Generic properties of homeomorphisms preserving a given dynamical simplex

2021/05/16 by Melleray, Julien
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.07456

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^*.

Related

• Dynamical simplices and minimal homeomorphisms
• Propriétés dynamiques génériques des homéomorphismes conservatifs
• Invariant measures for Cantor dynamical systems
• From invariant measures to orbit equivalence, via locally finite groups
• Generically there is but one self homeomorphism of the Cantor set