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\[ \int {fd}\left( {\widetilde{T}\mu }\right) = \int f \circ {Td\mu }\;\forall f \in C\left( X\right) . \] | Proof. It suffices to deal with real-valued \( f \in C\left( X\right) \) By definition of \( \widetilde{T} \) we have \( \int {\chi }_{B}d\left( {\widetilde{T}\mu }\right) = \int {\chi }_{B} \circ {Td\mu }\forall B \in \mathcal{B}\left( X\right) \) . Therefore \( \int {hd}\left( {\widetilde{T}\mu }\right) = \int h \cir... | Yes |
Theorem 6.7. The map \( \widetilde{T} : M\left( X\right) \rightarrow M\left( X\right) \) is continuous and affine. | Proof. If \( f \in C\left( X\right) \) then \( \int {fd}\widetilde{T}\mu = \int f - {Td\mu } \) . Therefore if \( {\mu }_{n} \rightarrow \mu \) in \( M\left( X\right) \) then \( \int {fd}\widetilde{T}{\mu }_{n} = \int f\;{Td}{\mu }_{n} \rightarrow \int f \circ {Td\mu } = \int {fd}\widetilde{T}\mu \) and so \( \widetild... | Yes |
Theorem 6.8. If \( T : X \rightarrow X \) is continuous and \( \mu \in M\left( X\right) \) then \( \mu \in M\left( {X, T}\right) \) iff \( \int f \circ {Td\mu } = \int {fd\mu }\;\forall f \in C\left( X\right) . | Proof. This is immediate from Lemma 6.6 and Theorem 6.2. | No |
Theorem 6.9. Let \( T : X \rightarrow X \) be continuous. If \( {\left\{ {\sigma }_{n}\right\} }_{n = 1}^{\infty } \) is a sequence in \( M\left( X\right) \) and we form the new sequence \( {\left\{ {\mu }_{n}\right\} }_{n = 1}^{\infty } \) by \( {\mu }_{n} = \left( {1/n}\right) \mathop{\sum }\limits_{{i = 0}}^{{n - 1}... | Proof. Let \( {\mu }_{{n}_{J}} \rightarrow \mu \) in \( M\left( X\right) \) . Let \( f \in C\left( X\right) \) . Then\n\n\[ \left| {\int f \circ {Td\mu }-\int {fd\mu }}\right| = \mathop{\lim }\limits_{{j \rightarrow \infty }}\left| {\int f \circ {Td}{\mu }_{{n}_{j}}-\int {fd}{\mu }_{{n}_{j}}}\right| \]\n\n\[ = \mathop{... | Yes |
Corollary 6.9.1 (Krylov and Bogolioubov). If \( T : X \rightarrow X \) is a continuous transformation of a compact metric space \( X \) then \( M\left( {X, T}\right) \) is non-empty. | Proof. We can make any choice for \( {\sigma }_{n} \) in Theorem 6.9; in particular choose \( y \in X \) and put \( {\sigma }_{n} = {\delta }_{y} \) for each \( n \) . | No |
Theorem 6.10. If \( T \) is a continuous transformation of the compact metric space \( X \) then\n\n(i) \( M\left( {X, T}\right) \) is a compact subset of \( M\left( X\right) \) . | (i) Suppose \( {\left\{ {\mu }_{n}\right\} }_{1}^{\infty } \) is a sequence of members of \( M\left( {X, T}\right) \) and \( {\mu }_{n} \rightarrow \mu \) in \( M\left( X\right) \) . Then \( \int {fd}\widetilde{T}\mu = \int f \circ {Td\mu } = \mathop{\lim }\limits_{{n \rightarrow \infty }}\int f \circ {Td}{\mu }_{n} = ... | Yes |
Lemma 6.11. Let \( \mu \in M\left( {X, T}\right) \) . Then\n\n(i) \( \mu \) is ergodic iff \( \forall f \in C\left( X\right) \forall g \in {L}^{1}\left( \mu \right) \)\n\n\[ \n\frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\int f\left( {{T}^{i}x}\right) g\left( x\right) {d\mu }\left( x\right) \rightarrow \int {fd\... | (i) Suppose the convergence condition holds and let \( F, G \in {L}^{2}\left( \mu \right) \) . Then\n\n\[ \nG \in {L}^{1}\left( \mu \right) \;\text{so}\frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\int f\left( {{T}^{i}x}\right) G\left( x\right) {d\mu }\left( x\right) \rightarrow \int {fd\mu }\int {Gd\mu }\;\foral... | Yes |
Theorem 6.12. Let \( T \) be a continuous transformation of a compact metric space. Let \( \mu \in M\left( {X, T}\right) \) . (i) \( \mu \) is ergodic iff whenever \( m \in M\left( X\right) \) and \( m \ll \mu \) then \[ \frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{\widetilde{T}}^{i}m \rightarrow \mu \] | (i) We use the ergodicity condition of Lemma 6.11. Let \( \mu \) be ergodic and suppose \( m \ll \mu, m \in M\left( X\right) \) . Let \( g = {dm}’{d\mu } \in {L}^{1}\left( \mu \right) \) . If \( f \in C\left( X\right) \) then \[ \int {fd}\left( {\frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{\widetilde{T}}^{i}m}\... | Yes |
Lemma 6.13. If \( T : X \rightarrow X \) is continuous and \( \mu \in M\left( {X, T}\right) \) is ergodic then there exists \( Y \in \mathcal{B}\left( X\right) \) with \( \mu \left( Y\right) = 1 \) such that \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}f\left( {{... | Proof. Choose a countable dense subset \( {\left\{ {f}_{k}\right\} }_{1}^{x} \) of \( C\left( X\right) \) . By the ergodic theorem there is \( {X}_{k} \in \mathcal{B}\left( X\right) \) such that \( \mu \left( {X}_{k}\right) = 1 \) and \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\mathop{\sum }\limits_{{... | Yes |
Theorem 6.14. Let \( T : X \rightarrow X \) be continuous and \( \mu \in M\left( {X, T}\right) \) . Then \( \mu \) is ergodic iff\n\n\[ \frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{\delta }_{{T}^{\prime }x} \rightarrow \mu \text{ a.e. } \] | Proof. If \( \mu \) is ergodic then Lemma 6.13 says \( \left( {1/n}\right) \mathop{\sum }\limits_{{1 = 0}}^{{n - 1}}{\delta }_{{T}^{i}x} \rightarrow \mu \forall x \in Y \) , where \( \mu \left( Y\right) = 1 \) .\n\nConversely suppose\n\n\[ \frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{\delta }_{{T}^{i}x} \righta... | Yes |
Theorem 6.15. Let \( T : X \rightarrow X \) be a continuous transformation of a compact metrisable space \( X \) . (i) For every \( \mu \in M\left( {X, T}\right) \) we have \( \mu \left( {\Omega \left( T\right) }\right) = 1 \) . (ii) If there is some \( \mu \in M\left( {X, T}\right) \) giving non-zero measure to every ... | ## Proof (i) Let \( {\left\{ {U}_{n}\right\} }_{1}^{L} \) be a base for the topology. Then \( X \smallsetminus \Omega \left( T\right) \) is the union of those \( {U}_{n} \) such that the sets \( {U}_{n},{T}^{-1}{U}_{n},{T}^{-2}{U}_{n}, \) . are pairwise disjoint. Such a set \( {U}_{n} \) must have measure zero for any ... | Yes |
Corollary 6.15.1. If \( {\Omega }_{\tau }\left( T\right) \) denotes the centre of \( T \) then \( \mu \left( {{\Omega }_{\tau }\left( T\right) }\right) = 1 \) for all \( \mu \in M\left( {X, T}\right) \) . | Proof. By Theorem 6.15(i) we can naturally identify \( M\left( {X, T}\right) \) with \( M(\Omega \left( T\right) \) , \( {\left. T\right| }_{\Omega \left( 1\right) } \) ). Applying Theorem 6.15(i) to \( {\left. T\right| }_{\Omega \left( T\right) } \) gives \( \mu \left( {{\Omega }_{2}\left( T\right) }\right) = 1\forall... | Yes |
Theorem 6.16. Let \( T : X \rightarrow X \) be a continuous transformation of a compact metrisable space \( X \) . Let \( N \geq 1 \) and \( x \in X \) . Then \( {T}^{N}\left( x\right) = x \) iff\n\n\[ \frac{1}{N}\mathop{\sum }\limits_{{i = 0}}^{{N - 1}}{\delta }_{{T}^{i}\left( x\right) } \in M\left( {X, T}\right) \] | Proof. If \( \mu \in M\left( X\right) \) then \( \mu \in M\left( {X, T}\right) \) iff \( \int f\;{Td\mu } = \int {fd\mu }\forall f \in C\left( X\right) \) . Therefore\n\n\[ \frac{1}{N}\mathop{\sum }\limits_{{i = 0}}^{{N - 1}}{\delta }_{{T}^{i}\left( x\right) } \in M\left( {X, T}\right) \;\text{ iff }\frac{1}{N}\mathop{... | Yes |
