On-line catalogue and orders (secure paiement, VISA or MASTERCARD only)

Journals available by subscription

Annales scientifiques de l'ENS

Astérisque

Bulletin de la SMF

Mémoires de la SMF

Revue d'Histoire des Mathématiques

Gazette des Mathématiciens

Books

Astérisque

Cours Spécialisés

Documents Mathématiques

Mémoires de la SMF

Panoramas et Synthèses

Séminaires et Congrès

Jean Morlet Chair Series

SMF/AMS Texts and Monographs

La Série T

Volumes "Journée Annuelle"

Other Books

Donald E. Knuth - French translations

Nicolas Bourbaki's seminar new edition

Jean Leray's scientific works new edition

Revue de l'Institut Elie Cartan

Electronic Editions

Annales scientifiques de l'ENS

Bulletin de la SMF

Revue d'Histoire des Mathématiques

Séminaires et Congrès

More information / Subscription

Publications for a general public

L'explosion des mathématiques (smf.emath.fr)

Mathématiques L'explosion continue (smf.emath.fr)

Zoom sur les métiers des maths (smf.emath.fr)

Zoom sur les métiers des mathématiques et de l'informatique (smf.emath.fr)

Où en sont les mathématiques ?

La Série T

For the authors

Submission of manuscripts

Formats and documentation

More info

Electronic distribution list (smf.emath.fr)

Information for bookselers and subscription agencies (smf.emath.fr)

Publications de la SMF
fr en
Your IP number: 54.80.87.62
Access to elec. publ.: SémCong

Annales scientifiques de l'ENS

Presentation of the publication

Titles

Last Titles

Editorial staff committee / Secretary

Serie 4:
Serie 3:
Serie 2:
Serie 1:

Search


Catalogue & orders

Annales scientifiques de l'ENS - Titles - série 4, 51 (2018)

Titles < série 4, 51

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 51, fascicule 1 (2018)

Shaobo Gan, Dawei Yang
Morse-Smale systems and horseshoes for three dimensional singular flows
Annales scientifiques de l'ENS 51, fascicule 1 (2018), 39-112

Télécharger cet article : Fichier PDF

Résumé :
Systèmes Morse-Smale et fers-à-cheval pour les flots singuliers en dimension 3
Nous montrons que tout champ de vecteurs en dimension trois peut être accumulé en topologie C^1 ou bien par un champ Morse-Smale, ou bien par un champ possédant une intersection homocline transverse associée à une orbite périodique hyperbolique. Contrairement au cas des difféomorphismes [15], la principale difficulté ici consiste à traiter les singularités. Nous progressons également en direction d'une autre conjecture de Palis.

Mots-clefs : Système de Morse-Smale, fer-à-cheval, champ de vecteurs, singularité.

Abstract:
We prove that every three-dimensional vector field can be C^1 accumulated by Morse-Smale ones, or by ones with a transverse homoclinic intersection of some hyperbolic periodic orbit. In contrast to the case of diffeomorphisms [15], the main difficulty here is that we need to deal with singularities. We also make progress on another conjecture related to Palis in this paper.

Keywords: Morse-Smale system, horseshoe, vector field, singularity.

Class. math. : 37C10, 37C20, 37C29, 37D15, 37D30.


