Non-existence of a universal zero entropy system for non-periodic amenable group actions

TL;DR

This paper proves that for non-periodic amenable groups, there is no topological action whose ergodic measures encompass all zero-entropy ergodic systems, extending a previous result from the integers.

Contribution

It generalizes the non-existence of a universal zero-entropy system from the integers to all non-periodic amenable groups.

Findings

No universal zero-entropy system exists for non-periodic amenable groups.

Extends previous results from the integers to broader classes of groups.

Addresses a question posed by B. Weiss.

Abstract

Let $G$ be a non-periodic amenable group. We prove that there does not exist a topological action of $G$ for which the set of ergodic invariant measures coincides with the set of all ergodic measure-theoretic $G$-systems of entropy zero. Previously J. Serafin, answering a question by B. Weiss, proved the same for $G = \mathbb{Z}$.