Geodesic
Generic diffeomorphisms on compact surfaces
Flavio Abdenur 1   ; Christian Bonatti 2   ; Sylvain Crovisier 3   ; Lorenzo J. Díaz 4
1 IMPA Estrada dona Castorina 110 CEP 222460-320 Rio de Janeiro, RJ, Brazil
2 CNRS - IMB, UMR 5584 BP 47 870 21078 Dijon Cedex, France
3 CNRS - LAGA, UMR 7539 Université Paris 13 Av. J.-B. Clément 93430 Villetaneuse, France
4 Dep. Matemática PUC-Rio Marquês de S. Vicente 225 CEP 22453-900 Rio de Janeiro, RJ, Brazil
Fundamenta Mathematicae, Tome 187 (2005) no. 2, pp. 127-159 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/fm187-2-3
Keywords: discuss remaining obstacles prove smales conjecture about density hyperbolicity among surface diffeomorphisms using generic approach classify possible pathologies may obstruct density hyperbolicity there essentially types obstruction persistence infinitely many hyperbolic homoclinic classes existence single homoclinic class which robustly exhibits homoclinic tangencies course discussion obtain related results about generic properties surface diffeomorphisms involving homoclinic classes chain recurrence classes hyperbolicity particular shown connected surface generic diffeomorphisms whose non wandering sets have non empty interior anosov diffeomorphisms

Flavio Abdenur  1   ; Christian Bonatti  2   ; Sylvain Crovisier  3   ; Lorenzo J. Díaz  4

1 IMPA Estrada dona Castorina 110 CEP 222460-320 Rio de Janeiro, RJ, Brazil
2 CNRS - IMB, UMR 5584 BP 47 870 21078 Dijon Cedex, France
3 CNRS - LAGA, UMR 7539 Université Paris 13 Av. J.-B. Clément 93430 Villetaneuse, France
4 Dep. Matemática PUC-Rio Marquês de S. Vicente 225 CEP 22453-900 Rio de Janeiro, RJ, Brazil 
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     title = {Generic diffeomorphisms on compact surfaces},
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Flavio Abdenur; Christian Bonatti; Sylvain Crovisier; Lorenzo J. Díaz. Generic diffeomorphisms on compact surfaces. Fundamenta Mathematicae, Tome 187 (2005) no. 2, pp. 127-159. doi: 10.4064/fm187-2-3

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