Theorem 6.17. Suppose \( T : X \rightarrow X \) is a homeomorphism of the compact metris-able space \( X \) . Suppose \( T \) is uniquely ergodic and \( M\left( {X, T}\right) = \{ \mu \} \) . Then \( T \) is minimal \( {\psi f\mu }\left( U\right) > 0 \) for all non-empty open sets \( U \) . | Proof. Suppose \( T \) is minimal. If \( U \) is open, \( {l}^{\prime } \neq \phi \), then \( X = \mathop{\bigcup }\limits_{{n = - \infty }}^{r}{T}^{n}\left( U\right) \) , so if \( \mu \left( U\right) = 0 \) then \( m\left( X\right) = 0 \), a contradiction.\n\nConversely, suppose \( \mu \left( U\right) > 0 \) for all o... | Yes |
Theorem 6.19. Let \( T : X \rightarrow X \) be a continuous transformation of a compact metrisable space \( X \) . The following are equivalent:\n\n(i) For every \( f \in C\left( X\right) \;\left( {1/n}\right) \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}f\left( {{T}^{i}x}\right) \) converges uniformly to a constant.\n\n(i... | (i) \( \Rightarrow \) (ii) hold trivially.\n\n(ii) \( \Rightarrow \) (iii). Luine \( k : C\left( X\right) \rightarrow \mathbb{C} \) by\n\n\[ h\left( f\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\mathop{\sum }\limits_{{1 = 0}}^{{n - 1}}f{T}^{l}\left( x\right) \]\n\nObserve that \( k \) is a linea... | Yes |
Theorem 7.1. If \( \\alpha \) is an open cover of \( X \) and \( T : X \\rightarrow X \) is continuous then \( \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\left( {1/n}\\right) H\\left( {\\mathop{\\bigvee }\\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\\alpha }\\right) \) exists. | Proof. Recall that if we set\n\n\[ \n{a}_{n} = H\\left( {\\mathop{\\bigvee }\\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\\alpha }\\right) \n\] \nthen by Theorem 4.9 it suffices to show that\n\n\[ \n{a}_{n + k} \\leq {a}_{n} + {a}_{k}\\;\\text{ for }k, n \\geq 1. \n\] \n\nWe have\n\n\[ \n{a}_{n + k} = H\\left( {\\mathop{\\bigve... | Yes |
Theorem 7.2. If \( {X}_{1},{X}_{2} \) are compact spaces and \( {T}_{i} : {X}_{i} \rightarrow {X}_{i} \) are continuous for \( i = 1,2 \), and if \( \phi : {X}_{1} \rightarrow {X}_{2} \) is a continuous map with \( \phi {X}_{1} = {X}_{2} \) and \( \phi {T}_{1} = {T}_{2}\phi \) then \( h\left( {T}_{1}\right) \geq h\left... | Proof. Let \( \alpha \) be an open cover of \( {X}_{2} \) . Then\n\n\[ h\left( {{T}_{2},\alpha }\right) = \lim \frac{1}{n}H\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}_{2}^{-i}\alpha }\right) \]\n\n\[ = \mathop{\lim }\limits_{n}\frac{1}{n}H\left( {{\phi }^{-1}\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}_{... | Yes |
Theorem 7.3. If \( T : X \rightarrow X \) is a homeomorphism of a compact space \( X \) then \( h\left( T\right) = h\left( {T}^{-1}\right) \) . | \[ h\left( {T,\alpha }\right) = \lim \frac{1}{n}H\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\alpha }\right) \] \[ = \lim \frac{1}{n}H\left( {{T}^{n - 1}\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\alpha }\right) }\right) \text{by Remark (5)} \] \[ = \lim \frac{1}{n}H\left( {\mathop{\big... | Yes |
Theorem 7.4. If \( d \) and \( {d}^{\prime } \) are uniformly equivalent and \( T \in {UC}\left( {X, d}\right) \) then \( {h}_{d}\left( T\right) = {h}_{{d}^{\prime }}\left( T\right) \) | Proof. Let \( {\varepsilon }_{1} > 0 \) . Choose \( {\varepsilon }_{2} > 0 \) such that\n\n\[ \n{d}^{\prime }\left( {x, y}\right) < {\varepsilon }_{2} \Rightarrow d\left( {x, y}\right) < {\varepsilon }_{1} \n\] \n\nand choose \( {\varepsilon }_{3} > 0 \) such that\n\n\[ \nd\left( {x, y}\right) < {\varepsilon }_{3} \Rig... | Yes |
Theorem 7.5. Let \( \\left( {X, d}\\right) \) be a metric space and \( T \\in {UC}\\left( {X, d}\\right) \) . If \( K \\subset {K}_{1} \\cup \\cdots \\cup {K}_{m} \) are all compact subsets of \( X \) then \( h\\left( {T;K}\\right) \\leq \\mathop{\\max }\\limits_{{1 \\leq i \\leq m}}h\\left( {T : {K}_{i}}\\right) \) . | Proof. Certainly \( {s}_{n}\\left( {\\varepsilon, K}\\right) \\leq {s}_{n}\\left( {\\varepsilon ,{K}_{1}}\\right) + \\cdots + {s}_{n}\\left( {\\varepsilon ,{K}_{m}}\\right) \) . Fix \( \\varepsilon > 0 \) . For each \( n \) choose \( {K}_{i\\left( {n,\\varepsilon }\\right) } \) such that \( {s}_{n}\\left( {\\varepsilon... | Yes |
Corollary 7.5.1. Let \( \\left( {X, d}\\right) \) be a metric space and \( T \\in {UC}\\left( {X, d}\\right) \) . Let \( \\delta > 0 \) . In order to compute \( {h}_{d}\\left( T\\right) \) is suffices to take the supremum of \( h\\left( {T;K}\\right) \) over those compact sets of diameter less than \( \\delta \) . | Proof. If \( K \) is compact it can be covered by a finite number of balls \( {B}_{1},\\ldots ,{B}_{m} \) of diameter \( \\delta /2 \) and hence \( h\\left( {T;K}\\right) \\leq \\mathop{\\max }\\limits_{{1 \\leq i \\leq m}}h\\left( {T;K \\cap {\\bar{B}}_{i}}\\right) \) . | No |
Corollary 7.5.2. If \( X \) is a compact metrisable space and \( d \) is any metric on \( X \) then \( h\left( T\right) = {h}_{d}\left( T\right) = h\left( {T;X}\right) \) . | Proof. If \( K \) is a compact subset of \( X \) then \( h\left( {T;K}\right) \leq h\left( {T;X}\right) \) . It follows from Theorem 7.4 that \( {h}_{d}\left( T\right) \) does not depend on \( d \) . | No |
Theorem 7.6. Let \( \left( {X, d}\right) \) be a compact metric space. If \( {\left\{ {\alpha }_{n}\right\} }_{1}^{\infty } \) is a sequence of open covers of \( X \) with \( \operatorname{diam}\left( {\alpha }_{n}\right) \rightarrow 0 \) then if \( {h}^{ * }\left( T\right) < \infty \mathop{\lim }\limits_{{n \rightarro... | Proof. Suppose \( {h}^{ * }\left( T\right) < \infty \) . Let \( \varepsilon > 0 \) be given and choose an open cover \( \gamma \) with \( {h}^{ * }\left( {T,\gamma }\right) > {h}^{ * }\left( T\right) - \varepsilon \) . Let \( \delta \) be a Lebesgue number for \( \gamma \) . Choose \( N \) so that \( n \geq N \) implie... | Yes |
Theorem 7.7. Let \( T : X \rightarrow X \) be a continuous map of a compact metric space \( \left( {X, d}\right) \). (i) If \( \alpha \) is an open cover of \( X \) with Lebesgue number \( \delta \) then \[ N\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\alpha }\right) \leq {r}_{n}\left( {\delta /2, X}\ri... | Proof. We know from Remark 5 that \( {r}_{n}\left( {\varepsilon, X}\right) \leq {s}_{n}\left( {\varepsilon ,{X}^{\prime }}\right) \forall \varepsilon > 0 \). (i) Let \( F \) be a \( \left( {n,\delta /2}\right) \) spanning set for \( X \) of cardinality \( {r}_{n}\left( {\delta /2.T}\right) \). Then \[ X = \mathop{\bigc... | Yes |
