Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Corollary 1.10.1. Let \( A : {K}^{n} \rightarrow {K}^{n} \) be a surjective continuous endomorphism of the \( n \) -torus. Then \( A \) is ergodic iff the matrix \( \left\lbrack A\right\rbrack \) has no roots of unity as eigenvalues. | Proof. If \( A \) is not ergodic Theorem 1.10 gives the existence of \( q \in {Z}^{n}q \neq 0 \) and \( k > 0 \) with \( {\left\lbrack A\right\rbrack }_{t}^{k}q = q \) . Then \( {\left\lbrack A\right\rbrack }_{t}^{k} \) has 1 as an eigenvalue so that \( {\left\lbrack A\right\rbrack }_{t} \) , and hence \( \left\lbrack ... | Yes |
Theorem 1.11. If \( T\left( x\right) = a \cdot A\left( x\right) \) is an affine transformation of the compact, connected, metric, abelian group \( G \) then the following are equivalent:\n\n(i) \( T \) is ergodic (relative to Haar measure).\n\n(ii) (a) whenever \( \gamma \circ {A}^{k} = \gamma \) for \( k > 0 \) then \... | Proof. First note that \( B \) is an endomorphism of \( G \) (but maybe nonsurjective) and commutes with \( A \) .\n\n(ii) \( \Rightarrow \) (i). Suppose (a) and (b) of (ii) hold. If \( f \circ T = f, f \in {L}^{2}\left( m\right) \), let \( f = \sum {b}_{i}{\gamma }_{i},{\gamma }_{i} \in \widehat{G} \), be the Fourier ... | Yes |
Theorem 1.12. The two-sided \( \left( {{p}_{0},\ldots ,{p}_{k - 1}}\right) \) shift is ergodic. | Proof. Let \( \mathcal{A} \) denote the algebra of all finite unions of measurable rectangles. Suppose \( {T}^{-1}E = E, E \in \mathcal{B} \) . Let \( \varepsilon > 0 \) be given, and choose \( A \in \mathcal{A} \) with \( m\left( {E\bigtriangleup A}\right) < \varepsilon \) . Then\n\n\[ \left| {m\left( E\right) - m\lef... | Yes |
Theorem 1.13. If \( T \) is the \( \left( {\mathbf{p}, P}\right) \) Markov shift (either one-sided or two-sided) then \( T \) is ergodic iff the matrix \( P \) is irreducible (i.e. \( \forall i, j\exists n > 0 \) with \( {p}_{ij}^{\left( u\right) } > 0 \) where \( {p}_{i, j}^{\left( n\right) } \) is the \( \left( {i, j... | We shall give the proof of this theorem in \( §{1.7} \) (Theorem 1.19) after we have used the ergodic theorem to derive another way of expressing ergodicity. We shall then only have to check a condition on measurable rectangles. | No |
Theorem 1.15 (Borel’s Theorem on Normal Numbers). Almost all numbers in \( \\lbrack 0,1) \) are normal to base 2, i.e., for a.e. \( x \in \\lbrack 0,1) \) the frequency of 1’s in the binary expansion of \( x \) is \( \\frac{1}{2} \) . | Proof. Let \( T : \\lbrack 0,1) \\rightarrow \\lbrack 0,1) \) be defined by \( T\\left( x\\right) = {2x}{\\;\\operatorname{mod}\\;1} \) . We know that \( T \) preserves Lebesgue measure \( m \) and is ergodic, by Example 4 at the end of \( \\$ {1.5} \) . Let \( Y \) denote the set of points of \( \\lbrack 0,1) \) that ... | Yes |
Corollary 1.14.1 ( \( {L}^{P} \) Ergodic Theorem of Von Neumann). Let \( 1 \leq p < \infty \) and let \( T \) be a measure-preserving transformation of the probability space \( \left( {X,\mathcal{B}, m}\right) \) . If \( f \in {L}^{P}\left( m\right) \) there exists \( {f}^{ * } \in {L}^{P}\left( m\right) \) with \( {f}... | Proof. If \( g \) is bounded and measurable then \( g \in {L}^{p} \) and by the ergodic theorem we have that \( \left( {1/n}\right) \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}g\left( {{T}^{i}x}\right) \rightarrow {g}^{ * }\left( x\right) \) a.e. Clearly \( {g}^{ * } \in {L}^{\infty }\left( m\right) \) and hence \( {g}^{ ... | Yes |
Corollary 1.14.2. Let \( \left( {X,\mathcal{B}, m}\right) \) be a probability space and let \( T : X \rightarrow X \) be a measure-preserving transformation. Then \( T \) is ergodic iff \( \forall A, B \in \mathcal{B} \)\n\n\[ \frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}m\left( {{T}^{-i}A \cap B}\right) \righta... | Proof. Suppose \( T \) is ergodic. Putting \( f = {\chi }_{A} \) in Theorem 1.14 gives \( \left( {1/n}\right) \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{\chi }_{A}\left( {{T}^{i}\left( x\right) }\right) \rightarrow m\left( A\right) \) a.e. Multiplying by \( {\chi }_{B} \) gives\n\n\[ \frac{1}{n}\mathop{\sum }\limits_{{1... | Yes |
Theorem 1.16 (Maximal Ergodic Theorem). Let \( U : {L}_{K}^{1}\left( m\right) \rightarrow {L}_{R}^{1}\left( m\right) \) be a positive linear operator with \( \parallel U\parallel \leq 1 \) . Let \( N > 0 \) be an integer and let \( f \in {L}_{R}^{1}\left( m\right) \) . Define \( {f}_{0} = 0,{f}_{n} = f + {Uf} + {U}^{2}... | Proof. (due to A. Garsia) Clearly \( {F}_{N} \in {L}_{R}^{1}\left( m\right) \) . For \( 0 \leq n \leq N \) we have \( {F}_{N} \geq {f}_{n} \) so \( U{F}_{N} \geq U{f}_{n} \) by positivity, and hence \( U{F}_{N} + f \geq {f}_{n + 1} \) . Therefore\n\n\[ U{F}_{v}\left( x\right) + f\left( x\right) \geq \mathop{\max }\limi... | Yes |
Corollary 1.16.1. Let \( T : X \rightarrow X \) be measure-preserving. If \( g \in {L}_{R}^{1}\left( m\right) \) and\n\n\[ \n{B}_{x} = \left\{ {x \in X : \mathop{\sup }\limits_{{n \geq 1}}\frac{1}{n}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}g\left( {{T}^{i}\left( x\right) }\right) > \alpha }\right\} \n\] \n\nthen \n\n\[... | Proof. We first prove this result under the assumptions \( m\left( X\right) < \infty \) and \( A = X \) . Let \( f = y - \alpha \), then \( {B}_{x} = \mathop{\bigcup }\limits_{{N = 0}}^{x}\left\{ {x : {F}_{N}\left( x\right) > 0}\right. \), so that \( {\int }_{{B}_{x}}{fdm} > 0 \) by Theorem 1.16 and therefore \( {\int ... | No |
Theorem 1.17. Let \( \left( {X,\mathcal{B}, m}\right) \) be a measure space and let \( \mathcal{S} \) be a semi-algebra that generates \( \mathcal{B} \) . Let \( T : X \rightarrow X \) be a measure-preserving transformation. Then\n\n(i) \( T \) is ergodic iff \( \forall A, B \in \mathcal{S} \)\n\n\[ \mathop{\lim }\limi... | Proof. Since each member of the algebra, \( \mathcal{A}\left( \mathcal{S}\right) \), generated by \( \mathcal{S} \) can be written as a finite disjoint union of members of \( \mathcal{S} \) it follows that if any of the three convergence properties hold for all members of \( \mathcal{S} \) then they hold for all member... | Yes |
Lemma 1.18. Let \( P \) be a stochastic matrix, having a strictly positive probability rector \( \mathrm{p} \) with \( \mathrm{p}P = \mathrm{p} \) . Then \( Q = \mathop{\lim }\limits_{{N \rightarrow s}}\left( {1/N}\right) \mathop{\sum }\limits_{{n = 0}}^{{N - 1}}{P}^{n} \) exists. The matrix \( Q \) is also stochastic ... | Proof. Let \( m \) denote the \( \left( {\mathbf{p}, P}\right) \) Markov measure and \( T \) be two-sided \( \left( {\mathbf{p}, P}\right) \) Markov shift. Let \( {\chi }_{i} \) denote the characteristic function of the cylinder \( {}_{0}{\left\lbrack i\right\rbrack }_{0} = \) \( \left\{ {{\left( {x}_{n}\right) }_{-n}^... | Yes |
