# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Generic diffeomorphisms on compact surfaces

### Tom 187 / 2005

Fundamenta Mathematicae 187 (2005), 127-159 MSC: 37C05, 37C20, 37C25, 37C29, 37C70. DOI: 10.4064/fm187-2-3

#### Streszczenie

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.

#### Autorzy

• Flavio AbdenurIMPA
Estrada dona Castorina 110
CEP 222460-320
Rio de Janeiro, RJ, Brazil
e-mail
• Christian BonattiCNRS - IMB, UMR 5584
BP 47 870
21078 Dijon Cedex, France
e-mail
• Sylvain CrovisierCNRS - LAGA, UMR 7539
Université Paris 13
Av. J.-B. Clément
93430 Villetaneuse, France
e-mail
• Lorenzo J. DíazDep. Matemática PUC-Rio
Marquês de S. Vicente 225
CEP 22453-900
Rio de Janeiro, RJ, Brazil
e-mail

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