Robustly expansive homoclinic classes

Abstract

Let $f: M \to M$ be a diffeomorphism defined in a three-dimensional compact boundary-less manifold M. We prove that for an open dense set, C¹-robustly expansive homoclinic classes H(p) for f are hyperbolic. A diffeomorphism f is $\alpha$-expansive on a compact invariant set K if there is $\alpha>0$ such that for all $x,y\in K$, if ${\rm dist}(f^n(x),f^n(y))\leq \alpha$ for all $n\in \mathbb Z$ then x = y. By ‘robustly’ we mean that there is $\alpha>0$ such that for all nearby diffeomorphisms g, the homoclinic class H(p_g) of the continuation of p is $\alpha$-expansive.