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Category Archives: ergodic theory
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
Posted in ergodic theory, theorems
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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
Posted in ergodic theory, smooth dynamics
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Alphabeta shifts
Given , the interval map given by is naturally semiconjugate to the shift . The shift space admits a natural description in terms of the lexicographic order, and another in terms of a countablestate directed graph. These have been used … Continue reading
Posted in ergodic theory, examples, Uncategorized
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Entropy of Sgap shifts
1. Sgap shifts Sgap 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
Slowly mixing sets
There are two equivalent definitions of mixing for a measurepreserving 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
Equidistribution for random rotations
Two very different types of dynamical behaviour are illustrated by a pair of very wellknown 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
Posted in ergodic theory, examples, random dynamics, statistical laws
Tagged space of invariant measures
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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
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
Posted in ergodic theory, statistical laws, theorems
Tagged central limit theorem, martingales
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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
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
Tagged interval maps, physical measures, spectral methods, transfer operator
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