Theorem 1.21. If \( T \) is a measure-preserving transformation of a probability space \( \left( {X,\mathcal{B}, m}\right) \) the following are equivalent:\n\n(i) \( T \) is weak-mixing.\n\n(ii) For every pair of elements \( A, B \) of \( \mathcal{B} \) there is a subset \( J\left( {A, B}\right) \) of \( {Z}^{ + } \) o... | Proof. Apply Theorem 1.20 with \( {a}_{n} = m\left( {{T}^{-n}A \cap B}\right) - m\left( A\right) m\left( B\right) \). | Yes |
Theorem 1.22. Let \( \left( {X,\mathcal{B}, m}\right) \) be a probability space with a countable basis and let \( T : X \rightarrow X \) be a measure-preserving transformation. Then \( T \) is weak-mixing if there is a subset \( J \) of \( {Z}^{ + } \) of density zero such that for all \( A, B \in \mathcal{B} \) \( \ma... | Proof. It suffices to prove that the stated condition holds if \( T \) is weak-mixing. Let \( {\left\{ {B}_{k}\right\} }^{\infty } \) be a countable basis for \( \left( {X,\mathcal{B}, m}\right) \) . Put\n\n\[ \n{a}_{n} = \mathop{\sum }\limits_{{i, j = 1}}^{\infty }\frac{\left| m\left( {T}^{-n}{B}_{i} \cap {B}_{j}\righ... | Yes |
Theorem 1.23. Suppose \( \left( {X,\mathcal{B}, m}\right) \) is a probability space and \( T : X \rightarrow X \) is measure-preserving.\n\n(i) The following are equivalent:\n\n(1) \( T \) is ergodic.\n\n(2) For all \( f, g \in {L}^{2}\left( m\right) \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {1/n}\right) \m... | Proof. (i), (ii) and (iii) are proved using similar methods. We shall prove (iii) to illustrate the ideas. Slight modification of this proof will prove (i) and (ii).\n\n\( \left( 2\right) \Rightarrow \left( 1\right) \) . This follows by putting \( f = {\chi }_{A}, g = {\chi }_{B} \), for \( A, B \in \mathcal{B} \) .\n\... | Yes |
Theorem 1.31. If \( T \) is the \( \left( {\mathbf{p}, P}\right) \) Markov shift (either one-sided or two-sided) the following are equivalent:\n\n(i) \( T \) is weak-mixing.\n\n(ii) \( T \) is strong-mixing.\n\n(iii) The matrix \( P \) is irreducible and aperiodic (i.e. \( \exists N > 0 \) such that the matrix \( {P}^{... | Proof. That (iii) and (iv) are equivalent is a standard use of the renewal theorem in probability theory.\n\n(i) \( \Rightarrow \) (iii). Since \( \left( {1/N}\right) \mathop{\sum }\limits_{{i = 0}}^{{N - 1}}\left| {m\left( {{}_{0}{\left\lbrack i\right\rbrack }_{0} \cap {T}_{0}^{-n}{\left\lbrack j\right\rbrack }_{0}}\r... | Yes |
Theorem 2.1. Let \( X \) be a complete separable metric space, let \( \mathcal{D}\left( X\right) \) be its \( \sigma \) -algebra of Borel subsets and let \( m \) be a probability measure on \( \mathcal{B}\left( X\right) \) with \( m\left( {\{ x\} }\right) = 0 \) for each set \( \{ x\} \) consisting of a single point \(... | A proof is given in Theorem 9, page 327 of Royden's book (Royden [1]). | Yes |
Theorem 2.2. Let \( {X}_{1},{X}_{2} \) be complete separable metric spaces, let \( \mathcal{B}\left( {X}_{1}\right) \) , \( \mathcal{B}\left( {X}_{2}\right) \) be their \( \sigma \) -algebras of Borel subsets and let \( {m}_{1},{m}_{2} \) be probability measures on \( \mathcal{B}\left( {X}_{1}\right) ,\mathcal{B}\left(... | \[ {m}_{1}\left( \left\{ {x \in {X}_{1} \mid \phi \left( x\right) \neq \psi \left( x\right) }\right\} \right) = 0. \] | Yes |
Theorem 2.3. Let \( \left( {{X}_{1},{\mathcal{D}}_{1},{m}_{1}}\right) ,\left( {{X}_{2},{\mathcal{B}}_{2},{m}_{2}}\right) \) be probability spaces and let \( \Phi : \left( {{\widetilde{\mathcal{B}}}_{2},{\widetilde{m}}_{2}}\right) \rightarrow \left( {{\widetilde{\mathcal{B}}}_{2},{\widetilde{m}}_{1}}\right) \) be a meas... | Proof. For a characteristic function \( {\chi }_{\widehat{B}},\widetilde{B} \in {\widetilde{\mathcal{B}}}_{2} \) define \( V{\chi }_{\widehat{B}} = {\chi }_{\Phi \left( \widehat{B}\right) } \) . Notice \( {\begin{Vmatrix}V{\chi }_{\widetilde{B}}\end{Vmatrix}}_{2} = {\begin{Vmatrix}{\chi }_{\widetilde{B}}\end{Vmatrix}}_... | Yes |
Theorem 2.5. For \( i = 1,2 \) let \( {T}_{i} \) be a measure-preserving transformation of the probability space \( \left( {{X}_{i},{\mathcal{B}}_{i},{m}_{i}}\right) \) . If \( {T}_{1} \) is isomorphic to \( {T}_{2} \) then \( {T}_{1} \) is conjugate to \( {T}_{2} \) . | Proof. Let \( \phi : {M}_{1} \rightarrow {M}_{2} \) be the isomorphism as in Definition 2.4. Define \( \Phi : \left( {{\widetilde{\mathcal{B}}}_{2},{\widetilde{m}}_{2}}\right) \rightarrow \left( {{\widetilde{\mathcal{B}}}_{1},{\widetilde{m}}_{1}}\right) \) by \( \Phi \left( \widetilde{B}\right) = {\left( {\phi }^{-1}\l... | Yes |
Theorem 2.6. Let \( \left( {{X}_{1},{\mathcal{B}}_{1},{m}_{1}}\right) ,\left( {{X}_{2},{\mathcal{B}}_{2},{m}_{2}}\right) \) be probability spaces which are either both Lebesgue spaces or where each \( {X}_{i} \) is a complete separable metric space and \( {\mathcal{B}}_{i} \) is its \( \sigma \) -algebra of Borel sets.... | Proof. Suppose \( \Phi : \left( {{\widetilde{\mathcal{B}}}_{2},{\widetilde{m}}_{2}}\right) \rightarrow \left( {{\widetilde{\mathcal{B}}}_{1},{\widetilde{m}}_{1}}\right) \) is the isomorphism of measure-algebras with \( \Phi {\widetilde{T}}_{2}^{-1} = {\widetilde{T}}_{1}^{-1}\Phi \) . By Theorem 2.2 (or the correspondin... | Yes |
Theorem 2.7. Let \( \left( {X,\mathcal{B}, m}\right) \) be a Lebesgue space or a probability space where \( X \) is a complete separable metric space and \( \mathcal{B} \) its \( \sigma \) -algebra of Borel subsets. Let \( T : X \rightarrow X \) be measure-preserving. Then \( T \) is invertible mod 0 iff \( {\widetilde... | Proof. We know \( {\widetilde{T}}^{-1} : \widetilde{\mathcal{B}} \rightarrow \widetilde{\mathcal{D}} \) is always injective. If \( T \) is invertible mod 0 it induces the same map on the measure algebra as that induced by the invertible map. Therefore \( {\widetilde{T}}^{-1}\widetilde{\mathcal{B}} = \widetilde{\mathcal... | Yes |
Theorem 2.8 Let \( T \) be a measure-preserving transformation of the probability space \( \left( {X,\mathcal{B}, m}\right) \) . Then \( {U}_{T} : {L}^{2}\left( m\right) \rightarrow {L}^{2}\left( m\right) \) is surjective iff \( {\widetilde{T}}^{-1} : \left( {\widetilde{\mathcal{B}},\widetilde{m}}\right) \rightarrow \l... | Proof. Let \( B \in \mathcal{B} \) . Then \( {U}_{T}{\chi }_{\widetilde{B}} = {\chi }_{{\widetilde{T}}^{-1}\widetilde{B}} \) . If \( {\widetilde{T}}^{-1} \) is surjective then the image of \( {U}_{T} \) contains all characteristic functions and so \( {U}_{T} \) is surjective. Suppose now \( {U}_{T} \) is surjective and... | Yes |
Theorem 2.9. Let \( {T}_{i}\left( {i = 1,2}\right) \) be a measure-preserving transformation of a probability space \( \left( {{X}_{i},{\mathcal{B}}_{i},{m}_{i}}\right) \) . If \( {T}_{1} \) and \( {T}_{2} \) are conjugate then they are spectrally isomorphic. | Proof. Suppose \( \Phi : \left( {{\widetilde{\mathcal{D}}}_{2},{\widetilde{m}}_{2}}\right) \rightarrow \left( {{\widetilde{\mathcal{D}}}_{1},{\widetilde{m}}_{1}}\right) \) is an isomorphism of measure algebras such that \( \Phi {\widetilde{T}}_{2}^{-1} = {\widetilde{T}}_{1}^{-1}\Phi \) . Let \( V \) be defined as in Th... | Yes |
