Zero entropy actions of amenable groups are not dominant

TL;DR

This paper proves that for any discrete amenable group, actions with zero entropy are not dominant, extending previous results that linked positive entropy with dominance.

Contribution

It establishes that zero entropy actions of amenable groups are not dominant, completing the characterization of dominance in terms of entropy.

Findings

Zero entropy actions are not dominant for any amenable group.

Positive entropy actions are characterized as dominant.

The result generalizes previous findings from the case G=Z.

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.