Entropy and mixing for amenable group actions

Daniel J. Rudolph*, Benjamin Weiss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

60 Scopus citations

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 languageEnglish
Pages (from-to)1119-1150
Number of pages32
JournalAnnals of Mathematics
Volume151
Issue number3
DOIs
StatePublished - May 2000

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