Theorem 2.10. If \( {T}_{i}\left( {i = 1,2}\right) \) is a measure-preserving transformation of a probability space \( \left( {{X}_{i},{\mathcal{B}}_{i},{m}_{i}}\right) \) and if \( V : {L}^{2}\left( {m}_{2}\right) \rightarrow {L}^{2}\left( {m}_{1}\right) \) is an invertible linear isometry satisfying the conditions of... | Proof. By Theorem 2.4 \( V \) is induced by an isomorphism of measure-algebras \( \Phi : \left( {{\widetilde{\mathcal{B}}}_{2},{\widetilde{m}}_{2}}\right) \rightarrow \left( {{\widetilde{\mathcal{B}}}_{1},{\widetilde{m}}_{1}}\right) \) in the sense \( V\left( {\chi }_{\widetilde{B}}\right) = {\chi }_{\Phi \left( \widet... | Yes |
Theorem 2.11. Any two invertible measure-preserving transformations with countable Lebesgue spectrum are spectrally isomorphic. | Proof. Let \( \left( {{X}_{i},{\mathcal{B}}_{i},{m}_{i}}\right) i = 1,2 \) be a probability space and let \( {T}_{i} : {X}_{i} \rightarrow {X}_{i} \) be an invertible measure-preserving transformation. Suppose \( {L}^{2}\left( {m}_{1}\right) \) has a basis \( \left\{ {f}_{0}\right\} \cup \left\{ {{U}_{{T}_{1}}^{n}{f}_{... | Yes |
Theorem 2.12. If a measure-preserving transformation \( T \) of a probability space \( \left( {X,\mathcal{B}, m}\right) \) has countable Lebesgue spectrum it is strong-mixing. | Proof. Let \( \left\{ {f}_{0}\right\} \cup \left\{ {{U}_{T}^{n}{f}_{j}j \geq 1, n \in Z}\right\} \) be a basis of \( {L}^{2}\left( m\right) \) where \( {f}_{0} \equiv 1 \) . Then if \( j, q \geq 0\mathop{\lim }\limits_{{p \rightarrow \infty }}\left( {{U}_{T}^{p} \circ {U}_{T}^{n}{f}_{j},{U}_{T}^{k}{f}_{q}}\right) = \le... | Yes |
Theorem 2.13. The following are spectral invariants of measure-preserving transformations: (1) ergodicity, (2) weak-mixing, (3) strong-mixing. | (1) We know \( T \) is ergodic iff \( \\left\\{ {f \\in {L}^{2}\\left( m\\right) : {U}_{T}f = f}\\right\\} \) is a one-dimensional subspace, and the latter condition is preserved under spectral isomorphism.\n\n(2) We know \( T \) is weak-mixing iff 1 is the only eigenvalue and \( T \) is ergodic, and this is preserved ... | Yes |
Theorem 3.1. Let \( T \) be a measure-preserving transformation of a probability space \( \left( {X,\mathcal{B}, m}\right) \) and suppose \( T \) is ergodic. Then the following are true.\n\n(i) If \( {U}_{T}f = {\lambda f}, f \in {L}^{2}\left( m\right), f ≢ 0 \), then \( \left| \lambda \right| = 1 \) and \( \left| f\ri... | (i) We have \( \begin{Vmatrix}{{U}_{T}f}\end{Vmatrix} = \left| \lambda \right| \parallel f\parallel \) so that \( \parallel f\parallel = \left| \lambda \right| \parallel f\parallel \) . Since \( \parallel f\parallel \neq 0 \) we have \( \left| \lambda \right| = 1. \) Then we have \( \left| {{U}_{T}f}\right| = \left| \l... | Yes |
Lemma 3.2. Let \( \left( {X,\mathcal{D}, m}\right) \) be a probability space. Let \( h \in {L}^{2}\left( m\right) \) . Then \( h \) is bounded (i.e. \( \exists c \in R \) such that \( m\left( {\{ x\left| \right| h\left( x\right) \mid > c\} }\right) = 0 \) ) iff \( h \cdot f \in {L}^{2}\left( m\right) \) for all \( f \i... | Proof. If \( h \) is bounded then clearly \( h \cdot f \in {L}^{2}\left( m\right) \) when \( f \in {L}^{2}\left( m\right) \) . Now suppose \( h \) is such that \( h \cdot f \in {L}^{2}\left( m\right) \) whenever \( f \in {L}^{2}\left( m\right) \) . Let\n\n\[ \n{X}_{n} = \{ x \in X\left| {n - 1 \leq }\right| h\left( x\r... | Yes |
Lemma 3.3. Let \( H \) be a discrete abelian group and \( K \) a divisible subgroup of \( H \) (i.e., \( \forall k \in K \) and \( \forall n > 0\exists a \in K \) such that \( {a}^{n} = k \) ). Then there exists a homomorphism \( \phi : H \rightarrow K \) such that \( {\left. \phi \right| }_{K} = \) identity (i.e., \( ... | Proof. Let \( \mathcal{A} \) consist of all retracts onto \( K \) from supergroups of \( K \) in \( H \) , i.e., \( \mathcal{R} \) consists of all pairs \( \left( {M,\phi }\right) \) where \( K \subset M \subset H \) and \( \phi : M \rightarrow K \) is a homomorphism such that \( {\left. \phi \right| }_{K} = \) identit... | Yes |
Corollary 3.4.1. If \( T \) is an invertible ergodic measure-preserving transformation with discrete spectrum then \( T \) and \( {T}^{-1} \) are conjugate. | Proof. They have the same eigenvalues. | No |
Theorem 3.5. Let \( T \), given by \( T\left( g\right) = {ag} \), be an ergodic rotation of a compact abelian group \( G \) . Then \( T \) has discrete spectrum. Every eigenfunction of \( T \) is a constant multiple of a character, and the eigenvalues of \( T \) are \( \{ \gamma \left( a\right) : \gamma \in \widehat{G}... | Proof. Let \( \gamma \in \widehat{G} \) . Then\n\n\[ \gamma \left( {Tg}\right) = \gamma \left( {ag}\right) = \gamma \left( a\right) \gamma \left( g\right) \]\n\nTherefore each character is an eigenfunction and so \( T \) has discrete spectrum since the characters are an orthonormal basis of \( {L}^{2}\left( m\right) \)... | Yes |
Theorem 3.6 (Representation Theorem). An ergodic measure-preserving transformation \( T \) with discrete spectrum on a probability space \( \left( {X,\mathcal{B}, m}\right) \) is . conjugate to an ergodic rotation on some compact abelian group. The group will be metrisable iff \( \left( {X,\mathcal{B}, m}\right) \) has... | Proof. Let \( \Lambda \) be the group of all eigenvalues of \( T \) and give \( \Lambda \) the discrete topology. So \( \Lambda \) is an algebraic subgroup of \( K \) but has the discrete topology. (If \( {L}^{2}\left( m\right) \) is separable then \( \Lambda \) is countable). Let \( G = \widehat{\Lambda } \), the char... | Yes |
Theorem 3.7 (Existence Theorem). Every subgroup \( \Lambda \) of \( K \) is the group of eigenvalues of an ergodic measure-preserving transformation with discrete spectrum. | Proof. The desired transformation is the rotation \( S \) constructed in the proof of Theorem 3.6. | No |
Theorem 4.1. Let \( {\Delta }_{k} = \left\{ {\left( {{p}_{1},\ldots ,{p}_{k}}\right) \in {R}^{k} \mid {p}_{i} \geq 0,\;\mathop{\sum }\limits_{{i = 1}}^{k}{p}_{i} = 1}\right\} \) . Suppose \( H : \mathop{\bigcup }\limits_{{k = 1}}^{s}{\Delta }_{k} \rightarrow R \) has the following properties:\n\n(i) \( H\left( {{p}_{1}... | The (elementary) proof of this theorem can be found on page 9 of Khinchine's book [1]. This theorem motivates our definition of the entropy of a partition. We shall prove that entropy has the properties listed in Theorem 4.1. | No |
Theorem 4.2. The function \( \phi : \lbrack 0,\infty ) \rightarrow R \) defined by\n\n\[ \phi \left( x\right) = \left\{ \begin{array}{ll} 0 & \text{if }x = 0 \\ x \cdot \log x & \text{if }x \neq 0 \end{array}\right. \]\n\nis strictly convex, i.e., \( \phi \left( {{\alpha x} + {\beta y}}\right) \leq {\alpha \phi }\left(... | Proof. We have\n\n\[ {\phi }^{\prime }\left( x\right) = 1 + \log x \]\n\n\[ {\phi }^{\prime \prime }\left( x\right) = \frac{1}{x} > 0\;\text{ on }\left( {0,\infty }\right) . \]\n\nFix \( \alpha ,\beta \) with \( \alpha > 0,\beta > 0 \) . Suppose \( y > x \) . By the mean value theorem\n\n\( \phi \left( y\right) - \phi ... | Yes |
