Hyperbolicity versus weak periodic orbits inside homoclinic classes

TL;DR

This paper proves that for generic diffeomorphisms, homoclinic classes with all periodic orbits having Lyapunov exponents away from zero are necessarily hyperbolic, using novel perturbation techniques that do not rely on classical connecting lemmas.

Contribution

It establishes a new intrinsic criterion linking Lyapunov exponents and hyperbolicity within homoclinic classes, extending stability conjecture insights.

Findings

Homoclinic classes with bounded away from zero Lyapunov exponents are hyperbolic.

New perturbation methods are developed that do not depend on classical connecting lemmas.

The result applies to generic $C^1$-diffeomorphisms, broadening the scope of stability theory.

Abstract

We prove that, for $C^1$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class $H(p)$ have all their Lyapunov exponents bounded away from 0, then $H(p)$ must be (uniformly) hyperbolic. This is in sprit of the works of the stability conjecture, but with a significant difference that the homoclinic class $H(p)$ is not known isolated in advance, hence the "weak" periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an "intrinsic" nature, and the classical proof of the stability conjecture does not pass through. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas.