Generic diffeomorphisms on compact surfaces

Tom 187 / 2005

Flavio Abdenur, Christian Bonatti, Sylvain Crovisier, Lorenzo J. Díaz Fundamenta Mathematicae 187 (2005), 127-159 MSC: 37C05, 37C20, 37C25, 37C29, 37C70. DOI: 10.4064/fm187-2-3


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.


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

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