Corollary 7.7.1. Let \( T : X \rightarrow X \) be a continuous map of a compact metric space \( \left( {X, d}\right) \) . Let \( \varepsilon > 0 \) . Let \( {\alpha }_{\varepsilon } \) be the cover of \( X \) by all open balls of raduus \( {2\varepsilon } \) and let \( {\gamma }_{\varepsilon } \) be any cover of \( X \... | This leads directly to | No |
Theorem 7.8. If \( T : X \rightarrow X \) is a continuous map of the compact metric space \( \left( {X, d}\right) \) then \( h\left( T\right) = {h}^{ * }\left( T\right) \) i.e. the two definitions of topological entropy coincide. | Proof. If \( \varepsilon > 0 \) and \( {\alpha }_{\varepsilon },{\gamma }_{\varepsilon } \) are as in Corollary 7.7.1 then \( {h}^{ * }\left( {T,{\alpha }_{t}}\right) \leq r\left( {\varepsilon, X,\Gamma }\right) \leq \) \( s\left( {\varepsilon, X, T}\right) \leq {h}^{ * }\left( {T,{\gamma }_{\varepsilon }}\right) \) . ... | Yes |
Theorem 7.9. If \( T : X \rightarrow X \) is a continuous map of a compact metric space \( \\left( {X, d}\\right) \) then\n\n\[ h\\left( T\\right) = \\mathop{\\lim }\\limits_{{\\varepsilon \\rightarrow 0}}\\mathop{\\liminf }\\limits_{{n \\rightarrow \\infty }}\\frac{1}{n}\\log {r}_{n}\\left( {\\varepsilon, X}\\right) =... | Proof. Corollary 7.7.1 gives\n\n\[ {h}^{ * }\\left( {T,{\\alpha }_{\\varepsilon }}\\right) \\leq \\mathop{\\liminf }\\limits_{{n \\rightarrow \\infty }}\\frac{1}{n}\\log {r}_{n}\\left( {\\varepsilon, X}\\right) \\leq \\mathop{\\liminf }\\limits_{{n \\rightarrow \\infty }}\\frac{1}{n}\\log {s}_{n}\\left( {\\varepsilon, ... | Yes |
Theorem 7.11. Let \( T : X \rightarrow X \) be an expansive homeomorphism of the compact metric space \( \left( {X, d}\right) \) . (i) If \( x \) is a generator for \( T \) then \( h\left( T\right) = h\left( {T,\alpha }\right) \) . | (i) Let \( \beta \) be any open cover. Let \( \delta \) be a Lebesgue number for \( \beta \) . By Theorem 5.21 choose \( N > 0 \) so that each member of \( \mathop{\bigvee }\limits_{{-N}}^{N}{T}^{-n}\alpha \) has diameter less than \( \delta \) . Then \( \beta < \mathop{\bigvee }\limits_{{-N}}^{N}{T}^{-n}\alpha \), and... | Yes |
Theorem 7.12. The two-sided shift on \( X = \prod {}_{-\infty }^{\infty }Y \), where \( Y = \{ 0,1,\ldots, k - 1\} \) , has topological entropy \( \log \left( k\right) \) . | Proof. Let \( \alpha = \left\{ {{A}_{0},\ldots ,{A}_{k - 1}}\right\} \) be the natural generator, i.e.\n\n\[ \n{A}_{j} = \left\{ {{\left\{ {x}_{n}\right\} }_{-\infty }^{\infty } \mid {x}_{0} = j}\right\} .\n\]\nThen by Theorem 7.11,\n\n\[ \nh\left( T\right) = h\left( {T, x}\right) = \mathop{\lim }\limits_{{n \rightarro... | Yes |
Theorem 7.13. Let \( T : X \rightarrow X \) be the two-sided shift on \( X = \mathop{\prod }\limits_{{r = 1}}^{n}Y \) where \( Y = \{ 0,1,\ldots, k - 1\} . \n\n(i) If \( {X}_{1} \) is a closed subset of \( X \) with \( T{X}_{1} = {X}_{1} \) then \( h\left( {\left. T\right| }_{{x}_{1}}\right) = \) \( \mathop{\lim }\limi... | (i) Let \( \alpha \) be the natural generator for \( T : X \rightarrow X \), as in Theorem 7.12. Then \( \alpha \) is a generator for \( {\left. T\right| }_{{X}_{1}} \) and \( {\theta }_{n}\left( {X}_{1}\right) = N\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}_{1}^{-i}\alpha }\right) \) . The result follows fr... | Yes |
Corollary 7.14.1. Any homeomorphism of \( \left\lbrack {0,1}\right\rbrack \) has zero topological entropy. | Proof. \( T : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) has either \( \dot{T}\left( 0\right) = 0 \) and \( T\left( 1\right) = 1 \), or \( T\left( 0\right) = 1 \) and \( T\left( 1\right) = 0 \) . In both cases \( {T}^{2} \) fixes both 0 and 1 . Let \( S \) be any homeomorphism of \( ... | Yes |
Theorem 7.15. For a differentiable transformation \( T : M \rightarrow M \) of a p-dimensional Riemannian manifold \( M \) we have \( {h}_{d}\left( T\right) \leq \max \left\{ {0, p\log \left( \mathop{\sup }\limits_{{\mathbf{x} \in M}}\right) \begin{Vmatrix}{{\tau }_{\mathbf{x}}T}\end{Vmatrix}}\right\} \) ). | Proof. Let \( a = \mathop{\sup }\limits_{{x \in X}}\begin{Vmatrix}{{\tau }_{x}T}\end{Vmatrix} \) . If \( a = \infty \) there is nothing to prove. If \( a \leq 1 \) the mean-value theorem implies \( T \) satisfies \( d\left( {{Tx},{Ty}}\right) \leq d\left( {x, y}\right) \forall x, y \in M \) so that \( {h}_{d}\left( T\r... | Yes |
Theorem 8.3. Let \( T : X \rightarrow X \) be a continuous map of a compact metric space. Let \( {\left( {\xi }_{n}\right) }_{n = 1}^{x} \) be a sequence of partitions \( X \) such that \( \operatorname{diam}\left( {\xi }_{n}\right) \rightarrow 0 \) . For every \( \mu \in M\left( {X, T}\right) {h}_{\mu }\left( T\right)... | Proof. Let \( \mu \in M\left( {X, T}\right) \) . Let \( \varepsilon > 0 \) . Choose a finite partition \( \check{\zeta } = \left\{ {{A}_{1},\ldots ,{A}_{r}}\right\} \) such that \( {h}_{\mu }\left( {T,\xi }\right) > {h}_{\mu }\left( T\right) - \varepsilon \) if \( {h}_{\mu }\left( T\right) < \infty \), or \( {h}_{\mu }... | Yes |
Theorem 8.4 (Jacobs). Let \( T : X \rightarrow X \) be a continuous map of a compact metrisble space. If \( \mu \in M\left( {X, T}\right) \) and \( \mu = {\int }_{E\left( {X, T}\right) }{md\tau }\left( m\right) \) is the ergodic decomposition of \( \mu \) then we have:\n\n(i) if \( \xi \) is a finite partition of \( \l... | ## Proof\n\n(i) Let \( \xi = \left\{ {{A}_{1},\ldots ,{A}_{k}}\right\} \) . Let \( \sum = \lceil \rceil \) ., \( Y \) where \( Y = \{ 1,2,\ldots, k\} \) . Define \( \phi : X \rightarrow \sum \) by \( \phi \left( x\right) = {\left\{ {i}_{n}\right\} }_{-x}^{x} \) if \( {T}^{n}\left( x\right) \in {A}_{{i}_{n}} \) . We hav... | Yes |
Lemma 8.5. Let \( X \) be a compact metric space and \( \mu \in M\left( X\right) \) . (i) If \( x \in X \) and \( \delta > 0 \) there exists \( {\delta }^{\prime } < \delta \) such that \( \mu \left( {\partial B\left( {x;{\delta }^{\prime }}\right) }\right) = 0 \) . (ii) If \( \delta > 0 \) there is a finite partition ... | (i) This is clear since we cannot have an uncountable collection of disjoint sets of positive measure. (ii) By (i) there is a finite open cover \( \beta = \left\{ {{B}_{1},\ldots ,{B}_{r}}\right\} \) of \( \mathrm{Y} \) by balls of radius less than \( \delta /2 \) with \( \mu \left( {\widehat{c}{\widehat{B}}_{1}}\right... | Yes |
Corollary 8.6.1. Let \( T : X \rightarrow X \) be a continuous map of a compact metric space. Then\n\n(i) \( h\left( T\right) = \sup \left\{ {{h}_{\mu }\left( T\right) \mid \mu \in E\left( {X, T}\right) }\right\} \) . | (i) Let \( \varepsilon > 0 \) be given. Choose \( \mu \in M\left( {X, T}\right) \) such that\n\n\[ \n{h}_{\mu }\left( T\right) > \left\{ \begin{array}{ll} h\left( T\right) - \varepsilon & \text{ if }h\left( T\right) < \infty \\ 1/\varepsilon & \text{ if }h\left( T\right) = \infty . \end{array}\right. \n\]\n\nIf \( \mu ... | Yes |
