Entropy and mixing for amenable group actions


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.


Daniel J. Rudolph

Benjamin Weiss