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arXiv · 2105.07456

Generic properties of homeomorphisms preserving a given dynamical simplex

Julien Melleray
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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 $G_K^*$.

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Julien Melleray. 2021-05-16. Generic properties of homeomorphisms preserving a given dynamical simplex. https://arxiv.org/abs/2105.07456

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