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	   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  			 		 $G_K^*$ 		.

Generic properties of homeomorphisms preserving a given dynamical simplex