Generic properties of homeomorphisms preserving a given dynamical simplex

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