no. 10
Dynamical Systems
Recurrence and genericity
[Récurrence et généricité]

Présenté par : Jean-Christophe Yoccoz

Bonatti, Christian1 ; Crovisier, Sylvain1
1 Institut de mathématiques de Bourgogne, UMR 5584, Université de Bourgogne, 9, av. A. Savary, BP 47870, 21078 Dijon cedex, France
Comptes Rendus. Mathématique, Tome 336 (2003) no. 10, pp. 839-844.

Nous montrons un lemme de connexion C1 pour les pseudo-orbites des difféomorphismes des variétés compactes. Nous explorons alors les conséquences pour les difféomorphismes C1-génériques. Par exemple, les difféomorphismes conservatifs C1-génériques sont transitifs.

We prove a C1-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C1-generic diffeomorphisms. For instance, C1-generic conservative diffeomorphisms are transitive.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00203-6
Bonatti, Christian 1 ; Crovisier, Sylvain 1

1 Institut de mathématiques de Bourgogne, UMR 5584, Université de Bourgogne, 9, av. A. Savary, BP 47870, 21078 Dijon cedex, France 
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Bonatti, Christian; Crovisier, Sylvain. Recurrence and genericity. Comptes Rendus. Mathématique, Tome 336 (2003) no. 10, pp. 839-844. doi : 10.1016/S1631-073X(03)00203-6. http://www.numdam.org/articles/10.1016/S1631-073X(03)00203-6/

[1] Arnaud, M.-C. Création de connexions en topologie C1, Ergodic Theory Dynamical Systems, Volume 21 (2001), pp. 339-381

[2] Ch. Bonatti, L.J. Dı́az, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., to appear

[3] Carballo, C.A.; Morales, C.M.; Pacı́fico, M.J. Homoclinic classes for 𝒞 1 -generic vector fields, Ergodic Theory Dynamical Systems, Volume 23 (2003), pp. 1-13

[4] Conley, C. Isolated invariant sets and Morse index, CBMS, 38, American Mathematical Society, 1978

[5] Hayashi, S. Connecting invariant manifolds and the solution of the C1-stability and Ω-stability conjectures for flows, Ann. of Math., Volume 145 (1997), pp. 81-137

[6] Herman, M. Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. II, Berlin, 1998, Doc. Math., Extra Vol. II (1998), pp. 797-808 (electronic)

[7] Hurley, M. Attractors: persistence, and density of their basins, Trans. Amer. Math. Soc., Volume 269 (1982), pp. 247-271

[8] Morales, C.M.; Pacı́fico, M.J. Lyapunopv stability of ω-limit sets, Discrete Contin. Dynam. Systems, Volume 8 (2002), pp. 671-674

[9] Pugh, C. The closing lemma, Ergodic Theory Dynamical Systems, Volume 3 (1983), pp. 261-314

[10] Pugh, C.; Robinson, C. The C1-closing lemma, including Hamiltonians, Ergodic Theory Dynamical Systems, Volume 3 (1983), pp. 261-314

[11] Wen, L.; Xia, Z. C1-connecting lemmas, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 5213-5230

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