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Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

Published online by Cambridge University Press:  07 November 2017

CHENG CHENG ,
SYLVAIN CROVISIER ,
SHAOBO GAN ,
XIAODONG WANG  and
DAWEI YANG
CHENG CHENG
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, PR China email [email protected], [email protected]
SYLVAIN CROVISIER
Affiliation:
CNRS - Laboratoire de Mathématiques d’Orsay, Université Paris-Sud 11, Orsay 91405, France email [email protected]
SHAOBO GAN
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, PR China email [email protected], [email protected]
XIAODONG WANG
Affiliation:
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, PR China email [email protected], [email protected]
DAWEI YANG
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou 215006, PR China email [email protected], [email protected]
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Abstract

We prove that, for $C^{1}$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-trivial non-hyperbolic ergodic measure supported on it. This proves a conjecture by Díaz and Gorodetski.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1805 - 1823
Copyright
© Cambridge University Press, 2017 

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References

Abdenur, F., Bonatti, C., Crovisier, S. and Díaz, L.. Generic diffeomorphisms on compact surfaces. Fund. Math. 187 (2005), 127–159.Google Scholar
Abdenur, F., Bonatti, C., Crovisier, S., Díaz, L. and Wen, L.. Periodic points and homoclinic classes. Ergod. Th. & Dynam. Sys. 27 (2007), 1–22.Google Scholar
Abraham, R. and Smale, S.. Nongenericity of 𝛺-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, 14) . American Mathematical Society, Providence, RI, 1970, pp. 5–8.Google Scholar
Aoki, N.. The set of Axiom A diffeomorphisms with no cycles. Bol. Soc. Brasil. Mat. 23 (1992), 21–65.Google Scholar
Baladi, V., Bonatti, C. and Schmitt, B.. Abnormal escape rates from nonuniformly hyperbolic sets. Ergod. Th. & Dynam. Sys. 19 (1999), 1111–1125.Google Scholar
Bochi, J. and Bonatti, C.. Perturbation of the Lyapunov spectra of periodic orbits. Proc. Lond. Math. Soc. 105 (2012), 1–48.Google Scholar
Bochi, J., Bonatti, C. and Díaz, L.. Robust criterion for the existence of nonhyperbolic ergodic measures. Comm. Math. Phys. 344 (2016), 751–795.Google Scholar
Bonatti, C.. Towards a global view of dynamical systems, for the C 1 -topology. Ergod. Th. & Dynam. Sys. 31 (2011), 959–993.Google Scholar
Bonatti, C. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158 (2004), 33–104.Google Scholar
Bonatti, C. and Crovisier, S.. Center manifolds for partially hyperbolic set without strong unstable connections. J. Inst. Math. Jussieu 15 (2016), 785–828.Google Scholar
Bonatti, C., Crovisier, S., Díaz, L. and Gourmelon, N.. Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication. Ergod. Th. & Dynam. Sys. 33 (2013), 739–776.Google Scholar
Bonatti, C., Crovisier, S. and Shinohara, K.. The C 1+𝛼 hypothesis in Pesin theory revisited. J. Mod. Dyn. 7 (2013), 605–618.Google Scholar
Bonatti, C. and Díaz, L.. Robust heterodimensional cycles and C 1 -generic dynamics. J. Inst. Math. Jussieu 7 (2008), 469–525.Google Scholar
Bonatti, C. and Díaz, L.. Abundance of C 1 -robust homoclinic tangencies. Trans. Amer. Math. Soc. 364 (2012), 5111–5148.Google Scholar
Bonatti, C., Díaz, L. and Gorodetski, A.. Non-hyperbolic ergodic measures with large support. Nonlinearity 23 (2010), 687–705.Google Scholar
Bonatti, C., Díaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III) . Springer, Berlin, 2005.Google Scholar
Bonatti, C., Gan, S. and Wen, L.. On the existence of non-trivial homoclinic classes. Ergod. Th. & Dynam. Sys. 27 (2007), 1473–1508.Google Scholar
Cao, Y., Luzzatto, S. and Rios, I.. Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: horseshoes with internal tangencies. Disc. Cont. Dyn. Syst. 15 (2006), 61–71.Google Scholar
Crovisier, S.. Periodic orbits and chain-transitive sets of C 1 -diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 104 (2006), 87–141.Google Scholar
Crovisier, S.. Partial hyperbolicity far from homoclinic bifurcations. Adv. Math. 226 (2011), 673–726.Google Scholar
Crovisier, S.. Dynamics of C 1 -diffeomorphisms: global description and prospect of classification. Proceedings of the International Congress of Mathematicians, Volume III. Kyung Moon, Seoul, 2014, pp. 571–595.Google Scholar
Crovisier, S. and Pujals, E.. Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Invent. Math. 201 (2015), 385–517.Google Scholar
Crovisier, S., Pujals, E. and Sambarino, M.. Hyperbolicity of the extremal bundles. In preparation.Google Scholar
Crovisier, S., Sambarino, M. and Yang, D.. Partial hyperbolicity and homoclinic tangencies. J. Eur. Math. Soc. 17 (2015), 1–49.Google Scholar
Díaz, L. and Gorodetski, A.. Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes. Ergod. Th. & Dynam. Sys. 29 (2009), 1479–1513.Google Scholar
Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301–308.Google Scholar
Gan, S.. A necessary and sufficient condition for the existence of dominated splitting with a given index. Trends Math. 7 (2004), 143–168.Google Scholar
Gorodetski, A., Ilyashenko, Yu., Kleptsyn, V. and Nalsky, M.. Nonremovable zero Lyapunov esponents. Funct. Anal. Appl. 39 (2005), 27–38.Google Scholar
Gourmelon, N.. A Frank’s lemma that preserves invariant manifolds. Ergod. Th. & Dynam. Sys. 36 (2016), 1167–1203.Google Scholar
Hayashi, S.. Diffeomorphisms in F1(M) satisfy Axiom A. Ergod. Th. & Dynam. Sys. 12 (1992), 233–253.Google Scholar
Klepstyn, V. and Nalsky, M.. Persistence of nonhyperbolic measures for C 1 -diffeomorphisms. Funct. Anal. Appl. 41 (2007), 271–283.Google Scholar
Liao, S.. Obstruction sets II. Acta Sci. Natur. Univ. Pekinensis 2 (1981), 1–36 (in Chinese).Google Scholar
Mañé, R.. An ergodic closing lemma. Ann. of Math. (2) 116 (1982), 503–540.Google Scholar
Palis, J.. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261 (2000), 335–347.Google Scholar
Pesin, Y.. Characteristic Lyapunov exponents and smooth ergodic theory. Uspekhi. Mat. Nauk. 32 (1977), 55–112.Google Scholar
Pujals, E. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for suface diffeomorphisms. Ann. of Math. (2) 151 (2000), 961–1023.Google Scholar
Rios, I.. Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies. Nonlinearity 14 (2001), 431–462.Google Scholar
Wang, X.. Hyperbolicity versus weak periodic orbits inside homoclinic classes. Ergod. Th. & Dynam. Sys. to appear, doi:10.1017/etds.2016.122, published online 14 March 2017.Google Scholar