Entropy and mixing for amenable group actions

Abstract

For $\Gamma$ a countable amenable group consider those actions of $\Gamma$ as measure-preserving transformations of a standard probability space, written as $\{T_\gamma\}_{\gamma\in\Gamma}$ acting on $(X,\mathcal{F},\mu)$. We say $\{T_\gamma\}_{\gamma\in\Gamma}$ has completely positive entropy (or simply cpe for short) if for any finite and nontrivial partition $P$ of $X$ the entropy $h(T,P)$ is not zero. Our goal is to demonstrate what is well known for actions of $\mathbb{Z}$ and even $\mathbb{Z}^d$, that actions of completely positive entropy have very strong mixing properties. Let $S_i$ be a list of finite subsets of $\Gamma$. We say that the $S_i$ spread if any particular $\gamma \ne \mathrm{id}$ belongs to at most finitely many of the sets $S_iS_i^{-1}$.

THEOREM 0.1. For $\{T_\gamma\}_{\gamma\in\Gamma}$ an action of $\Gamma$ of completely positive entropy and $P$ any finite partition, for any sequence of finite sets $S_i\subseteq \Gamma$ which spread we have
\[
\frac{1}{\#S_i} h(\underset{\gamma\in S_i}{\vee} T_{\gamma^{-1}}(P)) \underset i\to h(P).
\]

The proof uses orbit equivalence theory in an essential way and represents the first significant application of these methods to classical entropy and mixing.

DOI
https://doi.org/10.2307/121130

Authors

Daniel J. Rudolph

Benjamin Weiss