New criteria for hyperbolicity based on periodic sets

We prove some criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a C ¹-open set U then there exists an open and dense subset A ? U of Axiom A diffeomorphisms. Moreover, we also prove a noninvertible version of Ergodic Closing Lemma which we use to prove a counterpart of this result for local diffeomorphisms.