Résumé.
Nous montrons un lemme de connexion C 1 pour les pseudo-orbites des difféomorphismes des variétés compactes. Nous explorons alors les conséquences pour les difféomorphismes C 1-génériques. Par exemple, les difféomorphismes conservatifs C 1-génériques (d’une variété connexe) sont transitifs.
Abstract.
We prove a C 1-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C 1-generic diffeomorphisms. For instance, C 1-generic conservative diffeomorphisms (on connected manifolds) are transitive.
Références
Abdenur, F.: Generic robustness of a spectral decompositions. Ann. Sci. Éc. Norm. Supér., IV. Sér. 36, 213–224 (2003)
Arnaud, M.-C.: Création de connexions en topologie C 1. Ergodic Theory Dyn. Syst. 21, 339–381 (2001)
Arnaud, M.-C.: Le “closing lemma” en topologie C 1. Mém. Soc. Math. Fr., Nouv. Sér. 74 (1998), vi + 120pp.
Arnaud, M.-C., Bonatti, Ch., Crovisier, S.: Dynamiques symplectiques génériques. Prépublication 363, Institut de Mathématiques de Bourgogne (2004)
Birkhoff, G.D.: Dynamical systems. Am. Math. Soc. Colloq. Pub. 9. Providence, RI: Am. Math. Soc. 1927
Birkhoff, G.D.: Nouvelles recherches sur les systèmes dynamiques. Memoriae Pont. Acad. Sci. Novi Lyncaei 93, 85–216 (1935)
Bonatti, Ch., Crovisier, S.: Recurrence and genericity. C. R. Acad. Sci., Paris, Sér. I, Math. 336, 839–844 (2003)
Bonatti, Ch., Díaz, L.J.: Connexions hétéroclines et généricité d’une infinité de puits ou de sources. Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, 135–150 (1999)
Bonatti, Ch., Díaz, L.J.: On maximal transitive sets of generic diffeomorphisms. Publ. Math., Inst. Hautes Étud. Sci. 96, 171–197 (2003)
Bonatti, Ch., Díaz, L.J., Pujals, E.R.: A C 1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicicity or infinitely many sinks or sources. Ann. Math. 158, 355–418 (2003)
Bochi, J.: Genericity of zero Lyapunov exponents. Ergodic Theory Dyn. Syst. 22, 1667–1696 (2002)
Bochi, J., Viana, M.: Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps. Ann. Inst. Henri Poincaré Anal., Non Linéaire 19, 113–123 (2002)
Bowen, R.: Equilibrium states and the ergodic theory of Axiom A diffeomorphisms. Lect. Notes Math. 470. Springer-Verlag 1975
Carballo, C., Morales, C.: Homoclinic classes and finitude of attractors for vector fields on n-manifolds. Bull. Lond. Math. Soc. 35, 85–91 (2003)
Carballo, C., Morales, C., Pacífico, M.J.: Homoclinic classes for \(\mathcal{C}^1\)-generic vector fields. Ergodic Theory Dyn. Syst. 23, 1–13 (2003)
Conley, C.: Isolated invariant sets and Morse index. CBMS Regional Conf. Ser. Math. 38. Providence, RI: Am. Math. Soc. 1978
Crovisier, S.: Periodic orbits and chain transitive sets of C 1-diffeomorphisms. Prépublication, Institut de Mathématiques de Bourgogne (2004)
Franks, J.: Necessary conditions for stability of diffeomorphisms. Trans. Am. Math. Soc. 158, 301–308 (1971)
Gan, G., Wen, L.: Heteroclinic cycles and homoclinic closures for generic diffeomorphisms. Prépublication de Peking University
Hayashi, S.: Connecting invariant manifolds and the solution of the C 1-stability and Ω-stability conjectures for flows. Ann. Math. 145, 81–137 (1997); Ann. Math. 150, 353–356 (1999)
Herman, M.: Sur les courbes invariantes par les difféomorphismes de l’anneau. Astérisque 103–104 (1983), i + 221pp.
Herman, M.: Some open problems in dynamical systems. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 797–808 (electronic)
Hurley, M.: Attractors: persistence, and density of their basins. Trans. Am. Math. Soc. 269, 271–247 (1982)
Kechris, A.S.: Classical descriptive set theory. Grad. Texts Math. 156. New York: Springer-Verlag 1995
Morales, C., Pacífico, M.J.: Lyapunov stability of ω-limit sets. Discrete Contin. Dyn. Syst. 8, 671–674 (2002)
Moser, J.: Stable and random motions in dynamical systems. Ann. Math. Stud. 77. Princeton University Press 1973
Newhouse, S.: Diffeomorphisms with infinitely many sinks. Topology 13, 9–18 (1974)
Newhouse, S.: Quasi-elliptic periodic points in conservative dynamical systems. Am. J. Math. 99, 1061–1087 (1977)
Newhouse, S.: The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Publ. Math., Inst. Hautes Étud. Sci. 50, 101–151 (1979)
Oxtoby, J.C., Ulam, S.M.: Measure-preserving homeomorphisms and metrical transitivity. Ann. Math. 42, 874–920 (1941)
Palis, J.: A note on Ω-stability. Global Analysis. Proc. Sympos. Pure Math. 14, Providence, RI: Am. Math. Soc. 1970
Palis, J., Pugh, C.: Fifty problems in dynamical systems. Lect. Notes Math. 468, pp. 345–353. Springer-Verlag 1975
Poincaré, H.: Les méthodes nouvelles de la mécanique céleste (1899), réédité par Les grands classiques Gauthier-Villars. Paris: Librairie Blanchard 1987
Pugh, C.: The closing lemma. Am. J. Math. 89, 956–1009 (1967)
Pugh, C., Robinson, C.: The C 1-closing lemma, including Hamiltonians. Ergodic Theory Dyn. Syst. 3, 261–314 (1983)
Robinson, C.: Generic properties of conservative systems I & II. Am. J. Math. 92, 562–603, 897–906 (1970)
Robinson, C.: Dynamical systems: stability, symbolic dynamics, and chaos. Stud. Adv. Math., Boca Raton, FL: CRC Press 1999
Shub, M.: Dynamical systems, filtrations and entropy. Bull. Am. Math. Soc., New Ser. 80, 27–41 (1974)
Shub, M.: Stabilité globale des systèmes dynamiques. Astérisque 56 (1978)
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc., New Ser. 73, 747–817 (1967)
Takens, F.: Homoclinic points in conservative systems. Invent. Math. 18, 267–292 (1972)
Wen, L.: A uniform C 1 connecting lemma. Discrete Contin. Dyn. Syst. 8, 257–265 (2002)
Wen, L.: Homoclinic tangencies and dominated splittings. Nonlinearity 15, 1445–1469 (2002)
Wen, L., Xia, Z.: C 1-connecting lemmas. Trans. Am. Math. Soc. 352, 5213–5230 (2000)
Zehnder, E.: Note on smoothing symplectic and volume preserving diffeomorphisms. Lect. Notes Math. 597, 828–854. Springer-Verlag 1977
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Bonatti, C., Crovisier, S. Récurrence et généricité. Invent. math. 158, 33–104 (2004). https://doi.org/10.1007/s00222-004-0368-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-004-0368-1