Faces of simplices of invariant measures for actions of amenable groups
  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 $G$. 
Bartosz Frej, Dawid Huczek
math.DS