Corollary 4.2.1. If \( \xi = \left\{ {{A}_{1},\ldots ,{A}_{k}}\right\} \) then \( H\left( \xi \right) \leq \log k \), and \( H\left( \xi \right) = \log k \) only when \( m\left( {A}_{i}\right) = 1/k \) for all \( i \) . | Proof. Put \( {\alpha }_{i} = 1/k \) and \( {x}_{t} = m\left( {A}_{i}\right) ,1 \leq i \leq k \) . | No |
Theorem 4.3. Let \( \left( {X,\mathcal{B}, m}\right) \) be a probability space. If \( \mathcal{A},\mathcal{C},\mathcal{D} \) are finite subalgebras of \( \mathcal{B} \) then:\n\n(i) \( H\left( {\mathcal{A} \vee \mathcal{C}/\mathcal{D}}\right) = H\left( {\mathcal{A}/\mathcal{D}}\right) + H\left( {\mathcal{C}/\mathcal{A}... | Proof. Let \( \xi \left( \mathcal{A}\right) = \left\{ {A}_{i}\right\} ,\xi \left( \mathcal{C}\right) = \left\{ {C}_{j}\right\} ,\xi \left( \mathcal{D}\right) = \left\{ {D}_{k}\right\} \) and assume, without loss of generality, that all sets have strictly positive measure (since if \( \xi \left( \mathcal{A}\right) = \) ... | Yes |
Theorem 4.5. Let \( V \) denote the space of all finite sub-algebras of \( \mathcal{B} \) where two finite algebras \( \mathcal{A},\mathcal{C} \) are identified if \( \mathcal{A} \circeq \mathcal{C} \) . Then \( d\left( {\mathcal{A},\mathcal{C}}\right) = H\left( {\mathcal{A}/\mathcal{C}}\right) + H\left( {\mathcal{C}/\... | Proof. We have \( d\left( {\mathcal{A},\mathcal{C}}\right) \geq 0 \) and equality holds iff \( \mathcal{A} \doteq \mathcal{C} \) (Theorem 4.4). Also \( H\left( {\mathcal{A}\prime \mathcal{D}}\right) \leq H\left( {\mathcal{A} \vee \mathcal{C}/\mathcal{D}}\right) = H\left( {\mathcal{C}/\mathcal{D}}\right) + H\left( {\mat... | Yes |
Lemma 4.6. Let \( \\left( {X,\\mathcal{B}, m}\\right) \) be a probability space and let \( {\\left\\{ {\\mathcal{F}}_{n}\\right\\} }_{1}^{\\infty } \) be an increasing sequence of sub- \( \\sigma \) -algebras of \( \\mathcal{B} \) . Denote \( \\mathop{\\bigvee }\\limits_{{n = 1}}^{r}{\\mathcal{F}}_{n} \) by \( \\mathca... | Proof. Recall that \( E\\left( {\\cdot /{\\mathcal{F}}_{n}}\\right) \) is the orthogonal projection of \( {L}^{2}\\left( {X,\\mathcal{B}, m}\\right) \) onto \( {L}^{2}\\left( {X,{\\mathcal{F}}_{n}, m}\\right) \) . Let \( B \\in \\mathcal{F} \) . Choose \( {B}_{n} \\in {\\mathcal{F}}_{n} \) with \( m\\left( {{B}_{n}\\bi... | Yes |
Theorem 4.7. Let \( \left( {X,\mathcal{B}, m}\right) \) be a probability space. Let \( \mathcal{A} \) be a finite subalgebra of \( \mathcal{B} \) and let \( {\left\{ {\mathcal{F}}_{n}\right\} }_{1}^{\alpha } \) be an increasing sequence of sub- \( \sigma \) -algebras of \( \mathcal{B} \) with \( \mathop{\bigvee }\limit... | Proof. Let \( \xi \left( \mathcal{A}\right) = \left\{ {{A}_{1},\ldots ,{A}_{k}}\right\} \) . From Lemma 4.6. we know that for each \( \iota \)\n\n\[{\begin{Vmatrix}E\left( {\chi }_{{A}_{1}}/{\mathcal{F}}_{n}\right) - E\left( {\chi }_{{A}_{1}}/\mathcal{F}\right) \end{Vmatrix}}_{2} \rightarrow 0.\]\n\nTherefore \( E\left... | Yes |
Theorem 4.8. Let \( \\left( {X,\\mathcal{B}, m}\\right) \) be a probability space and let \( \\mathcal{A},\\mathcal{F} \) be sub- \( \\sigma \) - algebras of \( \\mathcal{B} \) with \( \\mathcal{A} \) finite. Then\n\n(i) \( H\\left( {\\mathcal{A}/\\mathcal{F}}\\right) = 0 \) iff \( \\mathcal{A} \\in \\mathcal{F} \) .\n... | Proof. Let \( \\xi \\left( \\mathcal{A}\\right) = \\left\\{ {{A}_{1},\\ldots ,{A}_{k}}\\right\\} \).\n\n(i) If \( \\mathcal{A} \\in \\mathcal{F} \) then \( E\\left( {{\\chi }_{{A}_{i}}/\\mathcal{F}}\\right) \\left( x\\right) \) takes only the values 0,1 so \( H\\left( {\\mathcal{A}//\\mathcal{F}}\\right) = 0 \) Convers... | Yes |
Theorem 4.9. If \( {\left\{ {a}_{n}\right\} }_{n \geq 1} \) is a sequence of real numbers such that \( {a}_{n + p} \leq {a}_{n} + {a}_{p} \) \( \forall n, p \) then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{a}_{n}/n \) exists and equals \( \mathop{\inf }\limits_{n}{a}_{n}/n \) . (The limit could be \( - \infty... | Proof. Fix \( p > 0 \) . Each \( n > 0 \) can be written \( n = {kp} + i \) with \( 0 \leq i < p \) . Then\n\n\[ \frac{{a}_{n}}{n} = \frac{{a}_{i + {kp}}}{i + {kp}} \leq \frac{{a}_{i}}{kp} + \frac{{a}_{kp}}{kp} \leq \frac{{a}_{i}}{kp} + \frac{k{a}_{p}}{kp} = \frac{{a}_{i}}{kp} + \frac{{a}_{p}}{p}. \]\n\nAs \( n \righta... | Yes |
Corollary 4.9.1. if \( T : X \rightarrow X \) is measure-preserving and \( \mathcal{L} \) is a finite sub-another of \( \delta \) then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {1 - n}\right) H\left( {\sqrt[{{2n} - 1}]{n - 1}{T}^{-1}d}\right) \) exists. | Proof. Let \( {a}_{n} = H\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-1}A}\right) \geq 0 \) . Then\n\n\[ \n{d}_{n + p} = H\left( {\mathop{\bigvee }\limits_{{1 = 0}}^{{n + p - 1}}{T}^{-1}x/}\right) \]\n\n\[ \n\leq H\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\mathcal{A}}\right) + H\left( {\m... | Yes |
Theorem 4.10. If \( T : X \rightarrow X \) is measure-preserving and \( \mathcal{A} \) is a finite sub-algebra of \( \mathcal{B} \) then \( \left( {1/n}\right) H\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\mathcal{A}}\right) \) decreases to \( h\left( {T,\mathcal{A}}\right) \) . | Proof. We first show, by induction, that\n\n\[ H\left( {\mathop{\sum }\limits_{{1 = 0}}^{{n - 1}}{T}^{-1}{sI}}\right) = H\left( {sI}\right) - \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}H\left( {{sI}\mathop{\bigvee }\limits_{{i = 1}}^{j}{T}^{-i}{sI}}\right) . \]\n\nFor \( n = 1 \) it is clear, and if we assume it true for... | Yes |
Theorem 4.11. Entropy is a conjugacy invariant and hence an isomorphism invariant. | Proof. Let \( {T}_{1} : {X}_{1} \rightarrow {X}_{1},{T}_{2} : {X}_{2} \rightarrow {X}_{2} \) be measure-preserving and let \( \Phi : \left( {{\widetilde{\mathcal{B}}}_{2},{\widetilde{m}}_{2}}\right) \rightarrow \left( {{\widetilde{\mathcal{B}}}_{1},{\widetilde{m}}_{1}}\right) \) be an isomorphism of measure algebras su... | Yes |
Corollary 4.12.1. If \( \mathcal{A} \) , \( \mathcal{C} \) are finite sub-algebras of \( \mathcal{B} \) we have \( \left| {h\left( {T,\mathcal{A}}\right) - h\left( {T,\mathcal{C}}\right) }\right| \leq d\left( {\mathcal{A},\mathcal{C}}\right) ,\; \) so that \( \;h\left( {T, \cdot }\right) \; \) is a continuous real-valu... | Proof. By (ii1)\n\n\[ \left| {h\left( {T,\mathcal{A}}\right) - h\left( {T,\mathcal{C}}\right) }\right| \leq \max \left( {H\left( {\mathcal{A}/\mathcal{C}}\right), H\left( {\mathcal{C}/\mathcal{A}}\right) }\right) \]\n\n\[ \leq d\left( {\mathcal{A},\mathcal{C}}\right) \] | Yes |
Theorem 4.13. Let \( T \) be a measure-preserving transformation of the probability space \( \left( {X,\mathcal{B}, m}\right) \). (i) For \( k > 0, h\left( {T}^{k}\right) = {kh}\left( T\right) \). | ## Proof (i) We first show that \[ h\left( {{T}^{k},\mathop{\bigvee }\limits_{{i = 0}}^{{k - 1}}{T}^{-i} \cdot \mathcal{A}}\right) = {kh}\left( {T,\mathcal{A}}\right) \;\text{ if }k > 0. \] This follows since \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}H\left( {\mathop{\bigvee }\limits_{{j = 0}}^{{k - ... | Yes |
