Dynamical systems
On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis
Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 85-88.

In this note we announce a result for vector fields on three-dimensional manifolds: those who are singular hyperbolic or exhibit a homoclinic tangency form a dense subset of the space of C1-vector fields. This answers a conjecture by Palis. The argument uses an extension for local fibred flows of Mañé and Pujals–Sambarino's theorems about the uniform contraction of one-dimensional dominated bundles.

Dans cette note, nous annonçons un résultat portant sur les champs de vecteurs des variétés de dimension 3 : ceux qui vérifient l'hyperbolicité singulière ou qui possèdent une tangence homocline forment un sous-ensemble dense de l'espace des champs de vecteurs C1. Ceci répond à une conjecture de Palis. La démonstration utilise une généralisation pour les flots fibrés locaux des théorèmes de Mañé et Pujals–Sambarino traitant de la contraction uniforme de fibrés unidimensionnels dominés.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.10.015
Crovisier, Sylvain 1; Yang, Dawei 2

1 CNRS – Laboratoire de mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay, France
2 School of Mathematical Sciences, Soochow University, Suzhou, 215006, PR China
@article{CRMATH_2015__353_1_85_0,
     author = {Crovisier, Sylvain and Yang, Dawei},
     title = {On the density of singular hyperbolic three-dimensional vector fields: a conjecture of {Palis}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {85--88},
     publisher = {Elsevier},
     volume = {353},
     number = {1},
     year = {2015},
     doi = {10.1016/j.crma.2014.10.015},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2014.10.015/}
}
TY  - JOUR
AU  - Crovisier, Sylvain
AU  - Yang, Dawei
TI  - On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 85
EP  - 88
VL  - 353
IS  - 1
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.crma.2014.10.015/
DO  - 10.1016/j.crma.2014.10.015
LA  - en
ID  - CRMATH_2015__353_1_85_0
ER  - 
%0 Journal Article
%A Crovisier, Sylvain
%A Yang, Dawei
%T On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis
%J Comptes Rendus. Mathématique
%D 2015
%P 85-88
%V 353
%N 1
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2014.10.015/
%R 10.1016/j.crma.2014.10.015
%G en
%F CRMATH_2015__353_1_85_0
Crovisier, Sylvain; Yang, Dawei. On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis. Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 85-88. doi : 10.1016/j.crma.2014.10.015. https://www.numdam.org/articles/10.1016/j.crma.2014.10.015/

[1] Afraĭmovič, V.; Bykov, V.; Silnikov, L. The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR, Volume 234 (1977), pp. 336-339

[2] Arroyo, A.; Rodriguez Hertz, F. Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 20 (2003), pp. 805-841

[3] Crovisier, S.; Pujals, E. Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, Invent. Math. (2014) (Preprint) | arXiv | DOI

[4] S. Crovisier, M. Sambarino, E. Pujals, Hyperbolicity of the extremal bundles, in preparation.

[5] Gan, S.; Yang, D. Morse–Smale systems and non-trivial horseshoes for three-dimensional singular flows (Preprint) | arXiv

[6] Guckenheimer, J. A strange, strange attractor, The Hopf Bifurcation Theorems and Its Applications, Applied Mathematical Series, vol. 19, Springer-Verlag, 1976, pp. 368-381

[7] Li, M.; Gan, S.; Wen, L. Robustly transitive singular sets via approach of extended linear Poincaré flow, Discrete Contin. Dyn. Syst., Volume 13 (2005), pp. 239-269

[8] Liao, S. On (η,d)-contractible orbits of vector fields, Syst. Sci. Math. Sci., Volume 2 (1989), pp. 193-227

[9] Mañé, R. Hyperbolicity, sinks and measure in one-dimensional dynamics, Commun. Math. Phys., Volume 100 (1985), pp. 495-524

[10] Metzger, R.; Morales, C. Sectional-hyperbolic systems, Ergod. Theory Dyn. Syst., Volume 28 (2008), pp. 1587-1597

[11] Morales, C.; Pacifico, M. A dichotomy for three-dimensional vector fields, Ergod. Theory Dyn. Syst., Volume 23 (2003), pp. 1575-1600

[12] Morales, C.; Pacifico, M.; Pujals, E. Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. Math., Volume 160 (2004), pp. 375-432

[13] Newhouse, S. The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. IHÉS, Volume 50 (1979), pp. 101-151

[14] Palis, J. Homoclinic bifurcations, sensitive-chaotic dynamics and strange attractors, Nagoya, 1990 (Adv. Ser. Dynam. Systems) (1991), pp. 466-472

[15] Palis, J. A global view of dynamics and a conjecture of the denseness of finitude of attractors, Astérisque, Volume 261 (2000), pp. 335-347

[16] Palis, J. A global perspective for non-conservative dynamics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 22 (2005), pp. 485-507

[17] Pujals, E.; Sambarino, M. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. Math. (2), Volume 151 (2000), pp. 961-1023

[18] Zhu, S.; Gan, S.; Wen, L. Indices of singularities of robustly transitive sets, Discrete Contin. Dyn. Syst., Volume 21 (2008), pp. 945-957

Cited by Sources: