Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T09:52:19.426Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

Published online by Cambridge University Press:  03 February 2009

LORENZO J. DÍAZ  and
ANTON GORODETSKI
LORENZO J. DÍAZ
Affiliation:
Departamento de Matemática, PUC-Rio, Marquês de S. Vicente 225, 22453-900 Rio de Janeiro RJ, Brazil (email: [email protected])
ANTON GORODETSKI
Affiliation:
Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that there is a residual subset 𝒮 in Diff1(M) such that, for every f∈𝒮, any homoclinic class of f containing saddles of different indices (dimension of the unstable bundle) contains also an uncountable support of an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of f.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1479 - 1513
Copyright
Copyright © Cambridge University Press 2009

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.)

References

[1]Abdenur, F., Bonatti, Ch. and Crovisier, S.. Nonuniform hyperbolicity for C 1-generic diffeomorphisms. Preprint.Google Scholar
[2]Abdenur, F., Bonatti, Ch., Crovisier, S. and Díaz, L. J.. Generic diffeomorphisms on compact surfaces. Fund. Math. 187 (2005), 127–159.CrossRefGoogle Scholar
[3]Abdenur, F., Bonatti, Ch., Crovisier, S., Díaz, L. J. and Wen, L.. Periodic points and homoclinic classes. Ergod. Th. & Dynam. Sys. 26 (2006), 1–22.Google Scholar
[4]Abdenur, F. and Díaz, L. J.. Pseudo-orbit shadowing in the C 1-topology. Discrete Contin. Dyn. Syst. 17 (2007), 223–245.CrossRefGoogle Scholar
[5]Abraham, R. and Smale, S.. Nongenericity of Ω-stability. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, CA, 1968). American Mathematical Society, Providence, RI, 1970, pp. 5–8.CrossRefGoogle Scholar
[6]Alves, J., Araújo, V. and Saussol, B.. On the uniform hyperbolicity of some nonuniformly hyperbolic systems. Proc. Amer. Math. Soc. 131 (2003), 1303–1309.CrossRefGoogle Scholar
[7]Aoki, N.. The set of Axiom A diffeomorphisms with no cycles. Bol. Soc. Brasil. Mat. (N.S.) 23 (1992), 21–65.CrossRefGoogle Scholar
[8]Baraviera, A. and Bonatti, Ch.. Removing zero Lyapunov exponents. Ergod. Th. & Dynam. Sys. 23 (2003), 1655–1670.CrossRefGoogle Scholar
[9]Bochi, J.. Genericity of zero Lyapunov exponents. Ergod. Th. & Dynam. Sys. 22 (2002), 1667–1696.CrossRefGoogle Scholar
[10]Bochi, J. and Viana, M.. Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps. Ann. Inst. H. Poincaré, Anal. Non Linéaire 19 (2002), 113–123.CrossRefGoogle Scholar
[11]Bonatti, Ch. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158 (2004), 33–104.CrossRefGoogle Scholar
[12]Bonatti, Ch. and Díaz, L. J.. Nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143 (1996), 357–396.CrossRefGoogle Scholar
[13]Bonatti, Ch. and Díaz, L. J.. Connexions hétéroclines et genericité d’une infinité de puits ou de sources. Ann. Sci. École Norm. Sup. 32 (1999), 135–150.CrossRefGoogle Scholar
[14]Bonatti, Ch. and Díaz, L. J.. On maximal transitive sets of generic diffeormophisms. Publ. Math. Inst. Hautes Études Sci. 96 (2002), 171–197.CrossRefGoogle Scholar
[15]Bonatti, Ch. and Díaz, L. J.. Robust heterodimensional cycles. J. Inst. Math. Jussieu 7(3) (2008), 469–525.CrossRefGoogle Scholar
[16]Bonatti, Ch., Díaz, L. J. and Fisher, T.. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete Contin. Dyn. Syst. 20 (2008), 589–604.CrossRefGoogle Scholar
[17]Bonatti, Ch., Díaz, L. J. and Pujals, E. R.. A 𝒞1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicicity or infinitely many sinks or sources. Ann. of Math. (2) 158 (2003), 355–418.CrossRefGoogle Scholar
[18]Bonatti, Ch., Díaz, L. J. and Turcat, G.. Pas de ‘Shadowing Lemma’ pour les dynamiques partiellement hyperboliques. C. R. Acad. Sci., Paris, Sér. I Math. 330(7) (2000), 587–592.CrossRefGoogle Scholar
[19]Bonatti, Ch., Díaz, L. J. and Viana, M.. Discontinuity of the Hausdorff dimension of hyperbolic sets. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 713–718.Google Scholar
[20]Bonatti, Ch., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences (Mathematical Physics), 102). Springer, Berlin, 2005.Google Scholar
[21]Bonatti, Ch. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.CrossRefGoogle Scholar
[22]Crovisier, S.. Periodic orbits and chain transitive sets of C 1-diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 104 (2006), 87–141.CrossRefGoogle Scholar
[23]Cao, Y.. Nonzero Lyapunov exponents and uniformly hyperbolicity. Nonlinearity 16 (2003), 1473–1479.CrossRefGoogle Scholar
[24]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. Discrete Contin. Dyn. Syst. 15 (2006), 61–71.CrossRefGoogle Scholar
[25]Cao, Y., Luzzatto, S. and Rios, I.. The boundary of hyperbolicity for Henon-like families. Ergod. Th. & Dynam. Sys. 28(4) (2008), 1049–1080.CrossRefGoogle Scholar
