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arXiv · 0901.1247

A topological lens for a measure-preserving system

Eli Glasner·Mariusz Lemanczyk·Benjamin Weiss
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Abstract

We introduce a functor which associates to every measure preserving system (X,B,μ,T) a topological system (C_2(μ),\tilde{T}) defined on the space of 2-fold couplings of μ, called the topological lens of T. We show that often the topological lens "magnifies" the basic measure dynamical properties of T in terms of the corresponding topological properties of \tilde{T}. Some of our main results are as follows: (i) T is weakly mixing iff \tilde{T} is topologically transitive (iff it is topologically weakly mixing). (ii) T has zero entropy iff \tilde{T} has zero topological entropy, and T has positive entropy iff \tilde{T} has infinite topological entropy. (iii) For T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

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Eli Glasner, Mariusz Lemanczyk, Benjamin Weiss. 2009-01-09. A topological lens for a measure-preserving system. https://arxiv.org/abs/0901.1247

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