Zero-dimensional extensions of amenable group actions

TL;DR

This paper proves that for any free action of a countable amenable group on a space, there exists a zero-dimensional extension that preserves measures uniquely and has zero conditional entropy, generalizing previous integer group results.

Contribution

It extends the existence of zero-dimensional faithful and principal extensions from integer actions to all countable amenable group actions.

Findings

Existence of zero-dimensional faithful extensions for amenable group actions

Extension preserves measure-theoretic properties uniquely

Generalizes earlier results from integer groups to amenable groups

Abstract

We prove that every dynamical system $X$ with free action of a countable amenable group $G$ by homeomorphisms has a zero-dimensional extension $Y$ which is faithful and principal, i.e. every $G$-invariant measure $\mu$ on $X$ has exactly one preimage $\nu$ on $Y$ and the conditional entropy of $\nu$ with respect to $X$ is zero. This is a version of an earlier result by T. Downarowicz and D. Huczek, which establishes the existence of zero-dimensional principal and faithful extensions for general actions of the group of integers.