Theorem 8.7. Let \( T : X \rightarrow X \) be a continuous transformation of a compact metrisahle space. Then\n\n(i) \( {M}_{\max }\left( {X, T}\right) \) is convex. | (i) This follows since the entropy map is affine (Theorem 8.1). | No |
Theorem 8.8. Let \( {T}_{i} : {X}_{i} \rightarrow {X}_{i}\left( {i = 1,2}\right) \) be a continuous transformation of a compact metrisable space and suppose \( {T}_{i} \) has a unique measure, \( {\mu }_{i} \), with maximal entropy. Suppose \( {h}_{{\mu }_{1}}\left( {T}_{1}\right) = {h}_{{\mu }_{2}}\left( {T}_{2}\right... | Proof. By Theorem 4.11 \( {h}_{{\mu }_{1}{\phi }^{-1}}\left( {T}_{2}\right) = {h}_{{\mu }_{1}}\left( {T}_{1}\right) \), so \( {h}_{{\mu }_{1}{\mu }_{2}}\left( {T}_{2}\right) = {h}_{{\mu }_{2}}\left( {T}_{2}\right) \) so \( {\mu }_{2} = {\mu }_{1} \cdot {\phi }^{-1} \) | Yes |
Theorem 8.9. Let \( Y = \{ 0,1,\ldots, k - 1\}, X = \mathop{\prod }\limits_{{-\infty }}^{c}Y \) and let \( T : X \rightarrow X \) be the two-sided shift. Then \( T \) has a unique measure with maximal entropy and this unique measure is the \( \left( {1/k,1/k,\ldots ,1/k}\right) \) -product measure. | Proof. We know \( h\left( T\right) = \log k \) . Suppose \( {h}_{\mu }\left( T\right) = \log k \) . Let \( \xi = \left\{ {{A}_{0},\ldots ,{A}_{k - 1}}\right\} \) be the natural generator (i.e. \( {A}_{j} = \left\{ {\left\{ {x}_{n}\right\} = x \mid {x}_{0} = j}\right\} \) ). Then \( \log k = {h}_{u}\left( \Gamma \right)... | Yes |
Theorem 8.15. Suppose \( T : {K}^{p} \rightarrow {K}^{p} \) is an affine transformation, \( {Tx} = a \cdot {.1}\left( x\right) \) , where \( a \in {K}^{p} \) and \( A \) is a surjective endomorphism of \( {K}^{p} \) . If \( m \) is Haar measure then\n\n\[ h\left( T\right) = {h}_{m}\left( T\right) = {h}_{m}\left( A\righ... | Proof. We know by Theorem 8.11 that\n\n\[ h\left( T\right) = {h}_{m}\left( T\right) = {h}_{m}\left( 4\right) = h\left( A\right) \]\n\nand by Corollary 8.12.1 that \( h\left( {.1}\right) = h\left( \widetilde{A}\right) \), where \( \widetilde{A} \) denotes the covering linear map of \( A \) . Since \( A \) is represented... | No |
Theorem 8.16. If \( T : X \rightarrow X \) is an expansive homeomorphism of a compact metric space then \( {N}_{n}\left( T\right) < \infty \forall n \geq 1 \) and \( h\left( T\right) \geq \mathop{\limsup }\limits_{{n \rightarrow n}}\left( {1/n}\right) \log {N}_{n}\left( T\right) \) . | Proof. Let \( \delta \) be an expansive constant for \( T \) . If \( {T}^{n}x = x,{T}^{n}y = y \) and \( x \neq y \) then if \( d\left( {{T}^{j}\left( x\right) ,{T}^{j}\left( y\right) }\right) \leq \delta ,0 \leq j \leq n - 1 \), then \( d\left( {{T}^{j}\left( x\right) ,{T}^{j}\left( y\right) }\right) \leq \delta \fora... | Yes |
Theorem 8.17. Suppose \( T : {X}_{A} \rightarrow {X}_{A} \) is a two sided topological Markov chain where \( A \) is an irreducible matrix. Then \( h\left( T\right) = \mathop{\lim }\limits_{{n \rightarrow x}}\left( {1/n}\right) \log {N}_{n}\left( T\right) \) and the unique measure with maximal entropy describes the dis... | Proof. If \( {\left\{ {x}_{j}\right\} }_{-x}^{x} \) is a point of \( {X}_{A} \) then it belongs to \( {F}_{n}\left( T\right) \) iff \( {x}_{j} = {x}_{j + n}\forall j \in Z \) . Therefore\n\n\[ \n{N}_{n}\left( T\right) = \mathop{\sum }\limits_{{{i}_{0},\ldots ,{i}_{n - 1} = 0}}^{{k - 1}}{a}_{{i}_{0}{i}_{1}}{a}_{{i}_{1}{... | Yes |
Theorem 8.18. Suppose \( 1 : {K}^{p} \rightarrow {K}^{p} \) is an automorphism of \( {K}^{ * } \) which is expansive (i.e. \( \left\lbrack A\right\rbrack \) has no eigenvalues of absolute value 1). Then \( h\left( {.1}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }} \) \( \left( {1\mathrm{\;n}}\right) \log {N}... | We now prove that \( h\left( A\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {1/n}\right) \log {N}_{n}\left( A\right) . \n\nProof. In the proof we shall use the fact that if \( B : {K}^{p} \rightarrow {K}^{p} \) is an endomorphism of \( {K}^{p} \) onto \( {K}^{p} \) then the kernel of \( B \) contains ... | Yes |
Theorem 8.19. Let \( T : X \rightarrow X \) be a homeomorphism of the compact metric space \( \left( {X, d}\right) \) . Let \( m \in M\left( {X, T}\right) \) and let \( m \) be ergodic. For \( \varepsilon > 0,\delta > 0 \) let \( {r}_{n}\left( {\varepsilon ,\delta, m}\right) \) denote the minimum number of \( \varepsil... | We refer to Katok [1] for the proof. | No |
Theorem 9.1. If \( f \in C\left( {X, R}\right) \) then \( P\left( {T, f}\right) = \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}P\left( {T, f,\varepsilon }\right) \) . | Proof. The limit exists by Remark 15. By Remark 13 we have \( P\left( {T, f}\right) \leq \) \( \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}P\left( {T, f,\varepsilon }\right) \) . By Remark 14, for any \( \delta > 0 \) we have \( \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}P\left( {T, f,\varepsilon }\right... | Yes |
Theorem 9.2. Let \( T : X \rightarrow X \) be continuous and \( f \in C\left( {X, R}\right) \). (i) If \( \alpha \) is an open cover of \( X \) with Lebesgue number \( \delta \) then \( {q}_{n}\left( {T, f, x}\right) \leq \) \( {Q}_{n}\left( {T, f,\delta /2}\right) \leq {P}_{n}\left( {T, f,\delta /2}\right) \n(ii) If \... | Proof. We know from Remark 13 that \( {Q}_{n}\left( {T, f,\varepsilon }\right) \leq {P}_{n}\left( {T, f,\varepsilon }\right) \) for all \( \varepsilon > 0 \). \n(i) If \( F \) is an \( \left( {n,\delta /2}\right) \) spanning set then \( X = \mathop{\bigcup }\limits_{{x \in F}}\mathop{\bigcap }\limits_{{i = 0}}^{{n - 1}... | Yes |
Lemma 9.3. If \( f \in C\left( {X, R}\right) \) and \( \alpha \) is an open cover of \( X \) then\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \alpha }}\frac{1}{n}\log {p}_{n}\left( {T, f,\alpha }\right) \]\n\nexists and equals \( \mathop{\inf }\limits_{n}\left( {1/n}\right) \log {p}_{n}\left( {T, f.\alpha }\right) \) . | Proof. By Theorem 4.9 it suffices to show \( {p}_{n + k}\left( {T, f, x}\right) \leq {p}_{n}\left( {T, f,\alpha }\right) \cdot {p}_{k}\left( {T, f, x}\right) \) . If \( \beta \) is a finite subcover of \( \mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\alpha \) and \( \gamma \) is a finite subcover of \( \mathop{\... | Yes |
Theorem 9.4. If \( T : X \rightarrow X \) is continuous and \( f \in C\left( {X, R}\right) \) then each of the following equals \( P\left( {T, f}\right) \). (i) \( \mathop{\lim }\limits_{{\delta \rightarrow 0}}\left\lbrack {\mathop{\sup }\limits_{\alpha }\left\{ {\mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {1... | (i) If \( \delta > 0 \) and \( \gamma \) is an open cover with \( \operatorname{diam}\left( \gamma \right) \leq \delta \) then \( {P}_{n}\left( {T, f,\delta }\right) \leq \) \( {p}_{n}\left( {T, f,\gamma }\right) \) (Theorem 9.2(ii)). Therefore \[ P\left( {T, f,\delta }\right) \leq \sup \left\{ {\mathop{\lim }\limits_{... | Yes |
