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.

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 | Leave a comment

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

Posted in ergodic theory, examples, Uncategorized | Leave a comment

Entropy bounds for equilibrium states

[Update 6/15/17: The original version of this post had a small error in it, which has been corrected in the present version; the definition of in the proof of the main theorem needed to be modified so that each is … Continue reading

Posted in Uncategorized | 2 Comments

Exponential decay of correlations

Let be a probability space and a measure-preserving transformation. Let be Banach spaces of measurable functions on , and the corresponding norms. Given and , the corresponding th correlation is We say that has exponential decay of correlations with respect … Continue reading

Posted in Uncategorized | Leave a comment

Lebesgue probability spaces, part II

This is a continuation of the previous post on the classification of complete probability spaces. Last time we set up the basic terminology and notation, and saw that for a complete probability space , the -algebra is countably generated mod … Continue reading

Posted in Uncategorized | 1 Comment

Lebesgue probability spaces, part I

In various areas of mathematics, classification theorems give a more or less complete understanding of what kinds of behaviour are possible. For example, in linear algebra we learn that up to isomorphism, is the only real vector space with dimension … Continue reading

Posted in Uncategorized | 4 Comments

Unique MMEs with specification – an alternate proof

The variational principle for topological entropy says that if is a compact metric space and is a continuous map, then , where is the topological entropy, and the supremum is taken over all -invariant Borel probability measures. A measure achieving … Continue reading

Posted in Uncategorized | Leave a comment