ISSN : 0012-9593
DOI : 10.24033/asens.2351
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Afrač, V. S. and Bykov, V. V. and Sil'nikov, L. P.
The origin and structure of the Lorenz attractor
Dokl. Akad. Nauk SSSR 234 (1977) 336–339
Math Reviews MR0462175
2
Anosov, D. V.
Dynamical systems in the 1960s: the hyperbolic revolution
in Mathematical events of the twentieth century
(2006) 1–17
Math Reviews MR2182776
3
Arnaud, Marie-Claude
Création de connexions en topologie C^1
Ergodic Theory Dynam. Systems 21 (2001) 339–381
Math Reviews MR1827109
4
Araújo, Vítor and Pacifico, Maria José
Three-dimensional flows
Springer, Heidelberg, 2010
Math Reviews MR2662317
5
Arroyo, Aubin and Rodriguez Hertz, Federico
Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows
Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003) 805–841
Math Reviews MR1995503
6
Birkhoff, G. D.
Nouvelles recherches sur les systèmes dynamiques
Mem. Pont. Acad. Sci. Nov. Lyncaei 53 (1935) 85–216
7
Bonatti, Christian and Crovisier, Sylvain
Récurrence et généricité
Invent. math. 158 (2004) 33–104
Math Reviews MR2090361
8
Bonatti, Christian and Díaz, Lorenzo J. and Pujals, Enrique Ramiro
A C^1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources
Ann. of Math. 158 (2003) 355–418
Math Reviews MR2018925
9
Bonatti, Christian and Díaz, Lorenzo J. and Viana, Marcelo
Dynamics beyond uniform hyperbolicity
Springer, Berlin, 2005
Math Reviews MR2105774
10
Bonatti, Christian and Gan, Shaobo and Wen, Lan
On the existence of non-trivial homoclinic classes
Ergodic Theory Dynam. Systems 27 (2007) 1473–1508
Math Reviews MR2358974
11
Bonatti, Christian and Gan, Shaobo and Yang, Dawei
Dominated chain recurrent class with singularities
Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (2015) 83–99
Math Reviews MR3379488
12
Bonatti, Christian and Gourmelon, Nikolas and Vivier, Thérèse
Perturbations of the derivative along periodic orbits
Ergodic Theory Dynam. Systems 26 (2006) 1307–1337
Math Reviews MR2266363
13
Conley, Charles
Isolated invariant sets and the Morse index
Amer. Math. Soc., Providence, R.I., 1978
Math Reviews MR511133
14
Crovisier, Sylvain
Periodic orbits and chain-transitive sets of C^1-diffeomorphisms
Publ. Math. IHÉS 104 (2006) 87–141
Math Reviews MR2264835
15
Crovisier, Sylvain
Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems
Ann. of Math. 172 (2010) 1641–1677
Math Reviews MR2726096
16
Crovisier, Sylvain and Sambarino, Martin and Yang, Dawei
Partial hyperbolicity and homoclinic tangencies
J. Eur. Math. Soc. (JEMS) 17 (2015) 1–49
Math Reviews MR3312402
17
Franks, John
Necessary conditions for stability of diffeomorphisms
Trans. Amer. Math. Soc. 158 (1971) 301–308
Math Reviews MR0283812
18
Guckenheimer, John
A strange, strange attractor
in The Hopf bifurcation theorems and its applications
Applied Mathematical Series 19 (1976) 368–381
19
Guckenheimer, John and Williams, R. F.
Structural stability of Lorenz attractors
Publ. Math. IHÉS 50 (1979) 59–72
Math Reviews MR556582
20
Hayashi, Shuhei
Connecting invariant manifolds and the solution of the C^1 stability and -stability conjectures for flows
Ann. of Math. 145 (1997) 81–137
Math Reviews MR1432037
21
Hirsch, M. W. and Pugh, Charles C. and Shub, M.
Invariant manifolds
Springer, Berlin-New York, 1977
Math Reviews MR0501173
22
Kupka, Ivan
Contribution à la théorie des champs génériques
Contributions to Differential Equations 2 (1963) 457–484
Math Reviews MR0165536
23
Kupka, Ivan
Addendum et corrections au mémoire: ``Contributions à la théorie des champs génériques''
Contributions to Differential Equations 3 (1964) 411–420
Math Reviews MR0168878
24
Li, Ming and Gan, Shaobo and Wen, Lan
Robustly transitive singular sets via approach of an extended linear Poincaré flow
Discrete Contin. Dyn. Syst. 13 (2005) 239–269
Math Reviews MR2152388
25
Liao, Shan Tao
A basic property of a certain class of differential systems
Acta Math. Sinica 22 (1979) 316–343
Math Reviews MR549216
26
Liao, Shan Tao
On the stability conjecture
Chinese Ann. Math. 1 (1980) 9–30
Math Reviews MR591229
27
Liao, Shan Tao
Obstruction sets. II
Beijing Daxue Xuebao 2 (1981) 1–36
Math Reviews MR646519
28
Liao, Shan Tao
Some uniformity properties of ordinary differential systems and a generalization of an existence theorem for periodic orbits
Beijing Daxue Xuebao 2 (1985) 1–19
Math Reviews MR813931
29
Liao, Shan Tao
On (,d)-contractible orbits of vector fields
Systems Sci. Math. Sci. 2 (1989) 193–227
Math Reviews MR1109896
30
Lorenz, E. N.
Deterministic nonperiodic flow
J. Atmosph. Sci. 20 (1963) 130–141
