Definition of theSubject
Measure‐preserving systems are a common modelof processes which evolve in time and for which the rulesgoverning the time evolution don't change. For example, inNewtonian mechanics the planets in a solar system undergomotion according to Newton's laws of motion: the planets movebut the underlying rule governing the planets' motion remainsconstant. The model adopted here is to consider thetime‐evolution as a transformation (either a mapin discrete time or a flow in continuous time) ona probability space or more generally a measurespace. This is the setting of the subject called ergodictheory. Applications of this point of view include the areas ofstatistical physics, classical mechanics, number theory,population dynamics, statistics, information theory andeconomics. The purpose of this chapter is to presenta flavor of the diverse range of examples ofmeasure‐preserving transformations which have playeda role in the development and application of ergodic theoryand...
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Abbreviations
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A transformation T of a measure space \( { (X,\mathcal{B}, \mu) } \) is measure‐preserving if \( { \mu (T^{-1} A)=\mu (A) } \) for all measurable \( { A\in \mathcal{B} } \).
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A measure‐preserving transformation \( { (X,\mathcal{B},\mu,T) } \) is ergodic if \( { T^{-1}(A)=A } \) (mod μ) implies \( { \mu(A)=0 } \) or \( { \mu(A^c)=0 } \) for each measurable set \( { A \in \mathcal{B} } \).
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A measure‐preserving transformation \( { (X,\mathcal{B},\mu,T) } \) of a probability space is weak‐mixing if \( \lim_{n\to \infty} \frac{1}{n} \sum_{i=0}^{n-1} |\mu(T^{-i} A\cap B)-\mu(A)\mu(B)| =0 \) for all measurable sets \( { A,B\in \mathcal{B} } \).
A measure‐preserving transformation \( { (X,\mathcal{B},\mu,T) } \) of a probability space is strong‐mixing if \( \lim_{n\to \infty} \mu (T^{-n} A\cap B) =\mu(A)\mu(B) \) for all measurable sets \( { A,B\in \mathcal{B} } \).
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A continuous transformation T of a compact metric space X is uniquely ergodic if there is only one T‑invariant Borel probability measure on X. A continous transformation of a topological space X is topologically mixing for any two open sets \( { U, V \subset X } \) there exists \( { N > 0 } \)such that \( { T^{-n}(U) \cap V \neq \emptyset }\), for each \( { n \geq N } \).
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Suppose \( { (X,\mathcal{B},\mu) } \) is a probability space. A finite partition \( { \mathcal{P} } \) of X is a finite collection of disjoint (mod μ, i. e., up to sets of measure 0) measurable sets \( { \{P_1,\dots, P_n\} } \) such that \( { X=\cup P_i } \) (mod μ). The entropy of \( { \mathcal{P} } \) with respect to μ is \( { H(\mathcal{P})=-\sum_i \mu(P_i)\ln \mu(P_i) } \) (other bases are sometimes used for the logarithm).
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The metric (or measure‐theoretic) entropy of T with respect to \( { \mathcal{P} } \) is \( h_{\mu} (T,\mathcal{P})=\lim_{n\to\infty} \frac{1}{n} H(\mathcal{P} \vee \dots \vee T^{-n+1}(\mathcal{P})) \), where \( { \mathcal{P} \vee \dots \vee T^{-n+1}(\mathcal{P}) } \) is the partition of X into sets of points with the same coding with respect to \( { \mathcal{P} } \) under T i, \( { i=0,\dots, n-1 } \). That is x, y are in the same set of the partition \( { \mathcal{P} \vee \dots \vee T^{-n+1}(\mathcal{P}) } \) if and only if \( { T^i (x) } \) and \( { T^i (y) } \) lie in the same set of the partition \( { \mathcal{P} } \) for \( { i=0,\dots,n-1 } \).
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The metric entropy \( { h_{\mu}(T) } \) of \( { (X,\mathcal{B},\mu,T) } \) is the supremum of \( { h_{\mu} (T,\mathcal{P}) } \) over all finite measurable partitions \( { \mathcal{P} } \).
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If T is a continuous transformation of a compact metric space X, then the topological entropy of T is the supremum of the metric entropies \( { h_{\mu} (T) } \), where the supremum is taken over all T‑invariant Borel probability measures.
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A system \( { (X, \mathcal{B},\mu,T) } \) is loosely Bernoulli if it is isomorphic to the first‐return system to a subset of positive measure of an irrational rotation or a (positive or infinite entropy) Bernoulli system.
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Two systems are spectrally isomorphic if the unitary operators that they induce on their L 2 spaces are unitarily equivalent.
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A smooth dynamical system consists of a differentiable manifold M and a differentiable map \( { f\colon M \to M } \). The degree of differentiability may be specified.
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Two submanifolds S 1, S 2 of a manifold M intersect transversely at \( { p\in M } \) if \( { T_p(S_1)+T_p (S_2)=T_p (M) } \).
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An (ϵ-) small C r perturbation of a C r map f of a manifold M is a map g such that \( { d_{C^r}(f,g)<\epsilon } \) i. e.the distance between f and g is less than ϵ in the C r topology.
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A map T of an interval \( { I=[a,b] } \) is piecewise smooth (C k for \( { k\ge 1 } \)) if there is a finite set of points \( a=x_1<x_2<\dots<x_n=b \) such that \( T |(x_i,x_{i+1}) \) is C k for each i. The degree of differentiability may be specified.
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A measure μ on a measure space \( { (X,\mathcal{B}) } \) is absolutely continuous with respect to a measure ν on \( { (X,\mathcal{B}) } \) if \( { \nu(A)=0 } \) implies \( { \mu(A)=0 } \) for all measurable \( { A\in \mathcal{B} } \).
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A Borel measure μ on a Riemannian manifold M is absolutely continuous if it is absolutely continuous with respect to the Riemannian volume on M.
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A measure μ on a measure space \( { (X,\mathcal{B}) } \) is equivalent to a measure ν on \( { (X,\mathcal{B}) } \) if μ is absolutely continuous with respect to ν and ν is absolutely continuous with respect to μ.
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Nicol, M., Petersen, K. (2009). Ergodic Theory: Basic Examples and Constructions. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_177
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