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The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. In this note, we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the prefix of $\mathbb{T}$ of length $n$). |
We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehat{\mu}(\xi)| = O(|\xi|^{-\eta}), \quad \text{as } |\xi| \to \infty. $$ We show that this property ho... |
In this note, we will consider the question of local equivalence of analytic functions which fix the origin and are tangent to the identity, as well as the question of flows of analytic vector fields. All mappings and equivalences are considered in the non-archimedean context e.g. all norms can be considered $p$-adic ... |
We investigate the qualitative characteristics of a test particle attracted to an irregular elongated body, modeled as a non-homogeneous straight segment with a variable linear density. By deriving the potential function in closed form, we formulate the Hamiltonian equations of motion for this system. |
Considering the propagation characteristics of COVID-19 in different regions, the dynamics analysis and numerical demonstration of long-term and short-term models of COVID-19 are carried out, respectively. The long-term model is devoted to investigate the global stability of COVID-19 model with asymptomatic infections... |
Consider a self-similar space X. A typical situation is that X looks like several copies of itself glued to several copies of another space Y, and Y looks like several copies of itself glued to several copies of X, or the same kind of thing with more than two spaces. Thus, the self-similarity of X is described by a sy... |
We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial $f$ have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge\exp(|z|^\alpha)$ for some $\alpha>0$ and all $z$ outside a set of finite Lebesgue measure. |
Bratteli diagrams with countably infinite levels exhibit a new phenomenon: they can be horizontally stationary. The incidence matrices of these horizontally stationary Bratteli diagrams are infinite banded Toeplitz matrices. |
In this paper we present a proof of the Verjovsky conjecture: Every codimension-one Anosov flow on a manifold of dimension greater than three is topologically equivalent to the suspension of a hyperbolic toral automorphism. In fact, the conjecture is derived from possible more general result that says that for every c... |
We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute value of the Lyapunov exponents. As applications we prove finiteness for a cla... |
We study eigenvalues along periodic cycles of post-critically finite endomorphisms of $\mathbb{CP}^n$ in higher dimension. It is a classical result when $n = 1$ that those values are either $0$ or of modulus strictly bigger than $1$. |
We discuss how matrix-free/timestepper algorithms can efficiently be used with dynamic non-Newtonian fluid mechanics simulators in performing systematic stability/bifurcation analysis. The timestepper approach to bifurcation analysis of large scale systems is applied to the plane Poiseuille flow of an Oldroyd-B fluid ... |
We consider the topological behaviors of continuous maps with one topological attractor on compact metric space $X$. This kind of map is a generalization of maps such as topologically expansive Lorenz map, unimodal map without homtervals and so on. |
In this short and elementary note, we study some ergodic optimization problems for circle expanding maps. We first make an observation that if a function is not far from being convex, then its calibrated sub-actions are closer to convex functions in certain effective way. |
We prove that triangular configurations are plentiful in large subsets of cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving... |
Following [6,12], we study coupled map networks over arbitrary finite graphs. An estimate from below for a topological entropy of a perturbed coupled map network via a topological entropy of an unperturbed network by making use of the covering relations for coupled map networks is obtained. |
In this paper we give a kinematical illustration of some distributions called special multi-flags distributions. Precisely we define the kinematic model in angular coordinates of an articulated arm constituted of a series of (n + 1) segments in Rk+1 and construct the special multi-flag distribution associated to this ... |
Let $K$ be the attractor of the following IFS <br>$$\{f_1(x)=\lambda x, f_2(x)=\lambda x +c-\lambda,f_3(x)=\lambda x +1-\lambda\}, $$ <br>where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main results of this paper are as follows: $$\sqrt{K}... |
For three-body problem, <a href="http://R.Montgomery" rel="external noopener nofollow" class="link-external link-http">this http URL</a> [3] proved a reconstruction formula which calculates the overall rotation relating two similar triangle configurations if the initial triangular configuration is similar to the config... |
