Abstract
For Γ a countable amenable group consider those actions of Γ as measure-preserving transformations of a standard probability space, written as {Tγ}γ∈Γ acting on (X, ℱ, μ). We say {Tγ}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 ℤd, that actions of completely positive entropy have very strong mixing properties. Let Si be a list of finite subsets of Γ. We say the Si spread if any particular γ ≠ id belongs to at most finitely many of the sets SiS-1i. THEOREM 0.1. For {Tγ}γ∈Γ an action of Γ of completely positive entropy and P any finite partition, for any sequence of finite sets Si ⊆ Γ which spread we have 1/#Si h( Vγ∈Si Tγ-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.
Original language | English |
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Pages (from-to) | 1119-1150 |
Number of pages | 32 |
Journal | Annals of Mathematics |
Volume | 151 |
Issue number | 3 |
DOIs | |
State | Published - May 2000 |