On a theorem Dan Rudolph: Part II: Amenable groups

TL;DR

This paper extends Rudolph's theorem to actions of countable amenable groups, showing that typical invariant measures with high entropy are Bernoulli, and discusses a related relative version.

Contribution

It generalizes Rudolph's theorem from $bZ$-actions to countable amenable groups, establishing typical measure properties in this broader context.

Findings

Typical invariant measures with entropy ≥ c are Bernoulli.

The theorem applies to the $G$-shift for countable amenable groups.

A relative version of the theorem is also proved.

Abstract

We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the $G$-shift $(\Lambda^G,\sigma)$, a typical measure has entropy $c$ and is Bernoulli. We also address a relative version of this theorem.