Zero-dimensional extensions of amenable group actions
Volume 256 / 2021
Abstract
We prove that every dynamical system 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.