Entropy and mixing for amenable group actions

Daniel J. Rudolph, Benjamin Weiss

Published 2000-05-01Version 1

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,{\cal 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 \Bbb Z and even \Bbb 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 the S_i spread if any particular \gamma \neq id belongs to at most finitely many of the sets S_i S_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(\spans{S_i}{P}){\mathop{\to}_i} 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.