We consider here a dynamic model for a gas in which a variable number of particles $N \in \mathbb{N}_0 := \mathbb{N} \cup \{0\}$ can be located at a site. This point of view leads us to the grand-canonical framework and the need for a chemical potential. |
The purpose of this note is to present my understanding of Tim Austin's proof of the multiple ergodic theorem for commuting transformations, emphasizing on the use of joinings, extensions and factors. The existence of a sated extension, which is a key argument in the proof, is presented in a general context. |
We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that, with sufficient non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially from a locus of border-col... |
A sequential dynamical system, or SDS, consists of an undirected graph Y, a vertex-indexed list of local functions F_Y, and a permutation pi of the vertex set (or more generally, a word w over the vertex set) that describes the order in which these local functions are to be applied. In this article we investigate the ... |
We prove that an asymmetric unimodal map has no wandering intervals. |
In this article we prove that given a self-similar interval exchange transformation T, whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy c... |
In this paper we show how to compute algorithmically the full set of algebraically independent constraints for singular mechanical and field-theoretical models with polynomial Lagrangians. If a model under consideration is not singular as a whole but has domains of dynamical (field) variables where its Lagrangian beco... |
Data-driven learning is prevalent in many fields of science, mathematics and engineering. The goal of data-driven learning of dynamical systems is to interpret timeseries as a continuous observation of an underlying dynamical system. |
We give two general constructions of braid equivalences which exist between certain deformations of the 2-branched Horsehoe map. We then give numerical evidence suggesting that these constructions of braid equivalences are always realised in the Hénon family. |
We prove a fiberwise almost sure invariance principle for random piecewise expanding transformations in one and higher dimensions using recent developments on martingale techniques. |
For $ \beta>1 $ let $ T_\beta $ be the $\beta$-transformation on $ [0,1) $. Let $ \beta_1,\dots,\beta_d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $. |
This paper considers 1-dimensional generalized random walks in random scenery. That is, the steps of the walk are generated by an arbitrary stationary process, and also the scenery is a priori arbitrary stationary. |
We introduce a relative notion of the 'big images and preimages'-property for random topological Markov chains. This then implies that a relative version of the Ruelle-Perron-Frobenius theorem holds with respect to summable and locally Hoelder continuous potentials. |
We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in N. Mazurenko, M. Zarichnyi, Invariant idempotent measures, Carpathian Math. Publ. |
Recently, Xavier Buff and Arnaud Cheritat have provided an elegant proof of the existence of quadratic Siegel disks with smooth boundary. In this short note, we show how results of Yoccoz and Risler can be used to conclude the same result. |
Pattern-forming fronts are often controlled by an external stimulus which progresses through a stable medium at a fixed speed, rendering it unstable in its wake. By controlling the speed of excitation, such stimuli, or "triggers," can mediate pattern forming fronts which freely invade an unstable equilibrium a... |
We consider the space of $C^1$-diffeomorphims equipped with the $C^1$-topology on a three dimensional closed manifold. It is known that there are open sets in which $C^1$-generic diffeomorphisms display uncountably many chain recurrences classes, while only countably many of them may contain periodic orbits. |
We show that for a large class of potential functions and big coupling constant $\lambda$ the Schrödinger cocycle over the expanding map $x\mapsto bx ~( \text{mod} 1)$ on $\mathbb{T}$ has a Lyapunov exponent $>(\log\lambda)/4$ for all energies, provided that the integer $b\geq \lambda^3$. |
We give a numerical condition for right-handedness of a dynamically convex Reeb flow on the $3$-sphere. Our condition is stated in terms of an asymptotic ratio between the amount of rotation of the linearised flow and the linking number of trajectories with a periodic orbit that spans a disk-like global surface of sec... |
We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. |
We consider the concept of generalized measure-theoretic entropy, where instead of the Shannon entropy function we consider an arbitrary concave function defined on the unit interval, vanishing in the origin. Under mild assumptions on this function we show that this isomorphism invariant is linearly dependent on the K... |
In the paper we develop a method to evaluate the topological degree of the Poincare map of the mathematical model of narrow lagoon subject to a T-periodic forcing. Using the method developved we arrive to the conditions for the parameteres that guarantee the existence of T-periodic solutions in a given region. |
