Preprint D77/2010
ERGODIC CLOSING AND NEW CRITERIA FOR HYPERBOLICITY BASED ON PERIODIC SETS

Augusto Armando Castro Jr.

**Keywords: **
Nonuniform Hyperbolicity | shadowing | 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.