Lemma 9.5. If \( T : X \rightarrow X \) is a continuous transformation of a compact met-risable space and \( \alpha \) is an open cover of \( X \) then for \( k > 0 \) and \( f \in C\left( {X, R}\right) \) | Proof. One readily gets\n\n\[ {e}^{-\left( {k + 1}\right) \left| \right| f\left| \right| }{q}_{n + k}\left( {T, f,\alpha }\right) \leq {q}_{n}\left( {T, f,\mathop{\bigvee }\limits_{{i = 0}}^{k}{T}^{-i}\alpha }\right) \leq {e}^{\left( {k + 1}\right) \left| \right| f\left| \right| }{q}_{n + k}\left( {T,\dot{f},\alpha }\r... | Yes |
Theorem 9.6. Let \( T : X \rightarrow X \) be an expansive homeomorphism of a compact metric space \( \left( {X, d}\right) \) .\n\n(i) If \( \alpha \) is a generator for \( T \) then\n\n\[ P\left( {T, f}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log {p}_{n}\left( {T, f,\alpha }\right) \]\n\n\[... | (i) Let \( \alpha \) be a generator for \( T \) . By Theorem 5.21 we have\n\n\[ \operatorname{diam}\left( {\mathop{\bigvee }\limits_{{i = - k}}^{k}{T}^{-i}\alpha }\right) \rightarrow 0 \]\n\nand so by Theorem 9.4(ii)\n\n\[ P\left( {T, f}\right) = \mathop{\lim }\limits_{{k \rightarrow \infty }}\mathop{\lim }\limits_{{n ... | Yes |
Theorem 9.7. Let \( T : X \rightarrow X \) be a continuous transformation of a compact metrisable space \( X \) . If \( f, g \in C\left( {X, R}\right) ,\varepsilon > 0 \) and \( c \in R \) then the following are true.\n\n(i) \( P\left( {T,0}\right) = h\left( T\right) \) . | Proof. Several times in the proofs we shall use the simple inequality\n\n\[ \frac{\sup {a}_{j}}{\sup {b}_{j}} \leq \sup \left( \begin{matrix} {a}_{j} \\ {b}_{j} \end{matrix}\right) \]\n\nwhen \( \left( {a}_{j}\right) ,\left( {b}_{j}\right) \) are collections of positive real numbers.\n\n(i) and (ii) are clear from the ... | No |
Theorem 9.8. Let \( T : X \rightarrow X \) be a continuous transformation of a compact metrisable space and let \( f \in C\left( {X, R}\right) \) . The following are true.\n\n(i) If \( k > 0P\left( {{T}^{k},{S}_{k}f}\right) = {kP}\left( {T, f}\right) \) . (Here \( \left( {{S}_{k}f}\right) \left( x\right) = \mathop{\sum... | (i) If \( F \) is \( \left( {{nk},\varepsilon }\right) \) spanning for \( T \) then \( F \) is \( \left( {n,\varepsilon }\right) \) spanning for \( {T}^{k} \) . Hence \( {Q}_{n}\left( {{T}^{k},{S}_{k}f,\varepsilon }\right) \leq {Q}_{nk}\left( {T, f,\varepsilon }\right) \) so that \( P\left( {{T}^{k},{S}_{k}f}\right) \l... | Yes |
Lemma 9.9. Let \( {a}_{1},\ldots ,{a}_{k} \) be given real numbers. If \( {p}_{i} \geq 0 \) and \( \mathop{\sum }\limits_{{i = 1}}^{k}{p}_{i} = 1 \)\n\nthen\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{k}{p}_{i}\left( {{a}_{i} - \log {p}_{i}}\right) \leq \log \left( {\mathop{\sum }\limits_{{i = 1}}^{k}{e}^{{a}_{i}}}\right) \... | Proof. Let \( M = \mathop{\sum }\limits_{{j = 1}}^{k}{e}^{{a}_{j}} \) . In Theorem 4.2 put\n\n\[ {\alpha }_{i} = \frac{{e}^{{a}_{i}}}{M}\text{ and }{x}_{i} = \frac{{p}_{i}M}{{e}^{{a}_{i}}}. \]\n\nThen\n\n\[ 0 = \phi \left( 1\right) \leq \mathop{\sum }\limits_{{i = 1}}^{k}\frac{{e}^{{a}_{i}}}{M}\frac{{p}_{i}M}{{e}^{{a}_... | Yes |
Corollary 9.10.1. Let \( T : X \rightarrow X \) be a continuous map of a compact metrisable space and let \( f \in C\left( {X, R}\right) \) . Then\n\n(i) \( P\left( {T, f}\right) = \sup \left\{ {{h}_{\mu }\left( T\right) +\int {fd\mu } \mid \mu \in E\left( {X, T}\right) }\right\} \) .\n\n(ii) \( P\left( {T, f}\right) =... | Proof. The proofs are simple generalisations of the proofs of the corresponding statements in Corollary 8.6. | No |
Corollary 9.10.2. If \( T : X \rightarrow X \) is uniquely ergodic and \( M\left( {X, T}\right) = \{ m\} \) then \( P\left( {T, f}\right) = {h}_{m}\left( T\right) + \int {fdm} \) . | So for a rotation \( {Tz} = {az} \) of a compact metric group \( G \) with \( \left\{ {a}^{n}\right\} \) dense in \( G \) we have \( P\left( {T, f}\right) = \int {fdm} \) where \( m \) is Haar measure on \( G \) . | No |
Theorem 9.11. Let \( T : X \rightarrow X \) be a continuous map of a compact metrisable space with \( h\left( T\right) < \infty \) . Let \( \mu : \mathcal{B}\left( X\right) \rightarrow R \) be a finite signed measure. Then \( \mu \in M\left( {X, T}\right) \) iff \( \int {fd\mu } \leq P\left( {T, f}\right) \;\forall f \... | Proof. If \( \mu \in M\left( {X, T}\right) \) then \( \int {fd\mu } \leq P\left( {T, f}\right) \) by the variational principle.\n\nNow suppose \( \mu \) is a finite signed measure and \( \int {fd\mu } \leq P\left( {T, f}\right) \forall f \in C\left( {X, R}\right) \) . We first show \( \mu \) takes only non-negative val... | Yes |
Theorem 9.13. Let \( T : X \rightarrow X \) be a continuous map of a compact metrisable space and let \( f \in C\left( {X, R}\right) \) . Then\n\n(i) \( {M}_{f}\left( {X, T}\right) \) is convex.\n\n(ii) If \( h\left( T\right) < \infty \) the extreme points of \( {M}_{f}\left( {X, T}\right) \) are precisely the ergodic ... | Proof. The first four parts are proved in the same way as the corresponding parts of Theorem 8.7. To prove (v) we notice that \( \forall \mu \in M\left( {X, T}\right) \int {fd\mu } = \) \( \int {gd\mu } + c \) . Therefore \( {h}_{\mu }\left( T\right) + \int {fd\mu } = {h}_{\mu }\left( T\right) + \int {gd\mu } + c \) an... | Yes |
Theorem 9.14. Let \( T : X \rightarrow X \) be a continuous map of a compact metrisable space \( X \) with \( h\left( T\right) < \infty \) and let \( f \in C\left( {X, R}\right) \) . Then \( {M}_{f}\left( {X, T}\right) \subset {t}_{f}\left( {X, T}\right) \subset \) \( M\left( {X, T}\right) \) . | Proof. Let \( \mu \in {M}_{f}\left( {X, T}\right) \) . If \( g \in C\left( {X, R}\right) \) ,\n\n\[ P\left( {T, f + g}\right) - P\left( {T, f}\right) \geq {h}_{\mu }\left( T\right) + \int {fd\mu } + \int {gd\mu } - {h}_{\mu }\left( T\right) - \int {fd\mu } = \int {gd\mu } \]\n\nby the variational principle. Therefore \... | Yes |
Theorem 9.15. Let \( T : X \rightarrow X \) be a continuous map of a compact metrisable space with \( h\left( T\right) < \infty \) and let \( f \in C\left( {X, R}\right) \) . If the entropy map of \( T \) is upper semi-continuous at the members of \( {t}_{f}\left( {X, T}\right) \) then \( {t}_{f}\left( {X, T}\right) = ... | Proof. It remains to show \( {t}_{f}\left( {X, T}\right) \subset {M}_{f}\left( {X, T}\right) \) . Let \( \mu \in {t}_{f}\left( {X, T}\right) \) . Then \( P\left( {T, f + g}\right) - \int \left( {f + g}\right) {d\mu } \geq P\left( {T, f}\right) - \int {fd\mu }\forall g \in C\left( {X, R}\right) \) so \( P\left( {T, h}\r... | Yes |
