A topological lens for a measure-preserving system

TL;DR

This paper introduces a topological lens for measure-preserving systems that translates measure-theoretic properties into topological dynamics, revealing deep connections between entropy, mixing, and chaos.

Contribution

It defines a functorial topological system from measure-preserving systems and establishes how key dynamical properties are preserved or reflected in this topological lens.

Findings

Weak mixing corresponds to topological transitivity.

Zero entropy corresponds to zero topological entropy.

Positive entropy corresponds to infinite topological entropy.

Abstract

We introduce a functor which associates to every measure preserving system (X,B,\mu,T) a topological system (C_2(\mu),\tilde{T}) defined on the space of 2-fold couplings of \mu, 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).