Author Archives: Vaughn Climenhaga

About Vaughn Climenhaga

I'm an assistant professor of mathematics at the University of Houston. I'm interested in dynamical systems, ergodic theory, thermodynamic formalism, dimension theory, multifractal analysis, non-uniform hyperbolicity, and things along those lines.

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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Law of large numbers for dependent but uncorrelated random variables

One of the fundamental results in probability theory is the strong law of large numbers, which was discussed in an earlier post under the guise of the Birkhoff ergodic theorem. Suppose we have a sequence of random variables which take … 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

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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

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Sharkovsky’s theorem

This post is based on a talk given by Keith Burns in the UH dynamical systems seminar yesterday, in which he presented a streamlined proof of Sharkovsky’s theorem due to him and Boris Hasselblatt. 1. Background and statement of the … 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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The Perron-Frobenius theorem and the Hilbert metric

In the last post, we introduced basic properties of convex cones and the Hilbert metric. In this post, we look at how these tools can be used to obtain an explicit estimate on the rate of convergence in the Perron–Frobenius … Continue reading

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Convex cones and the Hilbert metric

Having spent some time discussing spectral methods and coupling techniques as tools for studying the statistical properties of dynamical systems, we turn now to a third approach, based on convex cones and the Hilbert metric. This post is based on … Continue reading

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Spectral methods 3 – central limit theorem

With the previous post on convergence of random variables, the law of large numbers, and Birkhoff’s ergodic theorem as background, we return to the spectral methods discussed in the first two posts in this series. This post is based on … Continue reading

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