Generic properties of homeomorphisms preserving a given dynamical simplex

Abstract

Given a dynamical simplex $K$ on a Cantor space $X$, we consider the set $G_K^{\ast }$ 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 $G_K^{\ast }$ the set of invariant measures of $g$ is equal to $K$. We also investigate when there exists a generic conjugacy class in $G_K^{\ast }$ 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 $G_K^{\ast }$.

Updated (and final) version

References in corpus (1)

• Dynamical simplices and Fraïssé theory