweski. W skokach startował głównie w Pucharze Kontynentalnym. Jego najlepszym sezonem był sezon 2003/2004, kiedy zajął 122. miejsce w klasyfikacji generalnej. Jego rekord życiowy to 131 metrów osiągnięte podczas zawodów Pucharu Kontynentalnego 17 grudnia 2004 roku w czeskim Harrachovie. Jego najlepszy rezultat w Pucharze Świata w skokach narciarskich to 44. miejsce w zawodach Pucharu Świata w Zakopanem 16 stycznia 1999.
Największym sukcesem tego skoczka jest srebrny medal mistrzostw świata juniorów wywalczony w Saalfelden am Steinernen Meer w 1999 roku, gdzie przegrał tylko z Kazukim Nishishitą.
Startował także w kombinacji norweskiej, w zawodach Pucharu Świata, Pucharu Świata B, Letniego Grand Prix, Mistrzostwach Świata i Igrzyskach Olimpijskich. Jego życiowe osiągnięcia w tych typach zawodów to 27. miejsce w zawodach Pucharu Świata w Reit im Winkl 7 stycznia 2001, 2. miejsce w zawodach Pucharu Świata B w Tarvisio 7 stycznia 2004 i 9. miejsce w zawodach Letniego Grand Prix w Harrachovie 27 sierpnia 2004. Na Mistrzostwach Świata w Lahti był 41. w Gundersenie i nie ukończył sprintu. Na Igrzyskach Olimpijskich w Salt Lake City był 38. w Gundersenie i 39. w sprincie. Mistrzostwa Świata w Val di Fiemme zakończył na 41. miejscu w Gundersenie i 42. w sprincie. Jego ostatnimi mistrzostwami był Mistrzostwa w Oberstdorfie w 2005, osiągnął tam w sprincie 46. miejsce.
Wicemistrz Słowacji w zawodach indywidualnych z 1999 i 2000 roku.
Osiągnięcia – kombinacja norweska
Igrzyska olimpijskie
Mistrzostwa świata
Mistrzostwa świata juniorów
Puchar Świata
Miejsca w klasyfikacji generalnej
sezon 1999/2000: 57.
sezon 2000/2001: 43.
sezon 2001/2002: niesklasyfikowany
sezon 2002/2003: nie brał udziału
sezon 2003/2004: 53.
sezon 2004/2005: niesklasyfikowany
Miejsca na podium chronologicznie
Pšenko nigdy nie stanął na podium zawodów PŚ.
Puchar Kontynentalny
Miejsca w klasyfikacji generalnej
sezon 1998/1999: 81.
sezon 1999/2000: 28.
sezon 2000/2001: 4.
sezon 2001/2002: 12.
sezon 2002/2003: 30.
sezon 2003/2004: 36.
sezon 2004/2005: 72.
sezon 2005/2006: 77.
Miejsca na podium chronologicznie
Letnie Grand Prix
Miejsca w klasyfikacji generalnej
1999: 42.
2000: 18.
2001: 21.
2002: 26.
2003: 19.
Miejsca na podium chronologicznie
Pšenko nigdy nie stanął na podium zawodów LGP.
Osiągnięcia – skoki narciarskie
Mistrzostwa świata
Indywidualnie
Mistrzostwa świata juniorów
Indywidualnie
Zimowy olimpijski festiwal młodzieży Europy
Indywidualnie
Drużynowo
Puchar Kontynentalny
Miejsca w klasyfikacji generalnej
sezon 1998/1999: 127.
sezon 1999/2000: 127.
sezon 2003/2004: 122.
sezon 2004/2005: 152.
Linki zewnętrzne
Profil na oficjalnej stronie FIS jako skoczek
Profil na oficjalnej stronie FIS jako kombinator norweski
Uwagi
Przypisy
Słowaccy skoczkowie narciarscy
Urodzeni w 1982
Słowaccy kombinatorzy norwescy | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,505 |
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} | 9,521 |
The SR Education Group released its list of "Top Online Colleges" for 2017 and ranked the EKU Online OTD program #3 in the "Best Value Occupational Therapy" category.
