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Ergodic Theory: Basic Examples and Constructions

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Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

Examples

Constructions

Future Directions

Bibliography

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Abbreviations

•:

A transformation T of a measure space \({(X,\mathcal{B}, \mu)}\) is measure‐preserving if \({\mu (T^{-1} A)=\mu (A)}\) for all measurable \({A\in \mathcal{B}}\).

•:

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 \({A \in \mathcal{B}}\).

•:

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 \({A,B\in \mathcal{B}}\).

•:

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 \({n \geq N}\).

•:

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).

•:

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 \({i=0,\dots,n-1}\).

•:

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 \({\mathcal{P}}\).

•:

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.

•:

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.

•:

Two systems are spectrally isomorphic if the unitary operators that they induce on their L 2 spaces are unitarily equivalent.

•:

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.

•:

Two submanifolds S 1, S 2 of a manifold M intersect transversely at \({p\in M}\) if \({T_p(S_1)+T_p (S_2)=T_p (M)}\).

•:

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.

•:

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.

•:

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 \({A\in \mathcal{B}}\).

•:

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.

•:

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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  1. Department of Mathematics, University of Houston, Houston, USA

    Matthew Nicol

  2. Department of Mathematics, University of North Carolina, Chapel Hill, USA

    Karl Petersen

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Nicol, M., Petersen, K. (2012). Ergodic Theory: Basic Examples and Constructions. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_17

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