Theorem 4.14. If \( \mathcal{A} \) is a finite sub-algebra of \( \mathcal{B} \) and \( T \) is a measure-preserving transformation of \( \left( {X,\mathcal{B}, m}\right) \) then\n\n\[ h\left( {T,\mathcal{A}}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}H\left( {\mathcal{A}/\left( {\mathop{\bigvee }\limits_{{... | Proof. The limit exists since the right hand side is non-increasing in \( n \) by virtue of Theorem 4.3(v). We know from the proof of Theorem 4.10 that for \( n \geq 1 \)\n\n\[ H\left( {\mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\mathcal{A}}\right) = H\left( \mathcal{A}\right) + \mathop{\sum }\limits_{{j = 1}}... | Yes |
Corollary 4.14.1. Let \( T \) be a measure-preserving transformation of the probability space \( \left( {X,\mathcal{B}, m}\right) \) . Let \( \mathcal{I} \) be a finite subalgebra of \( \mathcal{D} \) . Then \( h\left( {T,\mathcal{A}}\right) = 0 \) iff \( \mathcal{S} \subset \mathop{\bigvee }\limits_{{i = 1}}^{\infty }... | Proof. By Theorems 4.14 and 4.8. | No |
Corollary 4.14.3. Let \( T \) be a measure-preserving transformation of the probability space \( \left( {X,\mathcal{B}, m}\right) \) . If \( h\left( T\right) = 0 \) then \( {T}^{-1}\mathcal{B} \doteq \mathcal{B} \) (so \( T \) is invertible mod 0 if \( \left( {X,\mathcal{B}, m}\right) \) is a Lebesgue space or a comple... | Proof. Let \( B \in \mathcal{B} \) and let \( \mathcal{A} \) be the finite algebra \( \{ \phi, B, X \smallsetminus B, X\} \) . By Corollary 4.14.2 we have \( \mathcal{A} \) & \( \mathop{\bigvee }\limits_{{i = 1}}^{\infty }{T}^{-i}\mathcal{A} \subset {T}^{-1}\mathcal{B} \) . Since \( B \) is an arbitrary element of \( \... | Yes |
Corollary 4.14.4. Let \( T \) be a measure-preserving transformation of the probability space \( \left( {X,\mathcal{B}, m}\right) \). Suppose \( h\left( T\right) = 0 \). If \( \mathcal{F} \) is a sub- \( \sigma \)-algebra of \( \mathcal{B} \) with \( {T}^{-1}\mathcal{F} \in \mathcal{F} \) then \( {T}^{-1}\mathcal{F} \c... | Proof. The transformation \( T \) induces a measure-preserving transformation \( {\left. T\right| }_{\left( \lambda ,\mathcal{F}, m\right) } \) of \( \left( {X,\mathcal{F}, m}\right) \) and it clearly has zero entropy. Apply Corollary 4.14.3 to this transformation. | No |
Lemma 4.15. Let \( r \geq 1 \) be a fixed integer. For each \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that if \( \xi = \left\{ {{A}_{1},\ldots ,{A}_{r}}\right\} ,\eta = \left\{ {{C}_{1},\ldots ,{C}_{r}}\right\} \) are any two partitions of \( \left( {X,\mathcal{B}, m}\right) \) into \( r \) sets with \( ... | Proof. Let \( \varepsilon > 0 \) be given. Choose \( \delta > 0 \) so that \( \delta < \frac{1}{4} \) and \( - r\left( {r - 1}\right) \delta \log \) \( \delta - \left( {1 - \delta }\right) \log \left( {1 - \delta }\right) < \varepsilon /2 \) . Let \( \zeta \) be the partition into the sets \( {A}_{i} \cap {C}_{j}\left(... | Yes |
Theorem 4.16. Let \( \left( {X,\mathcal{B}, m}\right) \) be a probability space and \( {\mathcal{B}}_{0} \) be an algebra such that the \( \sigma \) -algebra generated by \( {\mathcal{D}}_{0} \) (denoted by \( \mathcal{B}\left( {\mathcal{D}}_{0}\right) \) ) satisfies \( \mathcal{B}\left( {\mathcal{B}}_{0}\right) \doteq... | Proof. Let \( \xi \left( \mathcal{C}\right) = \left\{ {{C}_{1},\ldots ,{C}_{r}}\right\} \) . Let \( \varepsilon > 0 \) and choose \( \delta \) to correspond to \( r \) and \( c \) in Lemma 4.15. It suffices to show that for each \( \sigma > 0 \) we can find a partition \( \left\{ {{D}_{1},\ldots ,{D}_{r}}\right\} \) wi... | Yes |
Corollary 4.16.1. If \( \left\{ {\mathcal{A}}_{n}\right\} \) is an increasing sequence of finite sub-algebras of \( \mathcal{B} \) and \( \mathcal{C} \) is a finite sub-algebra with \( \mathcal{C} \in \mathop{\bigvee }\limits_{n}{\mathcal{A}}_{n} \), then \( H\left( {\mathcal{C}/{\mathcal{A}}_{n}}\right) \rightarrow 0 ... | Proof. If \( {\mathcal{B}}_{0} = \mathop{\bigcup }\limits_{{j = 1}}^{x}{\mathcal{A}}_{j} \) then \( {\mathcal{B}}_{0} \) is an algebra and \( \mathcal{C} \triangleq \mathcal{A}\left( {\mathcal{A}}_{0}\right) \) by hypothesis. Let \( \varepsilon > 0 \) . By Theorem 4.16 there exists a finite sub-algebra \( {\mathcal{D}}... | Yes |
Theorem 4.17 (Kolmogorov-Sinai Theorem). Let \( T \) be an invertible measure-preserving transformation of the probability space \( \left( {{X}^{ * }\mathcal{B}, m}\right) \) and let \( \mathcal{A} \) be a finite sub-algebra of \( \mathcal{B} \) such that \( \mathop{\bigvee }\limits_{{n = - \infty }}^{\infty }{T}^{n}\m... | Proof. Let \( \mathcal{C} \subseteq \mathcal{B} \) be finite. We want to show that \( h\left( {T,\mathcal{C}}\right) \leq h\left( {T,\mathcal{A}}\right) \) . For \( n \geq 1 \)\n\n\[ h\left( {T,\mathcal{C}}\right) \leq h\left( {T,\mathop{\bigvee }\limits_{{i = - n}}^{n}{T}^{i}\mathcal{S}}\right) + H\left( {\mathcal{C}/... | Yes |
Theorem 4.18. If \( T \) is a measure-preserving transformation (but not necessarily invertible) of the probability space \( \left( {X,\mathcal{B}, m}\right) \) and if \( \mathcal{A} \) is a finite sub-algebra of \( \mathcal{B} \) with \( \mathop{\bigvee }\limits_{{i = 0}}^{\infty }{T}^{-i}\mathcal{A} \doteq \mathcal{B... | Proof. This is similar to the proof of the previous theorem; use \( \mathop{\bigvee }\limits_{{i = 0}}^{{n - 1}}{T}^{-i}\mathcal{A} \) in the place of \( \mathop{\bigvee }\limits_{{i = - n}}^{n}{T}^{i}\mathcal{A} \), and Theorem 4.12(vi). | No |
Corollary 4.18.1. If \( T \) is an invertible measure-preserving transformation of the probability space \( \left( {X,\mathcal{D}, m}\right) \) and \( \mathop{\bigvee }\limits_{{i = 0}}^{\alpha }{T}^{-i}\mathcal{A} \doteq \mathcal{B} \) for some finite subalgebra \( \mathcal{A} \) then \( h\left( T\right) = 0 \) . | Proof. By Theorem 4.18\n\n\[ h\left( T\right) = h\left( {T,\mathcal{A}}\right) \]\n\n\[ = \mathop{\lim }\limits_{{n \rightarrow \infty }}H\left( {.\& /\mathop{\bigvee }\limits_{{i = 1}}^{n}{T}^{-i}{sI}}\right) \text{ by Theorem 4.14. } \]\n\nBut \( \mathop{\bigvee }\limits_{{i = 1}}^{\infty }{T}^{-i}\mathcal{A} \doteq ... | Yes |
Theorem 4.19. (See Rohlin [3].) Suppose \( \left( {X,\mathcal{B}, m}\right) \) is a Lebesgue space (which we assume is not isomorphic to a finite set) and \( T \) is an invertible measure-preserving transformation of \( \left( {X,\mathcal{B}, m}\right) \). Then \( T \) has a generator \( \xi \) with \( H\left( \xi \rig... | Thus, if \( T \) is ergodic and \( h\left( T\right) < \infty \) then \( T \) has a generator \( \xi \) with \( H\left( \xi \right) < \infty \) . | No |
Theorem 4.20. If \( \\left( {X,\\mathcal{B}, m}\\right) \) is a Lebesgue space and \( T \) is an ergodic invertible measure-preserving transformation of \( \\left( {X,\\mathcal{B}, m}\\right) \) with \( h\\left( T\\right) < \\infty \) then \( T \) has a finite generator | \[ \n\\xi = \\left\\{ {{A}_{1},\\ldots ,{A}_{n}}\\right\\} \n\] \n\nIn fact \( \\xi \) may be taken so that \( {e}^{h\\left( \\Gamma \\right) } \\leq n \\leq {e}^{h\\left( T\\right) } + 1 \) . | No |
