On the hyperbolicity of [formula omitted]-generic homoclinic classes

Abstract

The works of Liao, Mañé, Franks, Aoki, and Hayashi characterized a lack of hyperbolicity for diffeomorphisms by the existence of weak periodic orbits. In this note, we announce a result that can be seen as a local version of these works: for C1-generic diffeomorphisms, a homoclinic class either is hyperbolic or contains a sequence of periodic orbits that have a Lyapunov exponent arbitrarily close to 0.