Some properties of surface diffeomorphisms

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We obtain some properties of C¹ generic surface diffeomorphisms as finiteness of non-trivial attractors, approximation by diffeomorphisms with only a finite number of hyperbolic homoclinic classes, equivalence between essential hyperbolicity and the hyperbolicity of all dissipative homoclinic classes (and the finiteness of spiral sinks). In particular, we obtain the equivalence between finiteness of sinks and finiteness of spiral sinks, abscence of domination in the set of accumulation points of the sinks, and the equivalence between Axiom A and the hyperbolicity of all homoclinic classes. These results improve A, a, m and settle a conjecture by Abdenur, Bonatti, Crovisier and Díaz abcd.