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Generic diffeomorphisms on compact surfaces. (English) Zbl 1089.37032
Summary: 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
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