New Criteria of Generic Hyperbolicity based on Periodic Points

TL;DR

This paper establishes new criteria for uniform hyperbolicity based on periodic points, including a residual subset condition and a noninvertible version of the Ergodic Closing Lemma, with implications for local diffeomorphisms.

Contribution

It introduces a novel hyperbolicity criterion based on periodic points and extends the Ergodic Closing Lemma to noninvertible maps, advancing the understanding of hyperbolic dynamics.

Findings

Residual hyperbolicity condition implies Axiom A diffeomorphisms.

Noninvertible Ergodic Closing Lemma proved for local diffeomorphisms.

Local diffeomorphisms are uniformly expanding on the closure of their 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 $\mathrm{Per}(f)$.