Generic diffeomorphisms on compact surfaces
Lorenzo J. Diaz | Abdenur, Flavio | Bonatti, Christian | Crovisier, Sylvain
chain recurrence class | Dominated splitting | filtration | homoclinic class | hyperbolicity | Surface Diffeomorphism
In this paper we shed some light on the remaining obstacles for proving Smale?s conjecture that hyperbolicity is $C^1$-dense among compact surface diffeomorphisms. Using a $C^1$-generic approach, we classify the the possible pathologies that may obstruct the $C^1$-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly presents homoclinic tangencies. In the course of our discussion, we obtain some related results on $C^1$-generic properties of surface diffeomorphisms involving homoclinic classes, chain-recurrence classes, and hyperbolicity. In particular, we show that on a connected compact surface any $C^1$-generic diffeomorphism whose non-wandering set has non-empty interior is an Anosov diffeomorphism.