Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / All issues

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.