Weak equivalence of stationary actions and the entropy realization problem

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.

The proof of Theorem 4.6 in the previous version contained a gap, and we withdraw that claim. All the other results are still valid, including in particular the invariance of Furstenberg entropy under weak equivalence