Corollary 9.15.1. Let \( T : X \rightarrow X \) be a continuous map of a compact metrisable space and suppose the entropy map of \( T \) is upper semi-continuous at each point of \( M\left( {X, T}\right) \) . Then there is a dense subset of \( C\left( {X, R}\right) \) such that each member \( f \) of this subset has a ... | Proof. We use the theorem that a convex function on a separable Banach space has a unique tangent functional at a dense set of points (Dunford and Schwartz [1], p. 450). This combined with Theorem 9.15 gives the result. | No |
Theorem 9.16. Let \( T : X \rightarrow X \) be the two-slded shift homeomorphism of the space \( X = \mathop{\prod }\limits_{{j = x}}^{x}Y, Y = \{ 0,1,\ldots, k - 1\} \) . Let \( {a}_{0},{a}_{1},\ldots ,{a}_{k - 1} \in R \) and define \( f \in C\left( {\widehat{X}, R}\right) \) by \( f\left( x\right) = {a}_{{x}_{0}} \)... | Proof. Let \( \xi = \left\{ {{A}_{0},\ldots ,{A}_{k - 1}}\right\} \) denote the natural generator i.e. \( {A}_{i} = \) \( \left\{ {{\left\{ {x}_{n}\right\} }_{-\alpha }^{x} \mid {x}_{0} = i}\right\} \) . We know \( {h}_{\mu }\left( T\right) = {h}_{\mu }\left( {T,\xi }\right) \leq {H}_{\mu }\left( \xi \right) \forall \m... | Yes |
Corollary 10.1.1. Let \( T \) be a measure-preserving transformation of the probability space \( \left( {X,\mathcal{A}, m}\right) \) . Let \( A : X \rightarrow L\left( {{R}^{k},{R}^{k}}\right) \) be a measurable function such that \( {\left( \log \parallel A\left( x\right) \parallel \right) }^{ + } \in {L}^{1}\left( m\... | \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log \begin{Vmatrix}{A\left( {{T}^{n - 1}x}\right) \circ \cdots \circ A\left( {Tx}\right) \circ A\left( x\right) }\end{Vmatrix} = \chi \left( x\right) \text{ a.e. } \] and \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\int \log \begin{Vmatrix}{... | Yes |
Corollary 10.1.2. Let \( T : M \rightarrow M \) be a \( {C}^{1} \) -differentiable map of the compact manifold \( M \) and take any Riemannian metric on \( M \) . There exists \( B \in \mathcal{B}\left( M\right) \) with \( {TB} \subset B \) and \( m\left( B\right) = 1\forall m \in M\left( {m, T}\right) \), and a measur... | We have \( \chi \left( x\right) \leq \sup \left\{ {\begin{Vmatrix}{{\tau }_{y}T}\end{Vmatrix} : y \in M}\right\} ,\chi \left( {Tx}\right) = \chi \left( x\right) \forall x \in B \) and\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\int \log \begin{Vmatrix}{{\tau }_{x}\left( {T}^{n}\right) }\end{Vmatrix... | Yes |
Theorem 10.3. Let \( T \) be an invertible measure-preserving transformation of the probability space \( \left( {X,\mathcal{B}, m}\right) \) . Let \( A : X \rightarrow {GL}\left( {R}^{k}\right) \) be a measurable function with \( {\left( \log \parallel A\left( x\right) \parallel \right) }^{ + } \in {L}^{1}\left( m\righ... | \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\log \begin{Vmatrix}{A\left( {{T}^{n - 1}x}\right) \circ \cdots \circ A\left( {Tx}\right) \circ A\left( x\right) \left( v\right) }\end{Vmatrix} = {\lambda }^{\left( i\right) }\left( x\right) \] and \[ \mathop{\lim }\limits_{{n \rightarrow x}}\frac{1}{n}\log \... | No |
Theorem 1.2. If \( \langle B, + , \cdot , - ,\mathbf{0},\mathbf{1}\rangle \) is a Boolean algebra then \( \forall a, b \in B \)\n\n1. \( a + a = a\;{aa} = a\; \) Idempotent Laws.\n\n2. \( a + {ab} = a\;a\left( {a + b}\right) = a\; \) Absorption Laws. | Proof.\n\n1. \( a + a = \left( {a + a}\right) \mathbf{1} = \left( {a + a}\right) \left( {a + {}^{ - }a}\right) = a + a\left( {{}^{ - }a}\right) = a + \mathbf{0} = a \) .\n\n2. \( a + {ab} = a\mathbf{1} + {ab} = a\left( {\mathbf{1} + b}\right) = a\left( {-b + b + b}\right) = a\left( {-b + b}\right) = a\mathbf{1} = a \) ... | Yes |
Theorem 1.3. If \( \langle B, + , \cdot , - ,\mathbf{0},\mathbf{1}\rangle \) is a Boolean algebra then\n\n1. \( - \mathbf{0} = \mathbf{1}, - \mathbf{1} = \mathbf{0} \) .\n\n2. \( \left( {\forall a \in B}\right) \left\lbrack {1 + a = 1 \land {0a} = 0}\right\rbrack \) . | Proof.\n\n1. \( - \mathbf{0} = \mathbf{0} + - \mathbf{0} = \mathbf{1} \) .\n\n2. \( 1 + a = \left( {-a + a}\right) + a = - a + \left( {a + a}\right) = - a + a = 1 \) . | No |
Theorem 1.4. If \( \langle B, + , \cdot , - ,\mathbf{0},\mathbf{1}\rangle \) is a Boolean algebra then \( \forall a, b \in B \)\n\n1. \( a + b = \mathbf{1} \land {ab} = \mathbf{0} \rightarrow b = {}^{ - }a \) . | 1. \( b = b\mathbf{1} = b\left( {a + {}^{ - }a}\right) = {ba} + b\left( {{}^{ - }a}\right) \)\n\n\[ = 0 + b\left( {-a}\right) = a\left( {-a}\right) + b\left( {-a}\right) \]\n\n\[ = \left( {a + b}\right) \left( {-a}\right) = \mathbf{1}\left( {-a}\right) = - a. \] | Yes |
Theorem 1.6. If \( \langle B, + , \cdot , - ,\mathbf{0},\mathbf{1}\rangle \) is a Boolean algebra with natural order \( \leq \) then \( \forall a, b, c \in B \)\n\n1. \( a \leq a \) .\n\n2. \( a \leq b \land b \leq a \rightarrow a = b \) .\n\n3. \( a \leq b \land b \leq c \rightarrow a \leq c \) . | Proof.\n\n1. \( {aa} = a \) .\n\n2. \( a = {ab} = {ba} = b \) .\n\n3. If \( a = {ab} \land b = {bc} \) then \( a = {ab} = a\left( {bc}\right) = \left( {ab}\right) c = {ac} \) . | No |
Theorem 1.7. If \( \langle B, + , \cdot , - ,\mathbf{0},\mathbf{1}\rangle \) is a Boolean algebra with natural order \( \leq \) then \( \forall a, b \in B \)\n\n1. \( a \leq b \leftrightarrow - b \leq - a \) .\n\n2. \( a \leq b \leftrightarrow a - b = \mathbf{0} \) .\n\n3. \( a \leq b \leftrightarrow \left( {a \Rightar... | Proof. 1. If \( a \leq b \) then \( a = {ab} \) . Therefore \( {}^{ - }a = {}^{ - }\left( {ab}\right) = {}^{ - }a + {}^{ - }b \) . Then by Theorem 1.4.4 \( \left( {-b}\right) \left( {-a}\right) = - b \), i.e., \( - b \leq - a \) . Conversely if \( - b \leq - a \) then\n\n\[ \n- \left( {-a}\right) \leq - \left( {-b}\rig... | Yes |
Theorem 1.8. If \( \\langle B, + , \\cdot , - ,\\mathbf{0},\\mathbf{1}\\rangle \) is a Boolean algebra with natural order \( \\leq \) then \( \\forall a, b, c, d \\in B \n\n1. \( 0 \\leq b \\leq 1 \) .\n\n2. \( \\left\\lbrack {a \\leq b}\\right\\rbrack \\land \\left\\lbrack {c \\leq d}\\right\\rbrack \\rightarrow \\lef... | Proof. 1. \( \\mathbf{0} = \\mathbf{0}b \\land b = b\\mathbf{1} \) .\n\n2. If \( a = {ab} \) and \( c = {cd} \) then \( \\left( {ac}\\right) \\left( {bd}\\right) = \\left( {ab}\\right) \\left( {cd}\\right) = {ac} \) and\n\n\\[ \n\\left( {a + c}\\right) \\left( {b + d}\\right) = {ab} + {ad} + {cb} + {cd} = a + {ad} + {c... | No |
Theorem 1.11. If \( \langle B, + , \cdot , - ,0,1\rangle \) is a Boolean algebra and \( A \subseteq B \) then\n\n1. \( - \mathop{\sum }\limits_{{a \in A}}a = \mathop{\prod }\limits_{{a \in A}}\left( {-a}\right) \) .\n\n2. \( - \mathop{\prod }\limits_{{a \in A}}a = \mathop{\sum }\limits_{{a \in A}}\left( {-a}\right) \) ... | Proof. 1. Since \( \left( {\forall b \in A}\right) \left\lbrack {b \leq \mathop{\sum }\limits_{{a \in A}}a}\right\rbrack \) we have \( - \mathop{\sum }\limits_{{a \in A}}a \leq - b \) and hence\n\n\[ \n- \mathop{\sum }\limits_{{a \in A}}a \leq \mathop{\prod }\limits_{{a \in A}}\left( {-a}\right) \n\]\n\nAlso \( \left( ... | No |
