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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-2-3,
author = {Flavio Abdenur and Christian Bonatti and Sylvain Crovisier and Lorenzo J. D\'\i az},
title = {Generic diffeomorphisms on compact surfaces},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {127-159},
zbl = {1089.37032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-2-3}
}
Flavio Abdenur; Christian Bonatti; Sylvain Crovisier; Lorenzo J. Díaz. Generic diffeomorphisms on compact surfaces. Fundamenta Mathematicae, Tome 185 (2005) pp. 127-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-2-3/