Weak equivalence of stationary actions and the entropy realization problem

TL;DR

This paper introduces a new concept of weak containment for stationary group actions, establishing a topology on their equivalence classes and showing that Furstenberg entropy is a continuous invariant within this framework.

Contribution

It defines weak containment for stationary actions, constructs a topology on their classes, and proves Furstenberg entropy's invariance and continuity under this topology.

Findings

Furstenberg entropy is invariant under weak equivalence.

A natural topology on the space of weak equivalence classes is established.

Furstenberg entropy varies continuously within this topology.

Abstract

We introduce the notion of weak containment for stationary actions of a countable group and define a natural topology on the space of weak equivalence classes. We prove that Furstenberg entropy is an invariant of weak equivalence, and moreover that it descends to a continuous function on the space of weak equivalence classes.