Comptes Rendus
Dynamical systems
On the hyperbolicity of C1-generic homoclinic classes
Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1047-1051.

The works of Liao, Mañé, Franks, Aoki, and Hayashi characterized a lack of hyperbolicity for diffeomorphisms by the existence of weak periodic orbits. In this note, we announce a result that can be seen as a local version of these works: for C1-generic diffeomorphisms, a homoclinic class either is hyperbolic or contains a sequence of periodic orbits that have a Lyapunov exponent arbitrarily close to 0.

Des travaux de Liao, Mañé, Franks, Aoki et Hayashi ont caractérisé le manque d'hyperbolicité des difféomorphismes par l'existence d'orbites périodiques faibles. Dans cette note, nous annonçons un résultat qui peut être considéré comme une version locale de ces travaux : pour les difféomorphismes C1-génériques, une classe homocline, ou bien est hyperbolique, ou bien contient une suite d'orbites périodiques qui ont un exposant de Lyapunov arbitrairement proche de 0.

Published online:
DOI: 10.1016/j.crma.2015.07.017

Xiaodong Wang 1, 2

1 School of Mathematical Sciences, Peking University, Beijing, 100871, China
2 Laboratoire de mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay, France
     author = {Xiaodong Wang},
     title = {On the hyperbolicity of $ {\mathrm{C}}^{1}$-generic homoclinic classes},
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Xiaodong Wang. On the hyperbolicity of $ {\mathrm{C}}^{1}$-generic homoclinic classes. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1047-1051. doi : 10.1016/j.crma.2015.07.017.

[1] N. Aoki The set of Axiom A diffeomorphisms with no cycles, Bol. Soc. Bras. Mat., Volume 23 (1992), pp. 21-65

[2] C. Bonatti; S. Crovisier Récurrence et généricité, Invent. Math., Volume 158 (2004), pp. 33-104

[3] C. Bonatti; S. Gan; D. Yang On the hyperbolicity of homoclinic classes, Discrete Contin. Dyn. Syst., Volume 25 (2009), pp. 1143-1162

[4] Y. Cao; S. Luzzatto; I. Rios Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: horseshoes with internal tangencies, Discrete Contin. Dyn. Syst., Volume 15 (2006), pp. 61-71

[5] S. Crovisier Periodic orbits and chain-transitive sets of C1-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., Volume 104 (2006), pp. 87-141

[6] S. Crovisier; M. Sambarino; D. Yang Partial hyperbolicity and homoclinic tangencies, J. Eur. Math. Soc., Volume 17 (2015), pp. 1-49

[7] J. Franks Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., Volume 158 (1971), pp. 301-308

[8] S. Gan; L. Wen Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dyn. Differ. Equ., Volume 15 (2003), pp. 451-471

[9] S. Gan; D. Yang Expansive homoclinic classes, Nonlinearity, Volume 22 (2009), pp. 729-734

[10] S. Hayashi Diffeomorphisms in F1(M) satisfy Axiom A, Ergod. Theory Dyn. Syst., Volume 12 (1992), pp. 233-253

[11] M. Hirsch; C. Pugh; M. Shub Invariant Manifolds, Lecture Notes in Mathematics, vol. 583, Springer-Verlag, Berlin, 1977

[12] S. Liao On the stability conjecture, Chinese Ann. Math., Volume 1 (1980), pp. 9-30

[13] R. Mañé An ergodic closing lemma, Ann. of Math. (2), Volume 116 (1982), pp. 503-540

[14] R. Mañé A proof of the C1 stability conjecture, Publ. Math. Inst. Hautes Études Sci., Volume 66 (1988), pp. 161-210

[15] V. Pliss On a conjecture due to Smale, Differ. Uravn., Volume 8 (1972), pp. 262-268

[16] I. Rios Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies, Nonlinearity, Volume 14 (2001), pp. 431-462

[17] L. Wen A uniform C1 connecting lemma, Discrete Contin. Dyn. Syst., Volume 8 (2002), pp. 257-265

[18] L. Wen The selecting lemma of Liao, Discrete Contin. Dyn. Syst., Volume 20 (2008), pp. 159-175

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

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