Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T15:26:24.306Z Has data issue: false hasContentIssue false
  • English
  • Français

NON-HYPERBOLIC ERGODIC MEASURES AND HORSESHOES IN PARTIALLY HYPERBOLIC HOMOCLINIC CLASSES

Part of: Ergodic theory Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  07 January 2019

Dawei Yang  and
Jinhua Zhang
Dawei Yang
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou, 215006, P.R. China (yangdw1981@gmail.com; yangdw@suda.edu.cn)
Jinhua Zhang
Affiliation:
Laboratoire de Mathématiques d’Orsay, CNRS - Université Paris-Sud, Orsay 91405, France (Jinhua.zhang@math.u-psud.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$\ast$-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures. As a consequence, we obtain the continuity of topological entropy.

Keywords

blender horseshoepartial hyperbolicityentropyperiodic measurenon-hyperbolic ergodic measure

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37A35: Entropy and other invariants, isomorphism, classification 37C40: Smooth ergodic theory, invariant measures
Secondary: 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.) 37D30: Partially hyperbolic systems and dominated splittings
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1765 - 1792
Check for updates
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was done when J. Zhang visited Soochow University in July 2017. J. Zhang would like to thank Soochow University for hospitality. D. Yang was partially supported by NSFC 11671288 and NSFC 11790274. J. Zhang was partially supported by the ERC project 692925 NUHGD. J. Zhang is the corresponding author.

