Ergodicity and Mixing Properties

Abstract

In this article, we discuss ergodicity (a form of irreducibility) and the many kinds of mixing (independence of behaviour in the long term) for measure-preserving transformations. We discuss the partially understood phenomenon of higher-order mixing and indicate some of the contrast between the situation for single measure-preserving transformations and systems of multiple commuting measure-preserving transformations. We include a complete proof in the special case of a continuous map on a compact metric space of the ergodic decomposition by which a measure-preserving transformation can be split into ergodic parts. 2. Glossary 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 × Y given by (T × S)(x, y) = (T (x), S(y)). 3. Definition Many physical phenomena in equilibrium can be modeled as measure-preserving transformations. Ergodic theory is the abstract study of these transformations, dealing in particular with their long term average behaviour. One of the basic steps in analysing a measure-preserving transformation is to break it down into its simplest possible components. These simplest components are its ergodic components, and on each of these components, the system enjoys the ergodic property: the long-term time average of any measurement as the system evolves is equal to the average over the component. Ergodic decomposition gives a precise description of the manner in which a system can be split into ergodic components.