Category Archives: ergodic theory

Measure-preserving systems and their properties

Specification and the measure of maximal entropy

These are notes for a talk I am giving in Jon Chaika’s online working seminar in ergodic theory. The purpose of the talk is to outline Bowen’s proof of uniqueness of the measure of maximal entropy for shift spaces with … Continue reading

Gibbs measures have local product structure

Let be a compact smooth manifold and a transitive Anosov diffeomorphism. If is an -invariant Borel probability measure on that is absolutely continuous with respect to volume, then the Hopf argument can be used to show that is ergodic. In … Continue reading

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Alpha-beta shifts

Given , the interval map given by is naturally semi-conjugate to the -shift . The shift space admits a natural description in terms of the lexicographic order, and another in terms of a countable-state directed graph. These have been used … Continue reading

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Entropy of S-gap shifts

1. S-gap shifts S-gap shifts are a useful example for studying dynamics of shift spaces that are not subshifts of finite type but still exhibit some strong mixing properties. They are defined as follows: given , let be the set … Continue reading

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Slowly mixing sets

There are two equivalent definitions of mixing for a measure-preserving dynamical system . One is in terms of sets: for all measurable . The other is in terms of functions: for all . In both cases one may refer to … Continue reading

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Equidistribution for random rotations

Two very different types of dynamical behaviour are illustrated by a pair of very well-known examples on the circle: the doubling map and an irrational rotation. On the unit circle in , the doubling map is given by , while … Continue reading

Fubini foiled

An important issue in hyperbolic dynamics is that of absolute continuity. Suppose some neighbourhood of a smooth manifold is foliated by a collection of smooth submanifolds , where is some indexing set. (Here “smooth” may mean , or , or … Continue reading

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Central Limit Theorem for dynamical systems using martingales

This post is based on notes from Matt Nicol’s talk at the UH summer school in dynamical systems. The goal is to present the ideas behind a proof of the central limit theorem for dynamical systems using martingale approximations. 1. … Continue reading

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Laws of large numbers and Birkhoff’s ergodic theorem

In preparation for the next post on the central limit theorem, it’s worth recalling the fundamental results on convergence of the average of a sequence of random variables: the law of large numbers (both weak and strong), and its strengthening … Continue reading

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Spectral methods in dynamics (part 2)

This is a continuation of the last post, which were notes from the first in a series of talks at the Houston dynamics seminar on spectral methods for transfer operators as tools to establish statistical properties of dynamical systems. This … Continue reading

Posted in ergodic theory, examples, statistical laws | | 2 Comments