In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions; as well as variants of these systems--with right-hand sides no more homogeneous... |
This is the second part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper "Principal Lyapunov exponents and ... |
We investigate the behavior of itinerary sequence of each point of the Julia set of $z\mapsto z^2 + c$ when the parameter $c$ in the shift locus is allowed to pass through points in the bifurcation locus $\mathcal{P}_2$, which we call ``narrow", first proposed by Dierk Schleicher in \cite{schleicher2017internal}. ... |
Let $G$ be a group. Let $X$ be an algebraic group over an algebraically closed field $K$. |
This is an extended version of two lectures given during the Zagreb Dynamical Systems Workshop, October 22-26, 2018. |
For an ergodic map $T$ and a non-constant, real-valued $f \in L^1$, the ergodic averages $\mathbb{A}_N f(x) = \frac{1} {N} \sum_{n=1}^N f(T^n x)$ converge a.e., but the convergence is never monotone. Depending on particular properties of the function $f$, the averages $\mathbb{A}_N f(x)$ may or may not actually fluctu... |
We consider magnetic flows on 2-step nilmanifolds $M = \Gamma \backslash G$, where the Riemannian metric $g$ and the magnetic field $\sigma$ are left-invariant. Our first result is that when $\sigma$ represents a rational cohomology class and its restriction to $\mathfrak{g} = T_eG$ vanishes on the derived algebra, th... |
Quasilinear systems with piecewise constant arguments of generalized type are under investigation from the asymptotic point of view. The systems have discontinuous right-hand sides which are identified via a discrete-time map. |
Integrable Hamiltonian systems on symplectic manifolds have been well-studied. However, an intrinsic property of these kind of systems is that they can only live on even dimensional manifolds. |
We consider the complexification of the Henon family of quadratic diffeomorphisms of the plane, and the region of parameter space corresponding to complex horseshoes. We discuss some conjectures about the global topology of the complex horseshoe locus. |
We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host-Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. |
We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data is not sufficient for estimating time derivatives of state variables but enough to approximate the long-time behavior of the system through an approximation of it... |
In this paper, we mainly study the relation between regularity, independence and mean sensitivity for minimal systems. In the first part, we show that if a minimal system is incontractible, or local Bronstein with an invariant Borel probability measure, then the regularity is strictly bounded by the infinite independe... |
We consider families of multimodal interval maps with polynomial growth of the derivative along the critical orbits. For these maps Bruin and Todd have shown the existence and uniqueness of equilibrium states for the potential $\phi_t:x\mapsto-t\log|Df(x)|$, for $t$ close to 1. |
In this paper, we continue an investigation of the $p$-adic Ising-Vannimenus model on the Cayley tree of an arbitrary order $k$ $(k\geq 2$). We prove the existence of $p$-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional $p$-adic system of equations. |
The activation of a specific immune response takes place in the lymphoid organs such as the spleen. We present here a simplified model of the proliferation of specific immune cells in the form of a single delay equation. |
We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac{\log (c_n(X)/n)}{\log \log \log n} = 0$, then the quotient group $\textrm{Aut}(X,\sigma) / \langle \sigma\rangle$ of the automorphism group of $X$ by the subgroup generated by the shift $\sigma$ is locally finite. W... |
We study the twisted cohomoligical equation over the geodesic flow on $SL(2,\mathbb{R})/\Gamma$. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates for the solution, i.e, there is finite loss of regularity (with respect to Sob... |
We study the spatially homogeneous time dependent solutions and their bifurcations of the Gray-Scott model. We find the global map of bifurcations by a combination of rigorous verification of the existence of Takens Bogdanov and a Bautin bifurcations, in the space of two parameters k and F. With the aid of numerical c... |
This paper generalizes the classical notion of turbulence from dynamical systems generated by continuous functions to those defined by closed relations on compact metric spaces. Using the Mahavier product and the associated shift map, we introduce and explore CR-turbulence and reverse CR-turbulence, analyzing their re... |
We study geometrical and dynamical properties of the so-called discrete Lorenz-like attractors, that can be observed in three-dimensional diffeomorphisms. We propose new phenomenological scenarios of their appearance in one parameter families of such maps. |
We develop the theory of fractal homeomorphisms generated from pairs of overlapping affine iterated function systems. |
