Category Archives: examples

Specific examples that illustrate various phenomena

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

Posted in ergodic theory, examples, smooth dynamics | Tagged | 8 Comments

Markov chains and mixing times (part 2 – coupling)

This week’s post continues last week’s discussion of Markov chains and mixing times, and introduces the idea of coupling as a method for estimating mixing times. We remark that some nice notes on the subject of coupling (and others) can … Continue reading

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Markov chains and mixing times

Our seminar series is taking a hiatus from spectral methods for a couple weeks — these will return eventually, but in the meantime we’ll spend some time with the idea of coupling as a method for deriving statistical properties. In … 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

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A useful example for the space of ergodic measures

Last time I discussed the following three properties that may or may not be satisfied for a map on a compact metric space : (C) The set of ergodic measures is path-connected. (D) is dense in the set of all … Continue reading