Zero entropy actions of amenable groups are not dominant

Abstract

AbstractA 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, that is, that zero entropy implies non-dominance.