Faces of simplices of invariant measures for actions of amenable groups

TL;DR

This paper demonstrates that for zero-dimensional dynamical systems with free actions of amenable groups, every face of the invariant measure simplex can be represented as the entire invariant measure simplex of another such system.

Contribution

It establishes a modeling result linking faces of invariant measure simplices to entire simplices in related systems for amenable group actions.

Findings

Every face of the invariant measure simplex can be modeled as an entire simplex in another system.

The result applies to zero-dimensional systems with free actions of amenable groups.

Provides a structural understanding of invariant measures in these dynamical systems.

Abstract

We prove that every face in the simplex of invariant measures on a zero-dimensional dynamical system with free action of an amenable group $G$ can be modeled as the entire simplex of invariant measures on some other zero-dimensional dynamical system with free action of $G$.