Zero-dimensional extensions of amenable group actions
Volume 256 / 2021
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.
