Generic diffeomorphisms on compact surfaces.

*(English)*Zbl 1089.37032Summary: 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 nonwandering sets have nonempty interior are the Anosov diffeomorphisms.

##### MSC:

37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |

37C20 | Generic properties, structural stability of dynamical systems |

37C29 | Homoclinic and heteroclinic orbits for dynamical systems |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

37B20 | Notions of recurrence and recurrent behavior in dynamical systems |