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

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 = ℤ.