A doctorate in occupational therapy will be possible for more licensed OTs when Eastern Kentucky University launches a 100 percent online bachelor's-to-OTD degree path in Fall 2016. Occupational therapists will be able to complete the program in about 3 years, earning their master's degree along the way.
Can I Achieve My Professional Goals with an Online OTD Program?
Can I work on my professional goals while I am in the EKU OTD program?
Is the coursework really personalized for me?
The short answer is, "yes!
Dr. Leslie Hardman identifies as an OT and sees teaching as an extension of that role.
"I worked with clients for 30 years before getting my OTD," explained Hardman, who is currently an assistant professor at EKU.
The EKU Online Occupational Therapy Doctorate (OTD) placed second nationwide in the 2016 "Students Before Profits" award rankings by Nonprofit Colleges Online.
According to the organization's news release, the rankings recognize universities that prioritize providing students with quality educational opportunities over their bottom lines. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,556 |
This property sold for $630,000 almost 3 years ago.
The only home available in Lokahi Trovare. This beautiful and very comfortable 3/2.5 home is very spacious with vaulted ceilings in the living area, split a/c's through out, laminate flooring, relaxing tropical yard, sprinkler system, tile roof and much more. Conveniently located close to award winning Holomua School, parks, shopping, restaurants, and the beach. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,807 |
package cors
import (
"bytes"
"fmt"
"regexp"
"strings"
"github.com/raphael/goa"
)
type (
// CheckFunc is the signature of the user provided function invoked by the middleware to
// check whether to handle CORS headers.
CheckFunc func(*goa.Context) bool
// ResourceDefinition represents a CORS resource as defined by its path (or path prefix).
ResourceDefinition struct {
// Origin defines the origin that may access the CORS resource.
// One and only one of Origin or OriginRegexp must be set.
Origin string
// OriginRegexp defines the origins that may access the CORS resource.
// One and only one of Origin or OriginRegexp must be set.
OriginRegexp *regexp.Regexp
// Path is the resource URL path.
Path string
// IsPathPrefix is true if Path is a path prefix, false if it's an exact match.
IsPathPrefix bool
// Headers contains the allowed CORS request headers.
Headers []string
// Methods contains the allowed CORS request methods.
Methods []string
// Expose contains the headers that should be exposed to clients.
Expose []string
// MaxAge defines the value of the Access-Control-Max-Age header CORS requeets
// response header.
MaxAge int
// Credentials defines the value of the Access-Control-Allow-Credentials CORS
// requests response header.
Credentials bool
// Vary defines the value of the Vary response header.
// See https://www.fastly.com/blog/best-practices-for-using-the-vary-header.
Vary []string
// Check is an optional user provided functions that causes CORS handling to be
// bypassed when it return false.
Check CheckFunc
}
// Specification contains the information needed to handle CORS requests.
Specification []*ResourceDefinition
)
var (
// spec is the CORS specification being built by the DSL.
spec Specification
// dslErrors contain errors encountered when running the DSL.
dslErrors []error
)
// New runs the given CORS specification DSL and returns the built-up data structure.
func New(dsl func()) (Specification, error) {
spec = Specification{}
dslErrors = nil
if dsl == nil {
return spec, nil
}
dsl()
if len(dslErrors) > 0 {
msg := make([]string, len(dslErrors))
for i, e := range dslErrors {
msg[i] = e.Error()
}
return nil, fmt.Errorf("invalid CORS specification: %s", strings.Join(msg, ", "))
}
res := make([]*ResourceDefinition, len(spec))
for i, r := range spec {
res[i] = r
}
return Specification(res), nil
}
// Origin defines a group of CORS resources for the given origin.
func Origin(origin string, dsl func()) {
existing := spec
spec = Specification{}
dsl()
for _, res := range spec {
res.Origin = origin
}
spec = append(existing, spec...)
