Generic diffeomorphisms on compact surfaces

Flavio 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 -