Starting with the work of Lapidus and van Frankenhuysen a number of papers have introduced zeta functions as a way of capturing multifractal information. In this paper we propose a new multifractal zeta function and show that under certain conditions the abscissa of convergence yields the Hausdorff multifractal spectr... |
We answered a question by Barak Weiss on the uniform discreteness of the holonomy vectors of translation surfaces. |
In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and |
For diffeomorphisms or for non-singular flows, there are many results relating properties persistent under C1 perturbations and global structures for the dynamics ( such as hyperbolicity, partial hyperbolicity, dominated splitting). However, a difículty appears when a robust property of a flow holds on a set containin... |
For any $C^1$ diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The mainline of our approach t... |
We study non-selfadjoint representations of a finite dimensional real Lie algebra $\fg$. To this end we embed a non-selfadjoint representation of $\fg$ into a more complicated structure, that we call a $\fg$-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on th... |
The Solar System contains a population of dust and small particles originating from asteroids, comets, and other bodies. These particles have been studied using a number of techniques ranging from in-situ satellite detectors to analysis of lunar microcraters to ground-based observations of zodiacal light. |
The most accurate method for modelling planetary migration and hence the formation of resonant systems is using hydrodynamical simulations. Usually, the force (torque) acting on a planet is calculated using the forces from the gas disc and the star, while the gas accelerations are computed using the pressure gradient,... |
Among various observational techniques used for detection of large bolides on a global scale is a low frequency sound known as infrasound. Infrasound, which is also one of the four sensing modalities used by the International Monitoring System (IMS), offers continuous global monitoring, and can be leveraged for planet... |
We present the transmission spectrum of the original transiting hot Jupiter HD\,209458b from 2.3 -- 5.1 $\mu$m as observed with the NIRCam instrument on the James Webb Space Telescope (JWST). Previous studies of HD 209458b's atmosphere have given conflicting results on the abundance of H$_2$O and the presence of c... |
We investigate the behaviour of two recent methods for the computation of preliminary orbits. These methods are based on the conservation laws of Kepler's problem, and enable the linkage of very short arcs of optical observations even when they are separated in time by a few years. |
The discovery of a population of close-orbiting giant planets ($\le$ 1 au) has raised a number of questions about their origins and dynamical histories. These issues have still not yet been fully resolved, despite over 20 years of exoplanet detections and a large number of discovered exoplanets. |
The mechanical properties of minerals in planetary materials are not only interesting from a fundamental point of view but also critical to the development of future space missions. Here we present nanoindentation experiments to evaluate the hardness and reduced elastic modulus of olivine, (Mg, Fe)2SiO4, in meteorite ... |
The formation and early evolution of circumstellar discs often occurs within dense, newborn stellar clusters. For the first time, we apply the moving-mesh code AREPO, to circumstellar discs in 3-D, focusing on disc-disc interactions that result from stellar fly-bys. |
The surfaces of solid objects in our planetary system are dappled with craters. Some of them are due to impacts of various solid projectiles into the surfaces of the objects. |
The connection between the atmospheric composition of giant planets and their origin remains elusive. In this study, we explore how convective mixing can link the primordial planetary state to its atmospheric composition. |
A growing number of debris discs have been detected around metal-polluted white dwarfs. They are thought to be originated from tidally disrupted exoplanetary bodies and responsible for metal accretion onto host WDs. |
NASA Citizen Scientists have used Exoplanet Transit Interpretation Code (EXOTIC) to reduce 40 sets of time-series images of WASP-12 taken by privately owned telescopes and a 6-inch telescope operated by the Center for Astrophysics | Harvard & Smithsonian MicroObservatory (MOBs). Of these sets, 24 result in clean t... |
The accurate characterisation of the stellar magnetism of planetary host stars has been gaining momentum, especially in the context of transmission spectroscopy investigations of exoplanets. Indeed, the magnetic field regulates the amount of energetic radiation and stellar wind impinging on planets, as well as the pre... |
3I/ATLAS is the third confirmed interstellar object to visit our Solar System, and only the second to display a clear coma. Infrared spectroscopy with the James Webb Space Telescope (JWST) provides the opportunity to measure its coma composition and determine the primary activity drivers. |