}
// OriginRegex defines a group of CORS resources for the origins matching the given regex.
func OriginRegex(origin *regexp.Regexp, dsl func()) {
existing := spec
spec = Specification{}
dsl()
for _, res := range spec {
res.OriginRegexp = origin
}
spec = append(existing, spec...)
}
// Resource defines a resource subject to CORS requests. The resource is defined using its URL
// path. The path can finish with the "*" wildcard character to indicate that all path under the
// given prefix target the resource.
func Resource(path string, dsl func()) {
isPrefix := strings.HasSuffix(path, "*")
if isPrefix {
path = path[:len(path)-1]
}
res := &ResourceDefinition{Path: path, IsPathPrefix: isPrefix}
spec = append(spec, res)
dsl()
}
// Headers defines the HTTP headers that will be allowed in the CORS resource request.
// Use "*" to allow for any headerResources in the actual request.
func Headers(headers ...string) {
if len(spec) == 0 {
dslErrors = append(dslErrors, fmt.Errorf("invalid use of Headers, must define Origin and Resource first"))
} else {
res := spec[len(spec)-1]
res.Headers = append(res.Headers, headers...)
}
}
// Methods defines the HTTP methods allowed for the resource.
func Methods(methods ...string) {
if len(spec) == 0 {
dslErrors = append(dslErrors, fmt.Errorf("invalid use of Headers, must define Origin and Resource first"))
} else {
res := spec[len(spec)-1]
for _, m := range methods {
res.Methods = append(res.Methods, strings.ToUpper(m))
}
}
}
// Expose defines the HTTP headers in the resource response that can be exposed to the client.
func Expose(headers ...string) {
if len(spec) == 0 {
dslErrors = append(dslErrors, fmt.Errorf("invalid use of Headers, must define Origin and Resource first"))
} else {
res := spec[len(spec)-1]
res.Expose = append(res.Expose, headers...)
}
}
// MaxAge sets the Access-Control-Max-Age response header.
func MaxAge(age int) {
if len(spec) == 0 {
dslErrors = append(dslErrors, fmt.Errorf("invalid use of Headers, must define Origin and Resource first"))
} else {
res := spec[len(spec)-1]
res.MaxAge = age
}
}
// Credentials sets the Access-Control-Allow-Credentials response header.
func Credentials(val bool) {
if len(spec) == 0 {
dslErrors = append(dslErrors, fmt.Errorf("invalid use of Headers, must define Origin and Resource first"))
} else {
res := spec[len(spec)-1]
res.Credentials = val
}
}
// Vary is a list of HTTP headers to add to the 'Vary' header.
func Vary(headers ...string) {
if len(spec) == 0 {
dslErrors = append(dslErrors, fmt.Errorf("invalid use of Headers, must define Origin and Resource first"))
} else {
res := spec[len(spec)-1]
res.Vary = append(res.Vary, headers...)
}
}
// Check sets a function that must return true if the request is to be treated as a valid CORS
// request.
func Check(check CheckFunc) {
if len(spec) == 0 {
dslErrors = append(dslErrors, fmt.Errorf("invalid use of Headers, must define Origin and Resource first"))
} else {
res := spec[len(spec)-1]
res.Check = check
}
}
// String returns a human friendly representation of the CORS specification.
func (v Specification) String() string {
if len(v) == 0 {
return "<empty CORS specification>"
}
var origin string
b := &bytes.Buffer{}
for _, res := range v {
o := res.Origin
if o == "" {
o = res.OriginRegexp.String()
}
if o != origin {
b.WriteString("Origin: ")
b.WriteString(o)
b.WriteString("\n")
o = origin
}
b.WriteString("\tPath: ")
b.WriteString(res.Path)
if res.IsPathPrefix {
b.WriteString("*")
}
if len(res.Headers) > 0 {
b.WriteString("\n\tHeaders: ")
b.WriteString(strings.Join(res.Headers, ", "))
}
if len(res.Methods) > 0 {
b.WriteString("\n\tMethods: ")
b.WriteString(strings.Join(res.Methods, ", "))
}
if len(res.Expose) > 0 {
b.WriteString("\n\tExpose: ")
b.WriteString(strings.Join(res.Expose, ", "))
}
if res.MaxAge > 0 {
b.WriteString(fmt.Sprintf("\n\tMaxAge: %d", res.MaxAge))
}
if res.MaxAge > 0 {
b.WriteString(fmt.Sprintf("\n\tMaxAge: %d", res.MaxAge))
}
if len(res.Vary) > 0 {
b.WriteString("\n\tVary: ")
b.WriteString(strings.Join(res.Vary, ", "))
}
}
return b.String()
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,711 |
Игнат Канев () е канадски строителен магнат и филантроп от български произход.
