Zero entropy actions of amenable groups are not dominant

Abstract

A probability measure preserving action of a discrete amenable group $G$ is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for $G = \mathbb{Z}$, an action is dominant if and only if it has positive entropy and that for any $G$, positive entropy implies dominance. In this paper, we show that the converse also holds for any $G$, i.e. that zero entropy implies non-dominance.