References

Abdenur, F., Bonatti, C. and Crovisier, S., Nonumiform hyperbolicity for C 1 generic diffeomorphisms, Israel J. Math. 183 (2011), 1–60.Google Scholar
Abraham, R. and Smale, S., Nongenericity of 𝛺-stability, in Global Analysis I, Proceedings of Symposia in Pure Mathematics, Volume 14, pp. 5–8 (American Mathematical Society, 1970).Google Scholar
Bochi, J., Bonatti, C. and Díaz, L. J., Robust criterion for the existence of nonhyperbolic ergodic measures, Comm. Math. Phys 344(3) (2016), 751–795.Google Scholar
Bochi, J., Bonatti, C. and Díaz, L. J., A criterion for zero averages and full support of ergodic measures, Mosc. Math. J. 18(1) (2018), 15–61.Google Scholar
Bonatti, C. and Crovisier, S., Récurrence et généricité, Invent. Math. 158(1) (2004), 33–104.Google Scholar
Bonatti, C., Crovisier, S., Díaz, L. J. and Wilkinson, A., What is ⋯ a blender? Notices Amer. Math. Soc. 63(10) (2016), 1175–1178.Google Scholar
Bonatti, C. and Díaz, L. J., Persistent nonhyperbolic transitive diffeomorphism, Ann. of Math. (2) 143(2) (1996), 357–396.Google Scholar
Bonatti, C. and Díaz, L. J., Robust heterodimensional cycles and C 1 generic dynamics, J. Inst. Math. Jussieu 7(3) (2008), 469–525.Google Scholar
Bonatti, C. and Díaz, L. J., Abundance of C 1 -robust homoclinic tangencies, Trans. Amer. Math. Soc 364(10) (2012), 5111–5148.Google Scholar
Bonatti, C., Díaz, L. J. and Gorodetski, A., Non-hyperbolic ergodic measures with large support, Nonlinearity 23(3) (2010), 687–705.Google Scholar
Bonatti, C., Díaz, L. J., Pujals, E. and Rocha, J., Robustly transitive sets and heterodimensional cycles. Geometric methods in dynamics. I, Astérisque 286 (2003), 187–222.Google Scholar
Bonatti, C., Díaz, L. J. and Ures, R., Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu 1(4) (2002), 513–541.Google Scholar
Bonatti, C., Gogolev, A. and Potrie, R., Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples, Invent. Math. 206(3) (2016), 801–836.Google Scholar
Bonatti, C., Gogolev, A., Hammerlindl, A. and Potrie, R., Anomalous partially hyperbolic diffeomorphisms III: abundance and incoherence, Preprint, 2017, arXiv:1706.04962.Google Scholar
Bonatti, C. and Zhang, J., On the existence of non-hyperbolic ergodic measures as the limit of periodic measures, Ergodic Theory Dynam. Systems (2018), 1–36 (Online) doi:10.1017/etds.2017.146.Google Scholar
Bonatti, C. and Zhang, J., Periodic measures and partially hyperbolic homoclinic classes, Trans. Amer. Math. Soc., to appear, Preprint, 2016, arXiv:1609.08489.Google Scholar
Burns, K. and Wilkinson, A., On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171(1) (2010), 451–489.Google Scholar
Buzzi, J. and Fisher, T., Entropic stability beyond partial hyperbolicity, J. Mod. Dyn. 7(4) (2013), 527–552.Google Scholar
Buzzi, J., Fisher, T., Sambarino, M. and Vásquez, C., Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems, Ergodic Theory Dynam. Systems 32(1) (2012), 63–79.Google Scholar
Crovisier, S., Partial hyperbolicity far from homoclinic bifurcations, Adv. Math. 226 (2011), 673–726.Google Scholar
Cheng, C., Crovisier, S., Gan, S., Wang, X. and Yang, D., Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes, Ergodic Theory Dynam. Systems (2017), 1–19 (Online), doi:10.1017/etds.2017.106.Google Scholar
Díaz, L. J., Gelfert, K. and Rams, M., Entropy spectrum of Lyapunov exponents for nonhyperbolic step skew-products and elliptic cocycles, Preprint, 2016,arXiv:1610.07167.Google Scholar
Díaz, L. J., Gelfert, K. and Santiago, B., Weak* and entropy approximation of nonhyperbolic measures: a geometrical approach, Preprint, 2018, arXiv:1804.05913.Google Scholar
Gan, S., A generalized shadowing lemma, Discrete Contin. Dyn. Syst. 8(3) (2002), 527–632.Google Scholar
Gelfert, K., Horseshoes for diffeomorphisms preserving hyperbolic measures, Math. Z. 283(3–4) (2016), 685–701.Google Scholar
Gorodetski, A., Ilyashenko, Yu. S., Kleptsyn, V. A. and Nalsky, M. B., Nonremovability of zero Lyapunov exponents, Funktsional. Anal. i Prilozhen. 39(1) (2005), 27–38, translation in Funct. Anal. Appl. 39(1) (2005), 21–30.Google Scholar
Gorodetski, A. and Pesin, Y., Path connectedness and entropy density of the space of hyperbolic ergodic measures, in Modern Theory of Dynamical Systems, Contemporary Mathematics, Volume 692, pp. 111–121 (American Mathematical Society, Providence, RI, 2017).Google Scholar
Hayashi, S., Connecting invariant manifolds and the solution of C 1 -stability and 𝛺-stability conjectures for flows, Ann. of Math. (2) 145(1) (1997), 81–137.Google Scholar
Hirsch, M., Pugh, C. and Shub, M., Invariant Manifolds, Lecture Notes in Mathematics, Volume 583 (Springer, New York, 1977).Google Scholar
Katok, A., Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.Google Scholar
Katok, A. and Hasselblatt, B., Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, Volume 54 (Cambridge University Press, Cambridge, 1995).Google Scholar
Liao, G., Viana, M. and Yang, J., The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc. (JEMS) 15(6) (2013), 2043–2060.Google Scholar
Liao, S. T., An existence theorem for periodic orbits, Acta Sci. Natur. Univ. Pekinensis 1 (1979), 1–20.Google Scholar
Oseledec, V., A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Tr. Mosk. Mat. Obs. 19 (1968), 179–210 (Russian).Google Scholar
Pesin, Y., Characteristic Ljapunov exponents, and smooth ergodic theory, Uspekhi Mat. Nauk 32(4) (287), 55–112 (196) (Russian).Google Scholar
Pliss, V., On a conjecture due to Smale, Differ. Uravn. 8 (1972), 262–268.Google Scholar
Saghin, R. and Yang, J., Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows, Israel J. Math. 215(2) (2016), 857–875.Google Scholar
Sigmund, K., Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math. 11(2) (1970), 99–109.Google Scholar
Sigmund, K., On the connectedness of ergodic systems, Manuscripta Math. 22(1) (1977), 27–32.Google Scholar
Tahzibi, A. and Yang, J., Invariance principle and rigidity of high entropy measures, Trans. Amer. Math. Soc., to appear, Preprint, 2016, arXiv:1606.09429.Google Scholar
Walters, P., An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, (Springer, New York, 1982).Google Scholar
Wen, L., Homoclinic tangencies and dominated splittings, Nonlinearity 15(5) (2002), 1445–1469.Google Scholar
Wen, L. and Xia, Z., C 1 connecting lemmas, Trans. Amer. Math. Soc. 352(11) (2000), 5213–5230.Google Scholar
Yomdin, Y., Volume growth and entropy, Israel J. Math. 57(3) (1987), 285–300.Google Scholar