Theorem 4.21. Let \( \left( {X,\mathcal{A}, m}\right) \) be a probability space. If \( {\mathcal{B}}_{0} \) is a sub-algebra of \( \mathcal{B} \) with \( \mathcal{B}\left( {\mathcal{B}}_{0}\right) \doteq \mathcal{B} \) then for each measure-preserving transformation \( T : X \rightarrow X \) we have \[ h\left( T\right)... | Proof. Let \( \varepsilon > 0 \) . Let \( \mathcal{C} \subseteq \mathcal{B} \) be finite. By Theorem 4.16 there exists a finite \( {\mathcal{D}}_{t} \subseteq {\mathcal{B}}_{0} \) such that \[ H\left( {\mathcal{C}/{\mathcal{D}}_{\varepsilon }}\right) < \varepsilon \] Thus \[ h\left( {T,\mathcal{C}}\right) \leq h\left( ... | Yes |
Theorem 4.22. Let \( \left( {X,\mathcal{B}, m}\right) \) be a probability space and let \( {\left\{ {\mathcal{A}}_{n}\right\} }_{1}^{\infty } \) be finite sub-algebras of \( \mathcal{D} \) such that \( {\mathcal{A}}_{1} \subseteq {\mathcal{A}}_{2} \subseteq \cdots \) and \( \mathop{\bigvee }\limits_{{n = 1}}^{\infty }{... | Proof. We note that \( h\left( {T,{\mathcal{A}}_{n}}\right) \) is an increasing sequence by Theorem 4.12(iii). Also \( {\mathcal{B}}_{0} = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{\mathcal{A}}_{n} \) is an algebra and \( \mathcal{B}\left( {\mathcal{B}}_{0}\right) \doteq \mathcal{D} \) . By Theorem 4.21 \( h\left( T... | Yes |
Corollary 4.25.1. Any ergodic transformation with discrete spectrum has zero entropy. | This follows from Theorem 3.6. (Actually we have shown the result only when \( \left( {X,\mathcal{B}, m}\right) \) has a countable basis since the above calculation was for a metric group G.) | No |
Theorem 4.28 (Ornstein). Let \( {T}_{1},{T}_{2} \) be Bernoulli shifts whose state spaces are Lebesgue spaces. If \( h\left( {T}_{1}\right) = h\left( {T}_{2}\right) \) then \( {T}_{1} \) is conjugate to \( {T}_{2} \), and hence isomorphic by the assumption on the state spaces (a countable direct product of Lehesgue spa... | The proof of this deep theorem is presented in Ornstein [1], Shields [1], and Moser et al [1]. Certain special cases had been worked out earlier by Meshalkin [1] and by Blum and Hanson [1]. This result reduces the conjugacy problem for Bernoulli shifts to their state spaces, since the entropy depends only on the state ... | No |
Theorem 4.30. Every Bernoulli automorphism is a Kolmogorov automorphism. | Proof. Let the state space for \( T \) be \( \left( {Y,\mathcal{F},\mu }\right) \) . If \( F \in \mathcal{F} \), let \( \widetilde{F} = \) \( \left\{ {\left\{ {x}_{n}\right\} \in X : {x}_{0} \in F}\right\} \in \mathcal{B} \) . Let \( \mathcal{G} = \{ \widetilde{F} : F \in \mathcal{F}\} \), which is called the time- \( ... | Yes |
Corollary 4.31.1. Entropy is not a complete invariant for the class of Kolmogorov automorphisms. | Proof. Let \( T \) be the example of Ornstein. By Corollary 4.14.4 \( h\left( T\right) > 0 \) . Choose a Bernoulli automorphism \( S \) with \( h\left( S\right) = h\left( T\right) \) . \( S \) and \( T \) are not isomorphic. | Yes |
Corollary 4.33.1 A Kolmogorov automorphism is strong-mixing. | Proof. By Theorem 2.12. | No |
Theorem 4.37. Let \( T : X \rightarrow X \) be an invertible measure-preserving transformation of a Lebesgue space \( \left( {X,\mathcal{D}, m}\right) \) . Suppose \( \mathcal{F} \) is a sub- \( \sigma \) -algebra of \( \mathcal{B} \) with \( T\mathcal{F} = \mathcal{F} \) and such that \( T \) has completely positive e... | For a proof see Parry ([2], Chapter 6). | No |
Theorem 5.1. The following are equivalent for a homeomorphism \( T : X \rightarrow X \) of a compact metric space.\n\n(i) \( T \) is minimal.\n\n(ii) The only closed subsets \( E \) of \( X \) with \( {TE} = E \) are \( \varnothing \) and \( X \) .\n\n(iii) For every non-empty open subset \( U \) of \( X \) we have \( ... | (i) \( \Rightarrow \) (ii). Suppose \( T \) is minimal and let \( E \) be closed, \( E \neq \varnothing \) and \( {TE} = E \) . If \( \mathrm{x} \in E \) then \( {0}_{T}\left( x\right) \subset E \) so \( X = \overline{{0}_{T}\left( x\right) } \subset E \) . Hence \( X = E \) .\n\n(ii) \( \Rightarrow \) (iii) If \( U \)... | Yes |
Theorem 5.2. Any homeomorphism \( T : X \rightarrow X \) has a minimal set. | Proof. Let \( \mathcal{E} \) denote the collection of all closed non-empty \( T \) -invariant subsets of \( X \) . Clearly \( \mathcal{E} \neq \varnothing \) since \( X \) belongs to \( \mathcal{E} \) . The set \( \mathcal{E} \) is a partially ordered set under inclusion. Every linearly ordered subset of \( \mathcal{E}... | Yes |
Theorem 5.3. If \( T : X \rightarrow X \) is minimal homeomorphism and \( f \in C\left( X\right) \) then \( f \circ T = f \) implies \( f \) is a constant. | Proof. Since \( f \circ T = f \) we have \( f \circ {T}^{n} = f\forall n \in Z \), so if we pick some \( x \in X \) we know \( f \) is constant on the dense set \( {0}_{T}\left( x\right) \) . Since \( f \) is continuous it must be constant. | Yes |
Theorem 5.5. Let \( T : X \rightarrow X \) be a continuous transformation of a compact metric space and \( x \in X \). Then\n\n(i) \( \omega \left( x\right) \neq \varnothing \).\n\n(ii) \( \omega \left( x\right) \) is a closed subset of \( X \).\n\n(iii) \( {T\omega }\left( x\right) = \omega \left( x\right) \). | Proof. (i) is clear.\n\n(ii) Let \( {y}_{k} \in \omega \left( x\right) \) for \( k \geq 1 \) and \( {y}_{k} \rightarrow y \in X \). We want to show \( y \in \omega \left( x\right) \). For \( \operatorname{each}j \geq 1 \) choose \( {k}_{j} \) with \( d\left( {{y}_{{k}_{j}}, y}\right) < \frac{1}{2}{\varkappa }_{j}. \) N... | Yes |
Theorem 5.6. Let \( T : X \rightarrow X \) be continuous. Then\n\n(i) \( \Omega \left( T\right) \) is closed.\n\n(ii) \( \mathop{\bigcup }\limits_{{x \in X}}\omega \left( x\right) \subset \Omega \left( T\right) \) (in particular \( \Omega \left( T\right) \neq \varnothing \) ).\n\n(iii) All periodic points belong to \( ... | ## Proof\n\n(i) From the definition of \( \Omega \left( T\right) \) it is clear that \( X \smallsetminus \Omega \left( T\right) \) is open.\n\n(ii) Let \( x \in X \) and \( y \in \omega \left( x\right) \) . We want to show \( y \in \Omega \left( T\right) \) . Let \( V \) be a neighbourhood of \( y \) . We want to find ... | Yes |
Theorem 5.7. If \( T : X \rightarrow X \) is a continuous transformation of a compact metric space then \( \Omega \left( T\right) = \{ x \in X \mid \) for every neighbourhood \( U \) of \( x \) and every \( N \geq 1 \) there exists \( n \geq N \) with \( {T}^{-n}U \cap U \neq \varnothing \} \) . | Proof. Clearly the stated set is a subset of \( \Omega \left( T\right) \) . Suppose \( x \in \Omega \left( T\right) \) and \( U \) is a neighbourhood of \( x \) and \( N \geq 1 \) . If \( x \) is a periodic point then clearly \( {T}^{-n}U \cap \) \( U \neq \varnothing \) for some \( n \geq N \) . Suppose \( x \) is not... | Yes |
