Polish Models and Sofic Entropy

TL;DR

This paper extends the computation of sofic entropy to actions on Polish spaces, utilizing topological methods, and explores implications for spectral properties and actions with positive or completely positive entropy.

Contribution

It introduces a new approach to compute sofic entropy for actions on Polish spaces, broadening the scope beyond compact spaces and linking entropy to topological features.

Findings

Entropy can be computed using topology for Polish space actions.

Applications to spectral analysis of actions with positive entropy.

Insights into actions with completely positive entropy.

Abstract

For actions of a sofic group on probability spaces, the entropy has been defined by Bowen, with an extension by Kerr-Li. In particular, when the action is by homeomorphisms of a compact space preserving a given measure, Kerr-Li show one can compute the measure-theoretic entropy in a manner which uses the topology of the space. We show how to compute the entropy of a an action of homeomorphisms of a Polish space preserving a given measure, also in a manner which uses the topology of the space. We give applications to spectral properties of actions with positive entropy, as well as for actions of completely positive entropy.