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Zero-dimensional extensions of amenable group actions

Volume 256 / 2021

Dawid Huczek Studia Mathematica 256 (2021), 121-145 MSC: Primary 37A35; Secondary 37B40. DOI: 10.4064/sm190203-5-11 Published online: 9 July 2020

Abstract

We prove that every dynamical system $X$ with a free action of a countable amenable group $G$ by homeomorphisms has a zero-dimensional extension $Y$ which is faithful and principal, i.e. every $G$-invariant measure $\mu $ on $X$ has exactly one preimage $\nu $ on $Y$ and the conditional entropy of $\nu $ with respect to $X$ is zero. This is a version of the result of Downarowicz and Huczek (2012) which establishes the existence of zero-dimensional principal and faithful extensions for general actions of the group of integers.

Authors

  • Dawid HuczekFaculty of Pure and Applied Mathematics
    Wrocław University of Science and Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail

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