Entropy and Mixing for Amenable Group Actions

Abstract

For i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space, written asfT∞g∞2i acting on (X;F;„). We sayfT∞g∞2i has completely positive entropy (or simply cpe for short) if for any flnite 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 and even d , that actions of completely positive entropy have very strong mixing properties. Let Si be a list of flnite subsets of i. We say the Si spread if any particular ∞6 id belongs to at most flnitely many of the sets SiS i1 i . Theorem 0.1. For fT∞g∞2i an action of i of completely positive entropy and P any flnite partition, for any sequence of flnite sets Siµ i which spread we have