Theorem 1.12. If \( \langle B, + , \cdot , - ,0,1\rangle \) is a Boolean algebra, if \( b, c \in B \) , \( A \subseteq B \), and\n\n\[ b = \mathop{\sum }\limits_{{a \in A}}a \]\n\nthen\n\n\[ {cb} = \mathop{\sum }\limits_{{a \in A}}{ca}. \] | Proof. If \( a \in A \) then by Definition 1.9, \( a \leq b \) and hence \( {ca} \leq {cb} \) . If for each \( a \in A,{ca} \leq d \) then since \( a = \left( {-c + c}\right) a = - {ca} + {ca} \leq - c + d \) it follows from Definition 1.9 that \( b \leq {}^{ - }c + d \) . Hence \( {cb} \leq d \) and again from Definit... | Yes |
Theorem 1.14. \( \mathcal{P}\left( X\right) \) is a topology on \( X \) . | Proof. Left to the reader. | No |
Theorem 1.17. If \( T \) is a topology on \( X \) and \( A \subseteq X \) then \( {A}^{0} \in T \) . | Proof. If \( B = \{ N \in T \mid N \subseteq A\} \) then \( B \subseteq T \) . Furthermore\n\n\[ \n x \in {A}^{0} \leftrightarrow \exists N\left( x\right) \subseteq A \n\]\n\n\[ \n \leftrightarrow \exists N\left( x\right) \in B \n\]\n\n\[ \n \leftrightarrow x \in \cup \left( B\right) \text{.} \n\]\n\nThen \( {A}^{0} = ... | Yes |
Theorem 1.19. If \( X \neq 0 \), if \( {T}^{\prime } \) is a collection of subsets of \( X \) with the properties\n\n1. \( \left( {\forall a \in X}\right) \left( {\exists A \in {T}^{\prime }}\right) \left\lbrack {a \in A}\right\rbrack \) .\n\n2. \( \left( {\forall a \in X}\right) \left( {\forall {A}_{1},{A}_{2} \in {T}... | Proof. If \( T = \left\{ {B \subseteq X \mid \left( {\exists C \subseteq {T}^{\prime }}\right) \left\lbrack {B = \bigcup \left( C\right) }\right\rbrack }\right\} \) then \( 0 = \bigcup \left( 0\right) \in T \) and from property \( 1, X = \bigcup \left( {T}^{\prime }\right) \in T \) . This establishes property 1 of Defi... | Yes |
Theorem 1.21. 1. In any topology on \( X \) both 0 and \( X \) are clopen. | Proof. Left to the reader. | No |
Theorem 1.22. If \( A \subseteq X \) and if \( B \subseteq X \) then\n\n1. \( {A}^{0} \subseteq A \subseteq {A}^{ - } \).\n\n2. \( {A}^{00} = {A}^{0} \land {A}^{- - } = {A}^{ - } \).\n\n3. \( A \subseteq B \rightarrow {A}^{0} \subseteq {B}^{0} \land {A}^{ - } \subseteq {B}^{ - } \).\n\n4. \( {\left( X - A\right) }^{ - ... | Proof.\n\n1. \( x \in {A}^{0} \rightarrow \exists N\left( x\right) \subseteq A \)\n\n\[ \rightarrow x \in A \]\n\n\[ x \in A \rightarrow \left( {\forall N\left( x\right) }\right) \left\lbrack {N\left( x\right) \cap A \neq 0}\right\rbrack \]\n\n\[ \rightarrow x \in {A}^{ - }\text{.} \]\n\n2. \( x \in {A}^{0} \rightarrow... | Yes |
Theorem 1.23. If \( A \subseteq X \) and if \( B \subseteq X \) then\n\n1. \( A \) regular open implies \( A \) open.\n\n2. \( A \) is open iff \( X - A \) is closed.\n\n3. \( A \) is closed iff \( X - A \) is open.\n\n4. \( A \subseteq B \) and \( A \) dense in \( X \) implies \( B \) dense in \( X \). | Proof.\n\n1. If \( A = {A}^{-0} \) then \( {A}^{0} = {A}^{-{00}} = {A}^{-0} = A \).\n\n2. \( A = {A}^{0} \leftrightarrow \left( {X - A}\right) = \left( {X - {A}^{0}}\right) \)\n\n\[ \leftrightarrow \left( {X - A}\right) = {\left( X - A\right) }^{ - }\text{.} \]\n\n3. Left to the reader.\n\n4. \( A \subseteq B \rightarr... | No |
Theorem 1.24. If \( C \) is a clopen set in the topological space \( X \) and \( {B}^{ - } - {B}^{0} \subseteq C \) then \( {B}^{ - } - C \) is clopen. | Proof. If \( x \in {B}^{ - } - C \) then since \( {B}^{ - } - {B}^{0} \subseteq C \)\n\n\[ x \in {B}^{0} \land x \notin C. \]\n\nSince \( C \) is closed \( X - C \) is open. Therefore \( {B}^{0} \cap \left( {X - C}\right) \) is open. Then \( x \in {B}^{0} \cap \left( {X - C}\right) \) implies \( \exists N\left( x\right... | Yes |
Theorem 1.26. The clopen sets of a topological space form a natural Boolean algebra. | Proof. Left to the reader. | No |
Theorem 1.27. If \( A \subseteq X \) and \( B \subseteq X \) then\n\n1. \( {\left( A \cup B\right) }^{ - } = {A}^{ - } \cup {B}^{ - },{\left( A \cap B\right) }^{0} = {A}^{0} \cap {B}^{0} \) ,\n\n2. \( {\left( A \cap B\right) }^{ - } \subseteq {A}^{ - } \cap {B}^{ - },{A}^{0} \cup {B}^{0} \subseteq {\left( A \cup B\righ... | Proof. 1. Since \( A \subseteq A \cup B \) and \( B \subseteq A \cup B \) we have \( {A}^{ - } \subseteq {\left( A \cup B\right) }^{ - } \)\n\nand \( {B}^{ - } \subseteq {\left( A \cup B\right) }^{ - } \) . Therefore \( \left( {{A}^{ - } \cup {B}^{ - }}\right) \subseteq {\left( A \cup B\right) }^{ - } \) .\n\n\( x \in ... | No |
Theorem 1.28. If \( A \subseteq X \), and if \( B \subseteq X \), then\n\n1. \( A = {A}^{0} \rightarrow A \subseteq {A}^{-0} \) . | Proof. 1. If \( A = {A}^{0} \) then since \( A \subseteq {A}^{ - } \) we have \( A = {A}^{0} \subseteq {A}^{-0} \) . | No |
If \( A \subseteq X \) and if \( B \subseteq X \) then 1. \( A \) and \( B \) are regular open implies \( A \cap B \) is regular open. | 1. If \( A \) and \( B \) are regular open then since \( A \cap B \subseteq A \land A \cap B \subseteq B \) we have \[{\left( A \cap B\right) }^{-0} \subseteq {A}^{-0} = A\] \[{\left( A \cap B\right) }^{-0} \subseteq {B}^{-0} = B.\] Therefore \( {\left( A \cap B\right) }^{-0} \subseteq A \cap B \) . But also \[ \left( ... | Yes |
Theorem 1.33. If \( \leq \) is a partial ordering of \( P \) then \( {T}^{\prime } \triangleq \{ \left\lbrack x\right\rbrack \mid x \in P\} \) is a base for a topology on \( P \) . Furthermore if \( A \subseteq P \) then in this topology\n\n1. \( x \in {A}^{0} \leftrightarrow \left\lbrack x\right\rbrack \subseteq A \) ... | Proof. \( \;\left( {\forall x \in P}\right) \left\lbrack {x \in \left\lbrack x\right\rbrack \in {T}^{\prime }}\right\rbrack \) .\n\n\[ \left( {\forall x \in P}\right) \left( {\forall \left\lbrack y\right\rbrack ,\left\lbrack z\right\rbrack \in {T}^{\prime }}\right) \left\lbrack {x \in \left\lbrack y\right\rbrack \cap \... | No |
Theorem 1.34. If \( \langle P, \leq \rangle \) is a partial order structure and \( A \) is a collection of open subsets of \( P \) then\n\n\[ \mathop{\bigcap }\limits_{{a \in A}}a \]\n\nis open. | Proof.\n\n\[ p \in \mathop{\bigcap }\limits_{{a \in A}}a \rightarrow \left( {\forall a \in A}\right) \left\lbrack {p \in a}\right\rbrack \]\n\n\[ \rightarrow \left( {\forall a \in A}\right) \left\lbrack {\left\lbrack p\right\rbrack \subseteq a}\right\rbrack \]\n\n\[ \rightarrow \left\lbrack p\right\rbrack \subseteq \ma... | No |
Theorem 1.35. If \( \\langle P, \\leq \\rangle \) is a partial order structure and \( A \) is a collection of regular open subsets of \( P \) then\n\n\[ \n\\mathop{\\prod }\\limits_{{a \\in A}}a = \\mathop{\\bigcap }\\limits_{{a \\in A}}a \n\] | Proof. By Theorem 1.34\n\n\[ \n\\mathop{\\bigcap }\\limits_{{a \\in A}}a \\subseteq {\\left( \\mathop{\\bigcap }\\limits_{{a \\in A}}a\\right) }^{-0} = \\mathop{\\prod }\\limits_{{a \\in A}}a \n\]\n\nOn the other hand, if \( a \\in A \) then\n\n\[ \n\\mathop{\\bigcap }\\limits_{{a \\in A}}a \\subseteq a \n\]\n\n\[ \n{\... | Yes |
