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Author Archives: Vaughn Climenhaga
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
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Cohomologous functions and the Livsic theorem
Given a dynamical system (typically is a compact metric space and is continuous), we commonly encounter realvalued functions in one of the following two roles. An observable function represents a measurement of the system, so that the sequence of functions … Continue reading
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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 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
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Exponential decay of correlations
Let be a probability space and a measurepreserving 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
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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
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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
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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
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De Bruijn graphs and entropy at finite scales
Let be a finite set, which we call an alphabet, and let be an infinite sequence of letters from . It is natural to ask how complex the sequence is: for example, if the alphabet is , then we expect … Continue reading
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