Entropy, virtual Abelianness, and Shannon orbit equivalence

TL;DR

The paper establishes that Shannon orbit equivalence preserves entropy for free actions of initely generated virtually Abelian groups, leading to classification results for Bernoulli actions and odometers.

Contribution

It proves entropy invariance under Shannon orbit equivalence for actions of virtually Abelian groups and classifies Bernoulli actions up to Shannon orbit equivalence.

Findings

Shannon orbit equivalence preserves entropy for initely generated virtually Abelian group actions.

Bernoulli actions of non-locally-finite amenable groups are classified by measure conjugacy via Shannon orbit equivalence.

Every Z-odometer is Shannon orbit equivalent to the universal Z-odometer.

Abstract

We prove that if two free p.m.p. $\mathbb{Z}$-actions are Shannon orbit equivalent then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein-Weiss and the entropy invariance results of Austin and Kerr-Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also show, at the opposite end of the stochastic spectrum, that every $\mathbb{Z}$-odometer is Shannon orbit equivalent to the universal $\mathbb{Z}$-odometer.