Theorem 1.37. If \( {\mathbf{B}}_{1} \) and \( {\mathbf{B}}_{2} \) are Boolean algebras and \( f : \left| {\mathbf{B}}_{1}\right| \rightarrow \left| {\mathbf{B}}_{2}\right| \) such that \( \forall x, y \in \left| {\mathbf{B}}_{1}\right| \n\n1. \( f\left( {x + y}\right) = f\left( x\right) + f\left( y\right) \).\n\n2. \(... | Proof.\n\n\[ f\left( {xy}\right) = f\left( {-\left( {-x + - y}\right) }\right) = - f\left( {-x + - y}\right) = - \left( {f\left( {-x}\right) + f\left( {-y}\right) }\right) \]\n\n\[ = - \left( {-f\left( x\right) + - f\left( y\right) }\right) = f\left( x\right) f\left( y\right) . \] | Yes |
Theorem 1.38. If \( f \) is a homomorphism from \( {\mathbf{B}}_{1} \) into \( {\mathbf{B}}_{2} \) then\n\n1. \( f\left( \mathbf{0}\right) = \mathbf{0} \) .\n\n2. \( f\left( 1\right) = 1 \) .\n\n3. \( \left( {\forall x, y \in \left| {\mathbf{B}}_{1}\right| }\right) \left\lbrack {x \leq y \rightarrow f\left( x\right) \l... | Proof.\n\n1. \( f\left( \mathbf{0}\right) = f\left( {\mathbf{0}\left( {-\mathbf{0}}\right) }\right) = f\left( \mathbf{0}\right) \left( {-f\left( \mathbf{0}\right) }\right) = \mathbf{0} \) .\n\n2. \( f\left( 1\right) = f\left( {1 + {}^{ - }1}\right) = f\left( 1\right) + {}^{ - }f\left( 1\right) = 1 \) .\n\n3. \( x \leq ... | Yes |
Theorem 1.39. If \( \mathbf{B} \) is a Boolean algebra with natural order \( \leq \) and if \( {B}_{0} = \left| \mathbf{B}\right| - \{ \mathbf{0}\} \) then \( \leq \) partially orders \( {B}_{0} \) and\n\n\[ \left( {\forall a, b \in {B}_{0}}\right) \left\lbrack {\left\lbrack a\right\rbrack \cap \left\lbrack b\right\rbr... | Proof. Left to the reader. | No |
Theorem 1.42. If \( I \) is an ideal in the Boolean algebra \( \mathbf{B} \) then\n\n1. \( \left( {\forall a, b \in \left| \mathbf{B}\right| }\right) \left\lbrack {a \leq b \in I \rightarrow a \in I}\right\rbrack \) .\n\n2. \( 1 \in I \rightarrow I = B \) . | Proof. Left to the reader. | No |
Theorem 1.44. If \( f \) is a Boolean homomorphism on \( \mathbf{B} \) then \( \ker \left( f\right) \) is a proper ideal in B. Furthermore if \( \ker \left( f\right) = \{ \mathbf{0}\} \) then \( f \) is an isomorphism. | Proof. Since \( f\left( \mathbf{0}\right) = \mathbf{0},\;\mathbf{0} \in \ker \left( f\right) \) . Furthermore \( \ker \left( f\right) \subseteq \left| \mathbf{B}\right| \) . If \( a, b \in \ker \left( f\right) \) then\n\n\[ f\left( {a + b}\right) = f\left( a\right) + f\left( b\right) = \mathbf{0} + \mathbf{0} = \mathbf... | Yes |
Theorem 1.45. If \( I \) is an ideal in the Boolean algebra \( \mathbf{B} \) then\n\n\[ \left( {\forall a, b \in \left| \mathbf{B}\right| }\right) \left\lbrack {a + b \in I \rightarrow a \in I \land b \in I}\right\rbrack . \] | Proof.\n\n\[ a = a + {ab} = a\left( {a + b}\right) \in I \]\n\n\[ b = b + {ab} = b\left( {a + b}\right) \in I. \] | Yes |
Theorem 1.47. If \( I \) is a proper ideal in the Boolean algebra \( \mathbf{B} \) then\n\n1. \( a/I = b/I \leftrightarrow a\left( {-b}\right) + b\left( {-a}\right) \in I \) . | Proof. 1. If \( a/I = b/I \) then since \( a\left( {-a}\right) + a\left( {-a}\right) = \mathbf{0} \in I \) we have \( a \in a/I \) and hence \( a \in b/I \) . Therefore \( a\left( {-b}\right) + b\left( {-a}\right) \in I \) . Conversely if \( a\left( {-b}\right) + \) \( b\left( {-a}\right) \in I \) and \( x \in a/I \) t... | Yes |
Theorem 1.48. If \( I \) is a proper ideal in \( \mathbf{B} \) then \( \left| \mathbf{B}\right| /I \) is the universe of a Boolean algebra, \( \mathbf{B}/I \), with operations\n\n\[ a/I + b/I = \left( {a + b}\right) /I,\;a/I \cdot b/I = {ab}/I,\; - \left( {a/I}\right) = \left( {-a}\right) /I \]\n\nand distinguished ele... | Proof. Left to the reader. | No |
Theorem 1.49. If \( I \) is a proper ideal in \( \mathbf{B} \) and\n\n\[ \left( {\forall a \in \left| \mathbf{B}\right| }\right) \left\lbrack {f\left( a\right) = a/I}\right\rbrack \]\n\nthen \( f \) is a Boolean homomorphism of \( \mathbf{B} \) onto \( \mathbf{B}/I \) and \( \ker \left( f\right) = I \) . | Proof. Left to the reader. | No |
Theorem 1.52. If \( \mathbf{B} = \langle B, \cup , \cap ,{}^{ - },\mathbf{0},\mathbf{1}\rangle \) is a complete Boolean algebra, if \( M \) is a standard transitive model of \( {ZF} \), and if \( 1 \in M \) then \( {B}^{M} \triangleq \langle B \cap M \) , \( \cup , \cap , - ,\mathbf{0},\mathbf{1}\rangle \) is an \( M \... | Proof. If \( a, b \in \left| \mathbf{B}\right| \cap M \) then\n\n\[ a \cup b \in \left| \mathbf{B}\right| \cap M,\;a \cap b \in \left| \mathbf{B}\right| \cap M\text{ and }{}^{ - }a = \mathbf{1} - a \in \left| \mathbf{B}\right| \cap M.\]\n\nSince \( \mathbf{0},\mathbf{1} \in \left| \mathbf{B}\right| \cap M \) it follows... | Yes |
Theorem 1.53. If \( \langle P, \leq \rangle \) is a partial order structure, if \( \langle P, \leq \rangle \in M \) , \( M \) a standard transitive model of \( {ZF} \), and if \( \mathbf{B} \) is the Boolean algebra of regular open subsets of \( P \), then \( \left| \mathbf{B}\right| \cap M \) is the universe of an \( ... | Proof. Since \( \langle P, \leq \rangle \in M \) and \( M \) is transitive\n\n\[ \left( {\forall p \in P}\right) \left\lbrack {\left\lbrack p\right\rbrack \in M}\right\rbrack \text{.} \]\n\nSince \( M \) satisfies the Axiom Schema of Replacement\n\n\[ \{ \langle p,\left\lbrack p\right\rbrack \rangle \mid p \in P\} \in ... | Yes |
1. The kernel of every complete Boolean homomorphism is a complete ideal. | 1. If \( f \) is a complete Boolean homomorphism on \( \mathbf{B} \) and \( A \subseteq \) \( \ker \left( f\right) \) then\n\n\[ f\left( {\mathop{\sum }\limits_{{a \in A}}a}\right) = \mathop{\sum }\limits_{{a \in A}}f\left( a\right) = \mathbf{0}. \]\n\nConsequently \( \mathop{\sum }\limits_{{a \in A}}a \in \ker \left( ... | Yes |
Theorem 1.55. Every complete Boolean ideal is principal. | Proof. If \( I \) is a complete ideal then, since \( I \subseteq I,\mathop{\sum }\limits_{{a \in I}}a \in I \) and \( \left( {\forall b \in I}\right) \left\lbrack {b \leq \mathop{\sum }\limits_{{a \in I}}a}\right\rbrack \) . Furthermore\n\n\[ \left( {\forall b \leq \mathop{\sum }\limits_{{a \in I}}a}\right) \left\lbrac... | Yes |
Theorem 1.57. \( I \) is a maximal ideal in the Boolean algebra \( \mathbf{B} \) iff\n\n\[ \left( {\forall b \in \left| \mathbf{B}\right| }\right) \left\lbrack {b \in I \leftrightarrow - b \notin I}\right\rbrack . \] | Proof. If \( b \in \left| \mathbf{B}\right| \land b \notin I \land - b \notin I \) and\n\n\[ J = \{ x + y \mid x \leq b \land y \in I\} \]\n\nthen \( \mathbf{0} \in J \) . If \( \left\lbrack {{x}_{1} \leq b}\right\rbrack \land \left\lbrack {{y}_{1} \in I}\right\rbrack \land \left\lbrack {{x}_{2} \leq b}\right\rbrack \l... | Yes |
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