Entropy and mixing for amenable group actions

For Γ a countable amenable group consider those actions of Γ as measure-preserving 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_iS^-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 have 1/#S_i h( V_γ∈Si T_γ-1 (P))→_ih(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.