Generic properties of homeomorphisms preserving a given dynamical simplex

TL;DR

This paper studies homeomorphisms on Cantor spaces that preserve a specified set of invariant measures, showing that generically such homeomorphisms have the prescribed measures and characterizing when a generic conjugacy class exists.

Contribution

It generalizes Yingst's theorem to characterize generic invariant measures and identifies conditions for the existence of a generic conjugacy class in the homeomorphism group.

Findings

For a generic homeomorphism in G_K^*, the invariant measures match the given simplex K.

A generic conjugacy class exists only when K has a single element, corresponding to an odometer.

The conjugacy class of this odometer is generic in G_K^*.

Abstract

Given a dynamical simplex $K$ on a Cantor space $X$, we consider the set $G_K^*$ of all homeomorphisms of $X$ which preserve all elements of $K$ and have no nontrivial clopen invariant subset. Generalising a theorem of Yingst, we prove that for a generic element $g$ of $G_K^*$ the set of invariant measures of $g$ is equal to $K$. We also investigate when there exists a generic conjugacy class in $G_K^*$ and prove that this happens exactly when $K$ has only one element, which is the unique invariant measure associated to some odometer; and that in that case the conjugacy class of this odometer is generic in $G_K^*$.