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 non-trivial clopen invariant subset. Generalizing 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 $G_K^*$ .