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Ergodicity and Mixing Properties

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Basics and Examples

Ergodicity

Ergodic Decomposition

Mixing

Hyperbolicity and Decay of Correlations

Future Directions

Bibliography

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Abbreviations

Bernoulli shift:

Mathematical abstraction of the scenario in statistics or probability in which one performs repeated independent identical experiments.

Markov chain:

A probability model describing a sequence of observations made at regularly spaced time intervals such that at each time, the probability distribution of the subsequent observation depends only on the current observation and not on prior observations.

Measure-preserving transformation:

A map from a measure space to itself such that for each measurable subset of the space, it has the same measure as its inverse image under the map.

Measure-theoretic entropy:

A non-negative (possibly infinite) real number describing the complexity of a measure‐preserving transformation.

Product transformation:

Given a pair of measure‐preserving transformations: T of X and S of Y, the product transformation is the map of \({X\times Y}\) given by \({(T\times S)(x,y)=(T(x),S(y))}\).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

    Anthony Quas

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  1. Anthony Quas
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Quas, A. (2012). Ergodicity and Mixing Properties. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_15

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