Theorem 5.8. The following are equivalent for a homeomorphism \( T : X \rightarrow X \) of a compact metric space.\n\n(i) \( T \) is topologically transitive.\n\n(ii) Whenever \( E \) is a closed subset of \( X \) and \( {TE} = E \) then either \( E = X \) or \( E \) is nowhere dense (or, equivalently, whenever \( U \)... | (i) \( \Rightarrow \) (ii). Suppose \( \overline{{0}_{T}\left( {x}_{0}\right) } = X \) and let \( E \neq \varnothing, E \) closed and \( {TE} = E \) . Suppose \( U \) is open and \( U \subset E, U \neq \varnothing \) . Then there exists \( p \) with \( {T}^{p}\left( {x}_{0}\right) \in U \subset E \) so that \( {0}_{T}\... | Yes |
Theorem 5.9. The following are equivalent for a continuous transformation \( T : X \rightarrow X \) with \( {TX} = X \) .\n\n(i) \( T \) is one-sided topologically transitive.\n\n(ii) Whenever \( E \) is a closed subset of \( X \) and \( E \subset {T}^{-1}E \) then either \( E = X \) or \( E \) is nowhere dense (equiva... | ## Proof\n\n(i) \( \Rightarrow \) (ii). Suppose \( \left\{ {{T}^{n}\left( {x}_{0}\right) \mid n \geq 0}\right\} \) is dense in \( X \), and suppose \( E \) is closed and \( {TE} \subset E \) . Suppose \( U \) is a non-empty open set with \( U \subset E \) . Then \( {T}^{p}\left( {x}_{0}\right) \in U \) for some \( p \g... | Yes |
Theorem 5.10. Let \( T : X \rightarrow X \) be a homeomorphism. Then \( T \) is one-sided topologically transitive iff \( T \) is topologically transitive and \( \Omega \left( T\right) = X \) . | Proof. Suppose \( \left\{ {{T}^{n}\left( {x}_{0}\right) \mid n \geq 0}\right\} \) is dense in \( X \) . Clearly \( T \) is topologically transitive. If \( \Omega \left( T\right) \neq X \) there is a non-empty open set \( U \) such that \( \left\{ {{T}^{n}U \mid n \in Z}\right\} \) are pairwise disjoint sets. For some \... | Yes |
Let \( A : {K}^{n} \rightarrow {K}^{n} \) be an ergodic automorphism of the n-torus \( {K}^{n} \). The periodic points of \( A \) are exactly those points \( \left( {{w}_{1},\ldots ,{w}_{n}}\right) \in {K}^{n} \) where each \( {w}_{i} \) is a root of unity. (In additive notation these are the points of \( {R}^{n}/{Z}^{... | Proof. Let \( A \) be an automorphism. Let \( w = \left( {{w}_{1},\ldots ,{w}_{n}}\right) \in {K}^{n} \) be so that each \( {w}_{i} \) is a root of unity. There is some \( k \geq 1 \) with \( {w}^{k} = e \), the identity element. For each fixed \( k \) the set \( {Y}_{k} = \left\{ {z \in {K}^{n} : {z}^{k} = e}\right\} ... | Yes |
Theorem 5.12. The two-sided and one-sided shifts have a dense set of periodic points. For the two-sided shift \( {\left\{ {x}_{n}\right\} }_{-\infty }^{\infty } \) is fixed by \( {T}^{p} \) iff \( {x}_{n} = {x}_{n + p}\forall n \in Z \) . For the one-sided shift \( {\left\{ {x}_{n}\right\} }_{0}^{\infty } \) is fixed b... | Proof. We shall consider only the two-sided case. If \( x = {\left\{ {x}_{n}\right\} }_{-\infty }^{\infty } \) and \( {T}^{p}x = x \) then \( {x}_{p + i} = {x}_{i} \) for each \( i \) . In other words, the points fixed by \( {T}^{p} \) have the form \( \left( {\ldots ,{x}_{p - 1}{x}_{0}{x}_{1},\ldots ,{x}_{p - 1}{x}_{0... | Yes |
Theorem 5.13. If \( X \) is a compact metrisable space, \( T : X \rightarrow X \) a topologically transitive homeomorphism, and if there exists a metric on \( X \) making \( T \) an isometry, then \( T \) is minimal. | Proof. Suppose \( d \) is such a metric, i.e., \( d\left( {{Tx},{Ty}}\right) = d\left( {x, y}\right) \) . Let \( \overline{{0}_{T}\left( {x}_{0}\right) } = X \) and consider \( x \in X \) . We want to show that \( \overline{{0}_{T}\left( x\right) } = X \) . Let \( y \in X \) and let \( \varepsilon > 0 \) . There exist ... | Yes |
Theorem 5.14. If \( T \) is a topologically transitive homeomorphism or a one-sided topologically transitive continuous transformation then \( T \) has no nonconstant invariant continuous function. | Proof. If \( f \circ T = f \) then \( f \circ {T}^{n} = f \) so \( f \) is constant on orbits of points. The result then follows. | Yes |
Theorem 5.15. Let \( T : X \rightarrow X \) be a homeomorphism of the compact metric space \( X \) and let \( m \) be a probability measure on the Borel subsets of \( X \) giving non-zero measure to every non-empty open set. If \( T \) is an ergodic measure-preserving transformation with respect to \( m \), then \( m\l... | Proof. Let \( {U}_{1},{U}_{2},\ldots \) be a countable base for the topology. Then\n\n\[ \left\{ {x : \overline{{0}_{T}\left( x\right) } = X}\right\} = \mathop{\bigcap }\limits_{{n = 1}}^{\infty }\mathop{\bigcup }\limits_{{k = - \infty }}^{\infty }{T}^{k}{U}_{n}. \]\n\nFor each \( n \) the open set \( \mathop{\bigcup }... | Yes |
Theorem 5.16. Let \( T : X \rightarrow X \) be continuous with \( {TX} = X \) and let \( m \) be a probability on the Borel subsets of \( X \) giving non-zero measure to every nonempty open set. If \( T \) is an ergodic measure-preserving transformation with respect to \( m \) then \( m\left( \left\{ {x \in X : {\left\... | Proof. If \( {\left\{ {U}_{n}\right\} }_{1}^{\infty } \) is a countable base for the topology then \( {\left\{ x \mid \left\{ {T}^{n}x\right\} \right\} }_{0}^{\infty } \) is \( {dense}\} = \mathop{\bigcap }\limits_{{n = 1}}^{\infty }\left( {\mathop{\bigcup }\limits_{{k = 0}}^{\infty }{T}^{-k}{U}_{n}.}\right. \) For eac... | Yes |
Theorem 5.17. Let \( T \) be a homeomorphism of a compact metric space \( X \) and suppose \( T \) is topologically transitive. Then\n\n(i) If \( f \circ T = {\lambda f} \) where \( 0 ≢ f \in C\left( X\right) \), then \( \left| \lambda \right| = 1 \) and \( \left| f\right| \) is constant.\n\n(ii) If \( f, g \) are both... | (i) Since \( \left| {f\left( {Tx}\right) }\right| = \left| \lambda \right| \left| {f\left( x\right) }\right| \) we have\n\n\[ \mathop{\sup }\limits_{{x \in X}}\left| {f\left( {Tx}\right) }\right| = \left| \lambda \right| \mathop{\sup }\limits_{{x \in X}}\left| {f\left( x\right) }\right| \]\n\nSince \( {TX} = X \) this ... | Yes |
Theorem 5.18 (Halmos and von Neumann). The following are equialent for a homeomorphism \( T \) of a compact metric space \( X \) .\n\n(i) \( T \) is topologically transitive and is an isometry for some n’tric on \( X \) .\n\n(ii) \( T \) is topologically conjugate to a minimal rotation on a comact abelian metric group.... | (i) \( \Rightarrow \) (ii). Let \( \rho \) be a metric on \( X \) for which \( T \) is an is@ery. Suppose \( {0}_{T}\left( {x}_{0}\right) = X \) . Define a multiplication \( * \) on \( {0}_{T}\left( {x}_{0}\right) \) by \( {T}^{n}{x}_{0} * {}^{m}{x}_{0} = {T}^{n + m}{x}_{0} \) . We have\n\n\[ \rho \left( {{T}^{n}{x}_{0... | No |
Theorem 5.19 (Topogical Discrete Spectrum Theorem). Two minimal homeomorphisms of upact metric spaces both having topological discrete spectrum are topologicly conjugate iff they have the same eigenvalues. | Proof. If \( T, S \) are to \( {p}_{D} \) gically conjugate they clearly have the same eigenvalues. We give the ou re of two proofs of the converse.\n\n(1) One proof is alo the lines of the proof of Theorem 3.4, but instead of using Theolem \( {2.10}_{c} \) use the Banach-Stone Theorem. This says that if \( X, Y \) are... | Yes |
