Generic diffeomorphisms on compact surfaces
We discuss the remaining obstacles to prove Smale's conjecture about the $C^1$-density of hyperbolicity among surface diffeomorphisms. Using a $C^1$-generic approach, we classify 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 exhibits homoclinic tangencies. In the course of our discussion, we obtain some related results about $C^1$-generic properties of surface diffeomorphisms involving homoclinic classes, chain-recurrence classes, and hyperbolicity. In particular, it is shown that on a connected surface the $C^1$-generic diffeomorphisms whose non-wandering sets have non-empty interior are the Anosov diffeomorphisms.