The distributions and abundances of molecules in protoplanetary disks are powerful tracers of the physical and chemical disk structures. The abundance ratios of HCN and its isomer HNC are known to be sensitive to gas temperature. |
(shorter version)The aim of this work is to search for planets around intermediate-mass stars in open clusters by using RV data obtained with HARPS from an extensive survey with more than 15 years of observations for a sample of 142 giant stars in 17 open clusters. We present the discovery of a periodic RV signal comp... |
We used GTC instrument OSIRIS to obtain long-slit spectra in the optical range (520-1040 nm) of the planetary host star WASP-43 (and a reference star) during a full primary transit event and four partial transit observations. We integrated the stellar flux of both stars in different wavelength regions producing severa... |
The equations of gas dynamics are extensively used to describe atmospheric loss from solar system bodies and exoplanets even though the boundary conditions at infinity are not uniquely defined. Using molecular-kinetic simulations that correctly treat the transition from the continuum to the rarefied region, we confirm... |
Debris discs are known to exist around many planet-host stars, but no debris dust has been found so far in systems with transiting planets. Using publicly available catalogues, we searched for infrared excesses in such systems. |
Seeing oceans, continents, quasi-static weather, and other surface features on exoplanets may allow us to detect and characterize life outside the solar system. The Proxima b planet resides within the stellar habitable zone allowing for liquid water on its surface, and it may be Earth-like. |
Ices, including water ice, prefer to recondense onto pre-existing nuclei rather than spontaneously forming grains from a cloud of vapor. Interestingly, different potential recondensation nuclei have very different propensities to actually nucleate water ice at the temperatures associated with freeze-out in protoplanet... |
I present RoadRunner, a fast exoplanet transit model that can use any radially symmetric function to model stellar limb darkening while still being faster to evaluate than the analytical transit model for quadratic limb darkening by Mandel & Agol (2002). CPU and GPU implementations of the model are available in th... |
We have detected H$_2$O and O$_2$ absorption against the far-UV continuum of stars located on lines of sight near the nucleus of Comet 67P/Churyumov-Gerasimenko using the Alice imaging spectrograph on Rosetta. These stellar appulses occurred at impact parameters of $\rho=4$-20 km, and heliocentric distances ranging fr... |
Giant planets embedded in protoplanetary disks (PPDs) can create annulus density gaps around their orbits in the type-II regime, potentially responsible for the ubiquity of annular substructures observed in PPDs. Despite of substantial amount of works studying type-II planet migration and gap properties, they are almo... |
Potential effects of sublimation of water ice from very slowly moving millimeter-sized and larger grains, a product of activity at ~10 AU or farther from the Sun driven presumably by annealing of amorphous water ice, are investigated by comparing 2I/Borisov with a nominal Oort Cloud comet of equal perihelion distance o... |
K2-19b and c were among the first planets discovered by NASA's K2 mission and together stand in stark contrast with the physical and orbital properties of the solar system planets. The planets are between the size of Uranus and Saturn at 7.0$\pm$0.2 R_E and 4.1$\pm$0.2 R_E, respectively, and reside a mere 0.1% out... |
The New Horizons spacecraft's encounter with the cold classical Kuiper belt object (486958) Arrokoth (formerly 2014 MU69) revealed a contact-binary planetesimal. We investigate how it formed, finding it is the product of a gentle, low-speed merger in the early Solar System. |
Emission of dust up to a few micrometer in size by impacts of sand grains during saltation is thought to be one source of dust within the Martian atmosphere. To study this dust fraction, we carried out laboratory impact experiments. |
Asteroids of size larger than 150 m generally do not have rotation periods smaller than 2.2 hours. This spin cutoff is believed to be due to the gravitationally bound rubble-pile structures of the asteroids. |
We report the electron density in a plasma tail of Comet ISON (C/2012 S1) derived from interplanetary scintillation (IPS) observations during November 1--28, 2013. Comet ISON showed a well-developed plasma tail (longer than 2.98 x 10^{7} km) before its perihelion passage on November 28. |
We numerically investigate the orbital dynamics of a spacecraft, or a comet, or an asteroid in the Pluto-Charon system in a scattering region around Charon using the planar circular restricted three-body problem. The test particle can move in bounded orbits around Charon or escape through the necks around the Lagrangi... |
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