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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
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
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
[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
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
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
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