Биография
Игнат Христов Кънев е роден на 6 октомври 1926 г. в с. Горно Абланово, Русенска област. Той е четвъртото от 7-те деца (Къньо, Ламби, Куна, Игнат, Мария, Симеон и Петър) на Христо и Мита Къневи.
Няма финансовата възможност да учи, а е принуден още на 14-годишна възраст да напусне България и да се установи в Австрия, за да търси препитание. През март 1941 г. с още няколко момчета отиват при Никола Парушчолаков, градинар от Горна Оряховица, който има зеленчукови градини в Австрия и търгува със земеделски стоки. След 1945 г. го убеждават да не се връща в България. След 5 години чиракуване вече има собствена търговия и малко производство на плодове и зеленчуци.
През 1951 г. зарязва бизнеса в Австрия и почти без средства, без да знае английски и без да има някакви връзки и приятели, заминава за Канада. Установява се в Торонто, където започва работа в строителния бизнес. Първоначално строи малки къщи за новопристигащите имигранти. Пет години по-късно, през 1956 г., вече основава там първата си строителна фирма. Следващата година строи първата си по-висока сграда – на 3 етажа с 9 апартамента, а 10 години по-късно вече вдига сграда с 262 жилища. Впоследствие годишният оборот на компанията му достига над 250 млн. долара. По време на своята активна дейност като строителен предприемач, той издига хиляди къщи, много обществени сгради и голф-съоръжения от световна класа.
Занимава се и с търговия на автомобили. Става представител на "General Motors" и продава хиляди коли и камиони. За успешния си бизнес той е поканен лично от президента на фирмата на вечеря в Детройт.
Собственик е на 6 голф игрища и голяма строителна фирма в Канада. Управител е на благотворителна фондация, носеща неговото име.
Богатството му се оценява на над 1 млрд. долара.
Въпреки че живее в Канада от много години, той поддържа силна връзка с България. През 2003 г. като признание за приноса му Игнат Канев е бил избран за почетен консул на Република България в Мисисага.
На 30 декември 2016 г., Игнат Канев е награден с една от най-високите награди в Канада – Ордена на Канада. Наградата е връчена на 12 май следващата година на специална церемония в официалната резиденция на генералния губернатор на Канада Дейвид Джонстън в Отава.
Семейство
През 1948 г. се жени за Катерина – австрийка от хърватски произход. За 28 години брак няма деца от нея, а тя умира от рак.
През 1977 г. се жени за Димитрина, която е 25 години по-млада от него. Запознават се, когато тя е на посещение в Канада при дядо си, емигрирал в страната преди години. През 1978 г. се ражда Анна-Мария Канева, а през 1980 – Кристина-Мария Канева. Анна-Мария завършва Колумбийския университет, а после икономика в Оксфорд. Кристина-Мария е юрист. И двете стават вицепрезиденти на семейната фирма. След оттеглянето на Игнат Канев от поста президент, тази длъжност се заема от майката, Димитрина Канева.
Дарителство
Благодарение на усърдието и предприемчивостта си Игнат Канев бързо натрупва голямо състояние, което му дава възможност да се заеме с благотворителна дейност. Дарява предимно за здравни и учебни цели, както и за утвърждаване на православната църква.
