Zero entropy is generic

TL;DR

This paper extends the concept of generic zero entropy actions from amenable groups to nonamenable groups, revealing that every action can be derived from a zero entropy action, highlighting entropy's complex behavior in nonamenable settings.

Contribution

It generalizes the zero entropy genericity result to nonamenable groups and demonstrates that all actions are factors of zero entropy actions, using advanced theorems and recent developments.

Findings

Every action is a factor of a zero entropy action in nonamenable groups.

Entropy can increase under factor maps in nonamenable group actions.

The result relies on Seward's Sinai's Factor Theorem generalization and Bernoulli shifts properties.

Abstract

Dan Rudolph showed that for an amenable group $\Gamma$, the generic measure-preserving action of $\Gamma$ on a Lebesgue space has zero entropy. Here this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in the presence of nonamenability, entropy can increase under a factor map. The proof uses Seward's recent generalization of Sinai's Factor Theorem, the Gaboriau-Lyons result and my theorem that for every nonabelian free group, all Bernoulli shifts factor onto each other.