31
Mañé, Ricardo
An ergodic closing lemma
Ann. of Math. 116 (1982) 503–540
Math Reviews MR678479
32
Mañé, Ricardo
Hyperbolicity, sinks and measure in one-dimensional dynamics
Comm. Math. Phys. 100 (1985) 495–524
Math Reviews MR806250
33
Morales, Carlos Arnoldo and Pacifico, Maria José
A dichotomy for three-dimensional vector fields
Ergodic Theory Dynam. Systems 23 (2003) 1575–1600
Math Reviews MR2018613
34
Morales, Carlos Arnoldo and Pacífico, Maria José and Pujals, Enrique Ramiro
On C^1 robust singular transitive sets for three-dimensional flows
C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 81–86
Math Reviews MR1649489
35
Morales, Carlos Arnoldo and Pacifico, Maria José and Pujals, Enrique Ramiro
Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers
Ann. of Math. 160 (2004) 375–432
Math Reviews MR2123928
36
Newhouse, Sheldon E.
Nondensity of axiom A(a) on S2
in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968)
(1970) 191–202
Math Reviews MR0277005
37
Newhouse, Sheldon E.
Diffeomorphisms with infinitely many sinks
Topology 13 (1974) 9–18
Math Reviews MR0339291
38
Newhouse, Sheldon E.
The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms
Publ. Math. IHÉS 50 (1979) 101–151
Math Reviews MR556584
39
Ohno, Taijiro
A weak equivalence and topological entropy
Publ. Res. Inst. Math. Sci. 16 (1980) 289–298
Math Reviews MR574037
40
Palis, Jacob
On the C^1 -stability conjecture
Publ. Math. IHÉS 66 (1988) 211–215
Math Reviews MR932139
41
Palis, Jacob
Homoclinic bifurcations, sensitive-chaotic dynamics and strange attractors
in Dynamical systems and related topics (Nagoya, 1990)
Adv. Ser. Dynam. Systems 9 (1991) 466–472
Math Reviews MR1164908
42
Palis, Jacob
A global view of dynamics and a conjecture on the denseness of finitude of attractors
Astérisque 261 (2000) 335–347
Math Reviews MR1755446
43
Palis, Jacob
A global perspective for non-conservative dynamics
Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 485–507
Math Reviews MR2145722
44
Palis, Jacob
Open questions leading to a global perspective in dynamics
Nonlinearity 21 (2008) 37–43
Math Reviews MR2399817
45
Palis, Jacob Jr. and de Melo, Welington
Geometric theory of dynamical systems
Springer, New York-Berlin, 1982
Math Reviews MR669541
46
Peixoto, M. M.
Structural stability on two-dimensional manifolds
Topology 1 (1962) 101–120
Math Reviews MR0142859
47
Pliss, V.
A hypothesis due to Smale
Diff. Eq. 8 (1972) 203–214
48
Poincaré, H.
Sur le problème des trois corps et les équations de la dynamique
Acta Math. 13 (1890) 1–270
49
Poincaré, H.
Les méthodes nouvelles de la mécanique céleste. Tome II
Librairie Scientifique et Technique Albert Blanchard, Paris, 1987
Math Reviews MR926907
50
Pugh, Charles C.
The closing lemma
Amer. J. Math. 89 (1967) 956–1009
Math Reviews MR0226669
51
Pujals, Enrique R. and Sambarino, Martín
Homoclinic tangencies and hyperbolicity for surface diffeomorphisms
Ann. of Math. 151 (2000) 961–1023
Math Reviews MR1779562
52
Smale, Stephen
Stable manifolds for differential equations and diffeomorphisms
Ann. Scuola Norm. Sup. Pisa 17 (1963) 97–116
Math Reviews MR0165537
53
Smale, Stephen
Diffeomorphisms with many periodic points
in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse)
(1965) 63–80
Math Reviews MR0182020
54
Smale, Steve
On how I got started in dynamical systems, 1959–1962
in From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990)
(1993) 22–26
Math Reviews MR1246104
55
Sun, Wenxiang and Young, Todd and Zhou, Yunhua
Topological entropies of equivalent smooth flows
Trans. Amer. Math. Soc. 361 (2009) 3071–3082
Math Reviews MR2485418
56
Thomas, Romeo F.
Topological entropy of fixed-point free flows
Trans. Amer. Math. Soc. 319 (1990) 601–618
Math Reviews MR1010414
57
Wen, Lan
On the C^1 stability conjecture for flows
J. Differential Equations 129 (1996) 334–357
Math Reviews MR1404387
58
Wen, Lan
Homoclinic tangencies and dominated splittings
Nonlinearity 15 (2002) 1445–1469
Math Reviews MR1925423
59
Wen, Lan
Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles
Bull. Braz. Math. Soc. (N.S.) 35 (2004) 419–452
Math Reviews MR2106314
60
Wen, Lan and Xia, Zhihong
C^1 connecting lemmas
Trans. Amer. Math. Soc. 352 (2000) 5213–5230
Math Reviews MR1694382
61
Wen, Lan
The selecting lemma of Liao
Discrete Contin. Dyn. Syst. 20 (2008) 159–175
Math Reviews MR2350064
62
Xi, R.
Kupka-Smale Theorem with obstacle
PhD Thesis, Peking University (2005) (in Chinese)
63
Yang, Dawei and Zhang, Yong
On the finiteness of uniform sinks
J. Differential Equations 257 (2014) 2102–2114
Math Reviews MR3227291