Theorem 5.20. If \( T : X \rightarrow X \) is a homeomorphism of a compact metrisable space then \( T \) has a generator iff \( T \) has a weak generator. | Proof. A generator is clearly a weak generator. Now suppose \( \beta \) is a weak generator for \( T,\beta = \left\{ {{B}_{1},\ldots ,{B}_{s}}\right\} \), and let \( \delta \) be a Lebesgue number for \( \beta \) (see Theorem 0.20). Let \( \alpha \) be a finite open cover by sets \( {A}_{i} \) having \( \operatorname{d... | Yes |
Theorem 5.21. Let \( T : X \rightarrow X \) be a homeomorphism of a compact metric space \( \left( {X, d}\right) \) . Let \( \alpha \) be a generator for \( T \) . Then \( \forall \varepsilon > 0\exists N > 0 \) such that each set in \( \mathop{\bigvee }\limits_{{-N}}^{N}{T}^{-n}\alpha \) has diameter less than \( \var... | Proof. Suppose the first part of the theorem does not hold. Then \( \exists \varepsilon > 0 \) such that \( \forall j > 0\exists {x}_{j},{y}_{j}, d\left( {{x}_{j},{y}_{j}}\right) > \varepsilon \) and \( \exists {A}_{j, i} \in \alpha , - j \leq i \leq j \) with \( {x}_{j},{y}_{j} \in \) \( \mathop{\bigcap }\limits_{{i =... | Yes |
Let \( T \) be a homeomorphism of a compact metric space \( \left( {X, d}\right) \) . Then \( T \) is expansive iff \( T \) has a generator iff \( T \) has a weak generator. | Proof. By Theorem 5.20 it suffices to show \( T \) is expansive iff \( T \) has a generator.\n\nLet \( \delta \) be an expansive constant for \( T \) and \( \alpha \) a finite cover by open balls of radius \( \delta /2 \) . Suppose \( x, y \in \mathop{\bigcap }\limits_{{-\infty }}^{\infty }{T}^{-i}{\bar{A}}_{n} \) wher... | Yes |
(i) Expansiveness is independent of the metric as long as the metric gives the topology of \( X \) . (However the expansive constant does change.) | (i) This is because the concept of generator does not depend on the metric. | No |
We want to show \( \psi \) is injective and the inverse map can be extended to a continuous map \( \phi : \bar{\Lambda } \rightarrow K \) . | We do this by proving for each \( \varepsilon > 0 \) there is an integer \( N \) such that if \( x, y \in K \smallsetminus \left\{ {{a}^{n}, - {a}^{n} : n \in Z}\right\} \) and \( {\left( \psi \left( x\right) \right) }_{n} = {\left( \psi \left( y\right) \right) }_{n} \) for \( \left| n\right| \leq N \) then \( d\left( ... | Yes |
Theorem 5.23. Let \( T \) be an expansive homeomorphism of a compact metric space \( \left( {X, d}\right) \) and let \( \delta \) be an expansive constant. Let \( \gamma \) be a finite cover of \( X \) (not necessarily an open cover) by sets \( \left\{ {{C}_{1},\ldots ,{C}_{r}}\right\} \) with \( \operatorname{diam}\le... | Proof. Suppose the conclusion is false. There exists \( {\varepsilon }_{0} > 0 \) a subsequence \( \left\{ {n}_{i}\right\} \) of natural numbers, points \( {x}_{i},{y}_{i} \) with \( d\left( {{x}_{i},{y}_{i}}\right) \geq {\varepsilon }_{0} \) and \( {x}_{i},{y}_{i} \in \mathop{\bigcap }\limits_{{j = - {n}_{i}}}^{{n}_{i... | Yes |
Theorem 5.25. Let \( T \) be an expansive homeomorphism of a compact metric space \( \left( {X, d}\right) \) and let \( \delta \) be an expansive constant for \( T \) . If \( \xi = \left\{ {{A}_{1},\ldots ,{A}_{k}}\right\} \) is a partition of \( X \) into Borel sets with \( \operatorname{diam}\left( {A}_{j}\right) \le... | Proof. Consider any open ball \( B\left( {x;\varepsilon }\right) \) . By Theorem 5.23 for each \( n \geq 1 \) choose \( {N}_{n} \) such that \( \operatorname{diam}\left( {\mathop{\bigvee }\limits_{{i = - {N}_{n}}}^{{N}_{n}}{T}^{-i}\xi }\right) < 1/n \) . Let \( {E}_{n} \) denote the union of all the members of \( \math... | Yes |
Theorem 5.26. Let \( T : X \rightarrow X \) be an expansive homeomorphism of a compact metric space. For each integer \( p > 0 \) the homeomorphism \( {T}^{p} \) has only a finite number of fixed points. | Proof. Let \( \delta \) be an expansive constant for \( {T}^{p} \) . Suppose \( {T}^{p}\left( x\right) = x \) and \( {T}^{p}\left( y\right) = y \) . Then either \( x = y \) or \( d\left( {x, y}\right) > \delta \) . | Yes |
Theorem 5.27. There are no expansive homeomorphisms of the unit circle \( K \) . | Proof. Suppose \( T : K \rightarrow K \) is a homeomorphism. By replacing \( T \) by \( {T}^{2} \), if necessary, we can assume \( T \) preserves orientation.\n\nCase 1. Suppose \( T \) has a periodic point so that \( {T}^{p} \) has a fixed point for some \( p > 0 \) . We know \( T \) is expansive iff \( {T}^{p} \) is.... | Yes |
Corollary 6.1.1. For a Borel probability measure m on a metric space \( X \) we have that for \( B \in \mathcal{B}\left( X\right) \) | \[ m\left( B\right) = \mathop{\sup }\limits_{\substack{{C\text{ closed }} \\ {C \subseteq B} }}m\left( C\right) ,\;\text{ and }m\left( B\right) = \mathop{\inf }\limits_{\substack{{U\text{ open }} \\ {U \geq B} }}m\left( U\right) . \] | Yes |
Theorem 6.2. Let \( m,\mu \) be two Borel probability measures on the metric space \( X \) . If \( {\int }_{X}{fdm} = {\int }_{X}{fd\mu }\forall f \in C\left( X\right) \) then \( m = \mu \) . | Proof. By the above corollary it suffices to show that \( m\left( C\right) = \mu \left( C\right) \) for all closed sets \( C = X \) . Suppose \( C \) is closed and let \( \varepsilon > 0 \) . By the regularity of \( m \) there exists an open set \( U \) with \( C \subset U \) and \( m\left( {UC}\right) < i \) .\n\nDefi... | Yes |
Theorem 6.3 (Riesz Representation Theorem). Let \( X \) be a compact metric space and \( J : C\left( X\right) \rightarrow C \) a continuous linear map such that \( J \) is a positive operator (i.e., if \( f \geq 0 \) then \( J\left( f\right) \geq 0 \) ) and \( J\left( 1\right) = 1 \) . Then there exists \( \mu \in M\le... | For the proof see Parthasarathy [2] p. 145. | No |
Theorem 6.4. If \( X \) is a compact metrisable space then the space \( M\left( X\right) \) is metrisable in the weak* topology. If \( {\left\{ {f}_{n}\right\} }_{n = 1}^{\infty } \) is a dense subset of \( C\left( X\right) \) then\n\n\[ D\left( {m,\mu }\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\left| \i... | Proof. The function \( D : M\left( X\right) \times M\left( X\right) \rightarrow R \) is clearly a metric. Consider the metric space \( \left( {M\left( X\right), D}\right) \) . For each fixed \( i \) the map \( \mu \rightarrow \int {f}_{i}{d\mu } \) is clearly continuous on \( \left( {X, D}\right) \) because \( \left| {... | Yes |
Theorem 6.5. If \( X \) is a compact metrisable space then \( M\left( X\right) \) is compact in the weak*-topology. | Proof. We shall write \( \mu \left( f\right) \) instead of \( \int {fd\mu } \) . Let \( {\left\{ {\mu }_{n}\right\} }_{1}^{f} \) be a sequence in \( M\left( X\right) \) and we shall show it has a convergent subsequence.\n\nChoose \( {f}_{1},{f}_{2},\ldots \) dense in \( C\left( X\right) \) . Consider the sequence of co... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.