Generic diffeomorphisms on compact surfaces

• Autorzy: Flavio Abdenur , Christian Bonatti , Sylvain Crovisier , Lorenzo J. Díaz
• Języki publikacji: EN

Abstrakty

EN

We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-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 C¹-generic properties of surface diffeomorphisms involving homoclinic classes, chain-recurrence classes, and hyperbolicity. In particular, it is shown that on a connected surface the C¹-generic diffeomorphisms whose non-wandering sets have non-empty interior are the Anosov diffeomorphisms.

Kategorie tematyczne

• 37C05: Smooth mappings and diffeomorphisms
• 37C29: Homoclinic and heteroclinic orbits
• 37C70: Attractors and repellers, topological structure
• 37C25: Fixed points, periodic points, fixed-point index theory
• 37C20: Generic properties, structural stability

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2005
• Tom: 187
• Numer: 2
• Strony: 127-159

Daty

wydano

2005

Twórcy

autor

Flavio Abdenur

• IMPA, Estrada dona Castorina 110, CEP 222460-320, Rio de Janeiro, RJ, Brazil

autor

Christian Bonatti

• CNRS - IMB, UMR 5584, BP 47 870, 21078 Dijon Cedex, France

autor

Sylvain Crovisier

• CNRS - LAGA, UMR 7539, Université Paris 13, Av. J.-B. Clément, 93430 Villetaneuse, France

autor

Lorenzo J. Díaz

• Dep. Matemática PUC-Rio, Marquês de S. Vicente 225, CEP 22453-900, Rio de Janeiro, RJ, Brazil

Identyfikatory

DOI

10.4064/fm187-2-3