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Funktsional. Anal. i Prilozhen., 2007, Volume 41, Issue 4, Pages 30–45 (Mi faa2877)  

This article is cited in 18 scientific papers (total in 18 papers)

Stability of Existence of Nonhyperbolic Measures for $C^1$-Diffeomorphisms

V. A. Kleptsynabcd, M. B. Nalskyab

a M. V. Lomonosov Moscow State University
b Independent University of Moscow
c CNRS — Unit of Mathematics, Pure and Applied
d University of Geneva

Abstract: In the space of diffeomorphisms of an arbitrary closed manifold of dimension $\ge3$, we construct an open set such that each diffeomorphism in this set has an invariant ergodic measure with respect to which one of the Lyapunov exponents is zero. These diffeomorphisms are constructed to have a partially hyperbolic invariant set on which the dynamics is conjugate to a soft skew product with fiber the circle. It is the central Lyapunov exponent that proves to be zero in this case, and the construction is based on an analysis of properties of the corresponding skew products.

Keywords: Lyapunov exponent, partial hyperbolicity, dynamical system, skew product

DOI: https://doi.org/10.4213/faa2877

Full text: PDF file (275 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2007, 41:4, 271–283

Bibliographic databases:

UDC: 519.987.5+517.938.5
Received: 10.04.2006

Citation: V. A. Kleptsyn, M. B. Nalsky, “Stability of Existence of Nonhyperbolic Measures for $C^1$-Diffeomorphisms”, Funktsional. Anal. i Prilozhen., 41:4 (2007), 30–45; Funct. Anal. Appl., 41:4 (2007), 271–283

Citation in format AMSBIB
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    Erratum

    This publication is cited in the following articles:
    1. A. S. Gorodetski, “The regularity of central leaves of partially hyperbolic sets and its applications”, Izv. Math., 70:6 (2006), 1093–1116  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Ilyashenko Yu., “Some open problems in real and complex dynamical systems”, Nonlinearity, 21:7 (2008), T101–T107  crossref  mathscinet  zmath  isi  scopus
    3. Zmarrou H., Homburg A.J., “Dynamics and bifurcations of random circle diffeomorphism”, Discrete Contin. Dyn. Syst. Ser. B, 10:2-3 (2008), 719–731  mathscinet  zmath  isi  elib
    4. Diaz L.J., Gorodetski A., “Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes”, Ergodic Theory Dynam. Systems, 29:5 (2009), 1479–1513  crossref  mathscinet  zmath  isi  elib  scopus
    5. Bonatti C., Grines V., Pécou E., “Non-hyperbolic ergodic measures with large support”, Nonlinearity, 23:3 (2010), 687–710  crossref  mathscinet  adsnasa  isi  elib  scopus
    6. A. V. Osipov, “Nondensity of the orbital shadowing property in $C^1$-topology”, St. Petersburg Math. J., 22:2 (2011), 267–292  mathnet  crossref  mathscinet  zmath  isi
    7. Ilyashenko Yu., Negut A., “Invisible parts of attractors”, Nonlinearity, 23:5 (2010), 1199–1219  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Diaz L.J. Gelfert K., “Porcupine-Like Horseshoes: Topological and Ergodic Aspects”, Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics, 54, ed. Ibanez S. DelRio J. Pumarino A. Rodriguez J., Springer-Verlag Berlin, 2013, 199–219  crossref  mathscinet  zmath  isi  scopus
    9. Bochi J. Bonatti Ch. Diaz L.J., “Robust Vanishing of All Lyapunov Exponents for Iterated Function Systems”, Math. Z., 276:1-2 (2014), 469–503  crossref  mathscinet  zmath  isi  scopus
    10. Bochi J. Bonatti Ch. Diaz L.J., “Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures”, Commun. Math. Phys., 344:3 (2016), 751–795  crossref  mathscinet  zmath  isi  scopus
    11. A. V. Okunev, I. S. Shilin, “On the attractors of step skew products over the Bernoulli shift”, Proc. Steklov Inst. Math., 297 (2017), 235–253  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. L. J. Díaz, K. Gelfert, M. Rams, “Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products”, Proc. Steklov Inst. Math., 297 (2017), 98–115  mathnet  crossref  crossref  mathscinet  isi  elib
    13. Gorodetski A. Pesin Ya., “Path Connectedness and Entropy Density of the Space of Hyperbolic Ergodic Measures”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, ed. Katok A. Pesin Y. Hertz F., Amer Mathematical Soc, 2017, 111–121  crossref  mathscinet  zmath  isi
    14. Ilyashenko Yu. Shilin I., “Attractors and Skew Products”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, ed. Katok A. Pesin Y. Hertz F., Amer Mathematical Soc, 2017, 155–175  crossref  mathscinet  zmath  isi  scopus
    15. Diaz L.J. Gelfert K. Rams M., “Nonhyperbolic Step Skew-Products: Ergodic Approximation”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 34:6 (2017), 1561–1598  crossref  mathscinet  zmath  isi  scopus
    16. Christian Bonatti, Lorenzo J. Díaz, Jairo Bochi, “A criterion for zero averages and full support of ergodic measures”, Mosc. Math. J., 18:1 (2018), 15–61  mathnet  crossref
    17. Bonatti Ch. Zhang J., “Periodic Measures and Partially Hyperbolic Homoclinic Classes”, Trans. Am. Math. Soc., 372:2 (2019), 755–802  crossref  isi
    18. Cheng Ch. Crovisier S. Gan Sh. Wang X. Yang D., “Hyperbolicity Versus Non-Hyperbolic Ergodic Measures Inside Homoclinic Classes”, Ergod. Theory Dyn. Syst., 39:7 (2019), 1805–1823  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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