Uniform mixing and completely positive sofic entropy

TL;DR

This paper introduces a new concept of uniform mixing for measure-preserving actions of sofic groups, showing it implies completely positive sofic entropy and constructing diverse non-Bernoulli examples.

Contribution

It defines uniform mixing for sofic group actions and demonstrates its implications for completely positive sofic entropy, including new non-Bernoulli examples.

Findings

Uniform mixing implies completely positive sofic entropy.

Existence of uncountably many nonisomorphic actions with this property.

Examples are not factors of Bernoulli shifts.

Abstract

Let $G$ be a countable discrete sofic group. We define a concept of uniform mixing for measure-preserving $G$-actions and show that it implies completely positive sofic entropy. When $G$ contains an element of infinite order, we use this to produce an uncountable family of pairwise nonisomorphic $G$-actions with completely positive sofic entropy. None of our examples is a factor of a Bernoulli shift.