New Criteria of Generic Hyperbolicity based on Periodic Points

Abstract

We prove a criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild (non uniform) hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a $C^1$-open set $\SU$ then there exists an open and dense subset $\SA\subset \SU$ 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. As a simple corollary of our techniques, we have that an arbitrary ${\mathrm{C}}^1$-class local diffeomorphism $f$ of a closed manifold $M^n$ is uniformly expanding on the closure ${\mathrm{Cl}}_{M^n}(\mathrm{Per}(f))$ of its periodic point set $\mathrm{Per}(f)$, if it is nonuniformly expanding on $\mathrm{Per}(f)$.

22 pages. This new version, published in Bull. of Braz. Math. Society in 2011, just covers the preprints arXiv:0906.2240 and ArXiv 0906.2031 published by the author in ArXiv in June 12, 2009. These results were announced by the author for the first time in the 2008 Summer School of Dyn. Systems, at ICTP, Trieste, Italy

References in corpus (2)

• SRB measures for weakly expanding maps
• The Ergodic Closing Lemma for Nonsingular Endomorphisms

Cited by in corpus (1)

• The Ergodic Closing Lemma for Nonsingular Endomorphisms