New Criteria of Generic Hyperbolicity Based on Periodic Points

We prove that, if a mild condition on the hyperbolicity of the periodic points holds for any diffeomorphism in a residual subset of a C-open set U , then such set U exhibits a residual subset A of Axiom A diffeomorphisms. We also prove an analogous result for nonsingular endomorphisms: if a mild expanding condition holds for the periodic set of local diffeomorphisms belonging in a residual subset of a C open set U , then U exhibits an open and dense subset of expanding maps. For this last result we use noninvertible versions of Ergodic Closing Lemma that we have proved in [5].