On the hyperbolicity of $C^1$-generic homoclinic classes

TL;DR

This paper proves that for C^1-generic diffeomorphisms, homoclinic classes are either hyperbolic or contain periodic orbits with Lyapunov exponents arbitrarily close to zero, extending classical characterizations of non-hyperbolicity.

Contribution

It establishes a local version of classical results, showing a dichotomy for C^1-generic homoclinic classes regarding hyperbolicity and the presence of near-zero Lyapunov exponents.

Findings

Homoclinic classes are either hyperbolic or contain periodic orbits with Lyapunov exponents close to zero.

The result provides a local characterization of non-hyperbolicity for C^1-generic diffeomorphisms.

Extends classical global characterizations to a local setting for homoclinic classes.

Abstract

Works of Liao, Ma\~n\'e, Franks, Aoki and Hayashi characterized lack of hyperbolicity for diffeomorphisms by the existence of weak periodic orbits. In this note we announce a result which can be seen as a local version of these works: for C$^1$-generic diffeomorphism, a homoclinic class either is hyperbolic or contains a sequence of periodic orbits that have a Lyapunov exponent arbitrarily close to 0. Des travaux de Liao, Ma\~n\'e, Franks, Aoki et Hayashi ont caract\'eris\'e le manque d'hyperbolicit\'e des diff\'eomorphismes par l'existence d'orbites p\'eriodiques faibles. Dans cette note, nous annon\c{c}ons un r\'esultat qui peut \^{e}tre consid\'{e}r\'{e} comme une version locale de ces travaux: pour les diff\'{e}omorphismes C$^1$-g\'{e}n\'{e}riques, une classe homocline ou bien est hyperbolique, ou bien contient une suite d'orbites p\'{e}riodiques qui ont un exposant de Lyapunov arbitrairement proche de 0.