Прави първото си голямо дарение през 1955 г. за болница в Торонто, когато дава непосилната за следвоенните години сума от 2000 долара.
Подпомага свои предприемчиви сънародници, дошли да търсят късмета си в Канада.
Дава земя и над 1 млн. долара за строеж на православния храм "Свети Димитър" в Торонто, осветен от бившия русенски митрополит, а впоследствие патриарх Неофит и от митрополита на Северна Америка и Австралия Йосиф.
През 1976 г. дарява средства за построяване на детска градина в родното си село Горно Абланово. А след демократичните промени спомага за ремонта на местната църква, обзавежда компютърен кабинет в училището и дарява линейка.
През 2001 и 2003 г. дарява оборудване за русенското училище "Отец Паисий".
През 2009 г. строителният магнат дава 2.5 млн. долара за реконструкция на сградата на Osgoode Hall Law School към Йоркския универститет в Торонто, която впоследствие се нарича на негово име Ignat Kaneff Building.
През 90-те години на 20 век Игнат Канев изпраща огромно количество образователна и бизнес литература в Русенския университет "Ангел Кънчев" в подкрепа на новосъздадения факултет по Бизнес и мениджмънт.
През 2012 г. дава два милиона лева (половината от необходимата сума) за построяването на модерен конферентен комплекс към университета, наречен на негово име "Канев Център". Центърът разполага с многофункционална зала за 800 места, сцена с площ 120 m², заседателна зала със 70 места, конферентна зала със 100 места, всичко това оборудвано с най-модерни системи за осветление, озвучаване, презентации и симултанен превод. Открит е тържествено на 10 октомври 2013 г. в присъствието на ректора на университета проф. Христо Белоев, патриарх Неофит, президента Росен Плевнелиев, бившия президент Георги Първанов, кмета на град Русе, областния управител, временно изпълняващия длъжността посланик на Канада в Букурещ и др. представителни лица.
За рождения му ден през 2014 г. с негово дарение от 300 хил. лв. е построен културен дом в родното му село Горно Абланово, наречен "Канев център", а читалището е преименувано на "Канев-Пробуда 1924".
Награди
През 2002 г. Игнат Канев е награден с орден "Стара планина първа степен".
През 2010 г. Игнат Канев е награден с орден на Онтарио.
През 2016 г. Игнат Канев е награден с орден на Канада.
За щедрото спомоществувателство и обновяването на сградата на юридическия колеж през 2010 г. Йоркският университет му присъжда званието почетен доктор по юридическите науки.
Като благодарност за щедрото дарение към Русенския университет строителят е удостоен с почетното звание "доктор хонорис кауза" на университета и става 38-ия почетен доктор на Русенската алма матер.
За щедрата финансова подкрепа към различни училища и учебни учреждения в Русе и Русенска област кметът на града Пламен Стоилов го удостоява със званието "Почетен гражданин на град Русе", а от областния управител Венцислав Калчев получава Знака на областния управител на Област Русе.
Източници
Милиардери от Канада
Милиардери от България
Канадски бизнесмени
Предприемачи
Строители
Дипломати на България в Канада
Почетни консули на България
Консули в Канада
Български благодетели
Почетни граждани на Русе
Доктор хонорис кауза на Русенския университет
Българи емигранти в Канада
Родени в област Русе
Починали в Торонто
Личности (Мисисага) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,205 |
title: registration
updated: 2016-11-04
img: assets/stuff/ita_registration.png
published: false
---
New sign up page for future researchers and businesses.
`Visualforce` `Bootstrap` `CSS` `JavaScript` | {
"redpajama_set_name": "RedPajamaGithub"
} | 586 |
About product and suppliers: offers 66,300 jaw crusher products. About 99% of these are crusher. A wide variety of jaw crusher options are available to you, such as free samples, paid samples.
cheap jaw crushers for sale Grinding Mill China. cheap jaw crushers for sale. Posted at: August 6, 2013 [ 5353 Ratings ] Jaw Crusher For Sale Rental – New Used Jaw .
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Rubber Product Manufacturers Distributor China Hardware stores, distributors,dealers of China, ... cheap jaw crusher machine, less dust jaw . | {
"redpajama_set_name": "RedPajamaC4"
} | 6,388 |
Whether you realize it or not, every moment of every day, your civil rights and obligations are constantly solicited. Indeed, should it be when experiencing problems with a newly acquired property (latent defect), or following the death of a loved one (Estate liquidation), you have rights that must be enforced and obligations that must be respected.
These are all stressful situations where you have to assert your legal rights.
If you need to assert your rights, we can guide you through your legal recourses. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,482 |
A Palazzo Calderoni Martini egykori nemesi palota Altamura történelmi városközpontjában.
Története
A palota a 16-17. században épült a Calderoni család számára. Ebben lakott Michele Martini di Sinarico az 1799-es megmozdulások egyik vezéregyénisége. Ezt követően a Sabini család tulajdonába került, akikr 1989-ben váltak meg tőle. A palotát nemrég restaurálták magánvállalkozók. A palota érdekességei a kovácsoltvas kapuja, volutás erkélye, valamint a homlokzaton elhelyezett márványból készült családi címerek.
Források
Palazzo Calderoni Martini
Altamura barokk építményei
Olaszország barokk kastélyai, palotái | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,790 |
SS Andes byl parník vybudovaný roku 1852 v loděnicích William Denny & Co. v Dumbartonu. 18. srpna 1852 byl spuštěn na vodu a 8. prosince téhož roku vyplul na svou první plavbu Liverpool-New York. V roce 1854 sloužil jako nemocniční loď v krymské válce a roku 1859 byl prodán do Španělska.
Reference
Osobní lodě
Parní lodě
Britské lodě
Lodě Cunard Line | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,157 |
begin
require 'iterm2_tab_formatter/base'
rescue LoadError
warn "Couldn't load base class for Iterm2TabFormatter. This might be okay if reading from the gemspec before running bundle install."
end
class Iterm2TabFormatter
VERSION = "0.8.0"
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,078 |
Q: Are new macOS versions retained in the recovery partition? If a Mac is reinstalled by booting in to the recovery partition,
and the Mac came with El Capitan but was upgraded to Sierra via wifi,
what os will the Mac have after reinstalling via recover mode?
A: Found the answer to my own question:
The simple answer is, whenever you perform a major OS X upgrade, the Recovery HD partition is also upgraded to the same version of OS X. So, an upgrade from Lion to Mountain Lion will result in a Recovery HD linked to OS X Mountain Lion.
source: https://www.lifewire.com/identify-os-x-version-on-recovery-hd-partition-2259968
A: Josh, it is even better: as soon as you want to (re)install OS from the recovery partition: it installs the actual OS version including the latest update: for example you have 10.12.3 on your mac and you install OS from the recovery partition, it will install 10.12.6
A: Coming from a windows background, this was too good to be true.
I wanted to check it out myself.
So I thought the upgrades are retained in the recovery partition. So I went to a local coffee shop (with slow internet), wiped my disc, and began reinstalling. Indeed the the installer bootstrapper was the latest Sierra version.
But it took me more than four hours. Why did it take so long to re-install? Fortunately the installer let's you peek in to the "activity log", and I found that despite the claims in the other answers to this question, the installer does actually end up downloading a very large amount of data from apple servers. It calls them chunks and I watched as all 400+ of them downloaded one by one. Mind you, this happened a few days after the recover partition was supposedly given the latest updates.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,411 |
The American Nurses Association: says a culture of purposeful communication can help with nurse stress management. When staff are able to share emotions openly , stress levels can decrease .
fewer misunderstandings and clearer overall communication.
The benefit of having a mentor has long been established for new grads . Why stop there ? Mentors are beneficial regardless of how long you have been a nurse .
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5. Take Time Away Sometimes the best nurse stress management technique is to simply get away for a few days .
Taking vacation allows downtime to invest in your health and well- being. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,791 |
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