Dynamical simplices and minimal homeomorphisms

TL;DR

This paper characterizes sets of probability measures on a Cantor space that can be realized as the invariant measures of some minimal homeomorphism, extending previous theorems with a new elementary approach.

Contribution

It provides a new elementary characterization of measure sets realizable by minimal homeomorphisms, generalizing prior results by Akin and Dahl.

Findings

Characterization of measure sets for minimal homeomorphisms

Extension of Akin's and Dahl's theorems

Elementary proof approach

Abstract

We give a characterization of sets K of probability measures on a Cantor space X with the property that there exists a minimal homeomorphism g of X such that the set of g-invariant probability measures on X coincides with K. This extends theorems of Akin (corresponding to the case when K is a singleton) and Dahl (when K is finite-dimensional). Our argument is elementary and different from both Akin's and Dahl's.