Entropy and Mixing for Amenable Group Actions

Abstract

For Γ a countable amenable group consider those actions of Γ as measurepreserving transformations of a standard probability space, written as {T γ } γ∈Γ acting on (X, F, µ).We say {T γ } γ∈Γ 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 Z and even Z d , that actions of completely positive entropy have very strong mixing properties.Let S i be a list of finite subsets of Γ.We say the S i spread if any particular γ = id belongs to at most finitely many of the sets S i S -1 i .Theorem 0.1.For {T γ } γ∈Γ an action of Γ of completely positive entropy and P any finite partition, for any sequence of finite sets S i ⊆ Γ which spread we haveThe proof uses orbit equivalence theory in an essential way and represents the first significant application of these methods to classical entropy and mixing.