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Ergodic Theorems

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Book cover Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Ergodic Theorems for Measure‐Preserving Maps

Generalizations to Continuous Time and Higher–Dimensional Time

Pointwise Ergodic Theorems for Operators

Subadditive and Multiplicative Ergodic Theorems

Entropy and the Shannon–McMillan–Breiman Theorem

Amenable Groups

Subsequence and Weighted Theorems

Ergodic Theorems and Multiple Recurrence

Rates of Convergence

Ergodic Theorems for Non‐amenable Groups

Future Directions

Bibliography

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Abbreviations

Dynamical system:

in its broadest sense, any set X, with a map \({T\colon X \to X}\). The classical example is: X is a set whose points are the states of some physical system and the state x is succeeded by the state Tx after one unit of time.

Iteration:

repeated applications of the map T above to arrive at the state of the system after n units of time.

Orbit of x :

the forward images \({x, Tx, T^{2}X \ldots}\) of \({x \in X}\) under iteration of T. When T is invertible one may consider the forward, backward or two-sided orbit of x.

Automorphism:

a dynamical system \({T\colon X \to X}\), where X is a measure space and T is an invertible map preserving measure.

Ergodic average:

if f is a function on X let \({A_{n}f(x) = n^{-1}\sum_{i=0}^{n-1} f(T^{i}x)}\); the average of the values of f over the first n points in the orbit of x.

Ergodic theorem:

an assertion that ergodic averages converge in some sense.

Mean ergodic theorem:

an assertion that ergodic averages converge with respect to some norm on a space of functions.

Pointwise ergodic theorem:

an assertion that ergodic averages \({A_{n}f(x)}\) converge for some or all \({x \in X}\), usually for a.e. x.

Stationary process:

a sequence \({(X_{1}, X_{2}, \ldots)}\) of random variables (real or complex‐valued measurable functions) on a probability space whose joint distributions are invariant under shifting \({(X_{1}, X_{2}, \ldots)}\) to \({(X_{2}, X_{3}, \ldots)}\).

Uniform distribution:

a sequence \({\{x_{n}\}}\) in \({[0,1]}\) is uniformly distributed if for each interval \({I \subset[0,1]}\), the time it spends in I is asymptotically proportional to the length of I.

Maximal inequality:

an inequality which allows one to bound the pointwise oscillation of a sequence of functions. An essential tool for proving pointwise ergodic theorems.

Operator:

any linear operator U on a vector space of functions on X, for example one arising from a dynamical system T by setting \({Uf(x) = f(Tx)}\). More generally any linear transformation on a real or complex vector space.

Positive contraction:

an operator T on a space of functions endowed with a norm \({\|\cdot\|}\) such that T maps positive functions to positive functions and \({\|Tf\| \leq |f|}\).

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  1. Department of Mathematics, University of Toronto, Toronto, Canada

    Andrés del Junco

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Junco, A. (2012). Ergodic Theorems. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_16

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