# Generic diffeomorphisms on compact surfaces

Flavio Abdenur; Christian Bonatti; Sylvain Crovisier; Lorenzo J. Díaz

Fundamenta Mathematicae (2005)

- Volume: 187, Issue: 2, page 127-159
- ISSN: 0016-2736

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topFlavio Abdenur, et al. "Generic diffeomorphisms on compact surfaces." Fundamenta Mathematicae 187.2 (2005): 127-159. <http://eudml.org/doc/282632>.

@article{FlavioAbdenur2005,

abstract = {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.},

author = {Flavio Abdenur, Christian Bonatti, Sylvain Crovisier, Lorenzo J. Díaz},

journal = {Fundamenta Mathematicae},

keywords = {chain-recurrence class; dominated splitting; filtration; homoclinic class; hyperbolicity; surface diffeomorphism; Smale's conjecture; -generic approach; homoclinic tangencies; Anosov diffeomorphisms},

language = {eng},

number = {2},

pages = {127-159},

title = {Generic diffeomorphisms on compact surfaces},

url = {http://eudml.org/doc/282632},

volume = {187},

year = {2005},

}

TY - JOUR

AU - Flavio Abdenur

AU - Christian Bonatti

AU - Sylvain Crovisier

AU - Lorenzo J. Díaz

TI - Generic diffeomorphisms on compact surfaces

JO - Fundamenta Mathematicae

PY - 2005

VL - 187

IS - 2

SP - 127

EP - 159

AB - 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.

LA - eng

KW - chain-recurrence class; dominated splitting; filtration; homoclinic class; hyperbolicity; surface diffeomorphism; Smale's conjecture; -generic approach; homoclinic tangencies; Anosov diffeomorphisms

UR - http://eudml.org/doc/282632

ER -

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