[26]Carballo, C., Morales, C. and Pacifico, M. J.. Homoclinic classes for generic C 1 vector fields. Ergod. Th. & Dynam. Sys. 23 (2003), 403–415.CrossRefGoogle Scholar
[27]Díaz, L. J.. Robust non-hyperbolic dynamics at heterodimensional cycles. Ergod. Th. & Dynam. Sys. 15 (1995), 291–315.CrossRefGoogle Scholar
[28]Dolgopyat, D. and Pesin, Ya.. Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod. Th. & Dynam. Sys. 22 (2002), 409–435.CrossRefGoogle Scholar
[29]Díaz, L. J., Pujals, E. R. and Ures, R.. Partial hyperbolicity and robust transitivity. Acta Math. 183 (1999), 1–43.CrossRefGoogle Scholar
[30]Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301–308.CrossRefGoogle Scholar
[31]Gorodetski, A.. Regularity of central leaves of partially hyperbolic sets and its applications. Izv. Ross. Akad. Nauk Ser. Mat. 70(6) (2006), 19–44  (Engl. transl. Izv. Math. 70(6) (2006), 1093–1116).Google Scholar
[32]Gorodetski, A. and Ilyashenko, Yu.. Some new robust properties of invariant sets and attractors of dynamical systems. Funct. Anal. Appl. 33 (1999), 16–30.CrossRefGoogle Scholar
[33]Gorodetski, A. and Ilyashenko, Yu.. Some properties of skew products over the horseshoe and solenoid. Proc. Steklov Inst. 231 (2000), 96–118.Google Scholar
[34]Gorodetski, A., Ilyashenko, Yu., Kleptsyn, V. and Nalsky, M.. Nonremovable zero Lyapunov esponents. Funct. Anal. Appl. 39 (2005), 27–38.CrossRefGoogle Scholar
[35]Gonchenko, S. V., Ovsyannikov, I. I., Simó, C. and Turaev, D. V.. Three-dimensional Hénon-like maps and wild Lorenz-like attractors. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), 3493–3508.CrossRefGoogle Scholar
[36]Gonchenko, S. V., Shilnikov, L. P. and Turaev, D. V.. On models with nonrough Poincaré homoclinic curves (Homoclinic chaos (Brussels, 1991)). Phys. D 62 (1993), 1–14.CrossRefGoogle Scholar
[37]Hayashi, S.. Connecting invariant manifolds and the solution of the C 1-stability and Ω-stability conjectures for flows. Ann. of Math. (2) 145 (1997), 81–137.CrossRefGoogle Scholar
[38]Kaloshin, V.. Generic diffeomorphisms with superexponential growth of number of periodic points. Comm. Math. Phys. 211 (2000), 253–271.CrossRefGoogle Scholar
[39]Kleptsyn, V. and Nalsky, M.. Robustness of nonhyperbolic measures for C 1-diffeomorphisms. Funct. Anal. Appl. 41(4) (2007), 30–45.CrossRefGoogle Scholar
[40]Katok, A. and Stepin, A.. Approximation of ergodic dynamical systems by periodic transformations. Dokl. Akad. Nauk. SSSR 171 (1966), 1268–1271.Google Scholar
[41]Katok, A. and Stepin, A.. Approximations in ergodic theory. Russian Math. Surveys 22 (1967), 81–106.CrossRefGoogle Scholar
[42]Liao, S. T.. Obstruction sets (II). Acta Sci. Nat. Univ. Pekin 2 (1981), 1–36.Google Scholar
[43]Mañé, R.. An ergodic closing lemma. Ann. of Math. (2) 116 (1982), 503–540.CrossRefGoogle Scholar
[44]Mañé, R.. Ergodic Theory and Differentiable Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 8). Springer, Berlin, 1987.CrossRefGoogle Scholar
[45]Nalsky, M.. Non-hyperbolic invariant measures on a maximal attractor, Preprint (arXiv: 0807.4963).Google Scholar
[46]Newhouse, S.. Non-density of Axiom A(a) on S 2. Proc. Amer. Math. Soc., Symp. Pure Math. 14 (1970), 191–202.CrossRefGoogle Scholar
[47]Newhouse, S.. Hyperbolic Limit Sets. Trans. Amer. Math. Soc. 167 (1972), 125–150.CrossRefGoogle Scholar
[48]Newhouse, S.. The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101–151.CrossRefGoogle Scholar
[49]Oseledec, V. I.. A multiplicative ergodic theorem. Characteristic Lyapunov exponents of dynamical systems. Trudy Moskov. Mat. Ob. 19 (1968), 179–210 (in Russian).Google Scholar
[50]Palis, J.. A global view of dynamics and a conjecture on the denseness of finitude of attractors (Géométrie complexe et systémes dynamiques (Orsay, 1995)). Asterisque 261 (2000), 335–347.Google Scholar
[51]Pesin, Ya.. Characteristic Lyapunov exponents and smooth ergodic theory. Usp. Mat. Nauk 32 (1977), 55–112.Google Scholar
[52]Pujals, E. R. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151 (2000), 961–1023.CrossRefGoogle Scholar
[53]Sakai, K.. C 1-stable shadowing chain components. Ergod. Th. & Dynam. Sys. 28(3) (2008), 987–1029.CrossRefGoogle Scholar
[54]Shub, M. and Wilkinson, A.. Pathological foliations and removable zero exponents. Invent. Math. 139 (2000), 495–508.CrossRefGoogle Scholar
[55]Takens, F.. Heteroclinic attractors: time averages and moduli of topological conjugacy. Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), 107–120.CrossRefGoogle Scholar
[56]Yang, J.. Ergodic measures far away from tangencies. PhD Thesis, IMPA.Google Scholar
[57]Yuan, G.-C. and Yorke, J. A.. An open set of maps for which every point is absolutely nonshadowable. Proc. Amer. Math. Soc. 128 (2000), 909–918.CrossRefGoogle Scholar