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Funktsional. Anal. i Prilozhen., 2005, Volume 39, Issue 1, Pages 27–38 (Mi faa29)  

This article is cited in 29 scientific papers (total in 29 papers)

Nonremovable Zero Lyapunov Exponents

A. S. Gorodetskiab, Yu. S. Ilyashenkocad, V. A. Kleptsyneaf, M. B. Nalskye

a Independent University of Moscow
b California Institute of Technology
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
d Cornell University
e M. V. Lomonosov Moscow State University
f CNRS — Unit of Mathematics, Pure and Applied

Abstract: Skew products over a Bernoulli shift with a circle fiber are studied. We prove that in the space of such products there exists a nonempty open set of mappings each of which possesses an invariant ergodic measure with one of the Lyapunov exponents equal to zero. The conjecture that the space of $C^2$-diffeomorphisms of the $3$-dimensional torus into itself has a similar property is discussed.

Keywords: Lyapunov exponent, partially hyperbolic system, nonuniform hyperbolicity, dynamical system, skew product, Bernoulli diffeomorphism


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English version:
Functional Analysis and Its Applications, 2005, 39:1, 21–30

Bibliographic databases:

UDC: 517.5
Received: 24.05.2004

Citation: A. S. Gorodetski, Yu. S. Ilyashenko, V. A. Kleptsyn, M. B. Nalsky, “Nonremovable Zero Lyapunov Exponents”, Funktsional. Anal. i Prilozhen., 39:1 (2005), 27–38; Funct. Anal. Appl., 39:1 (2005), 21–30

Citation in format AMSBIB
\by A.~S.~Gorodetski, Yu.~S.~Ilyashenko, V.~A.~Kleptsyn, M.~B.~Nalsky
\paper Nonremovable Zero Lyapunov Exponents
\jour Funktsional. Anal. i Prilozhen.
\yr 2005
\vol 39
\issue 1
\pages 27--38
\jour Funct. Anal. Appl.
\yr 2005
\vol 39
\issue 1
\pages 21--30

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    This publication is cited in the following articles:
    1. Gonchenko S.V., Ovsyannikov I.I., Simó C., Turaev D., “Three-dimensional Henon-like maps and wild Lorenz-like attractors”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15:11 (2005), 3493–3508  crossref  mathscinet  zmath  isi  scopus
    2. 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
    3. V. A. Kleptsyn, M. B. Nalsky, “Stability of Existence of Nonhyperbolic Measures for $C^1$-Diffeomorphisms”, Funct. Anal. Appl., 41:4 (2007), 271–283  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. 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
    5. Díaz 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
    6. Bonatti Ch., Díaz L.J., Gorodetski A., “Non-hyperbolic ergodic measures with large support”, Nonlinearity, 23:3 (2010), 687–705  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Ilyashenko Yu., Negut A., “Invisible parts of attractors”, Nonlinearity, 23:5 (2010), 1199–1219  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. V. A. Kleptsyn, P. S. Saltykov, “On $C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps”, Trans. Moscow Math. Soc., 72 (2011), 193–217  mathnet  crossref  zmath  elib
    9. Homburg A.J., “Circle Diffeomorphisms Forced by Expanding Circle Maps”, Ergod. Theory Dyn. Syst., 32:Part 6 (2012), 2011–2024  crossref  mathscinet  zmath  isi  elib  scopus
    10. Ilyashenko Yu., Negut A., “Holder Properties of Perturbed Skew Products and Fubini Regained”, Nonlinearity, 25:8 (2012), 2377–2399  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. Diaz L.J., Gelfert K., “Porcupine-Like Horseshoes: Transitivity, Lyapunov Spectrum, and Phase Transitions”, Fundam. Math., 216:1 (2012), 55–100  crossref  mathscinet  zmath  isi  elib  scopus
    12. Diaz L.J., Gelfert K., Rams M., “Almost Complete Lyapunov Spectrum in Step Skew-Products”, Dynam. Syst., 28:1 (2013), 76–110  crossref  mathscinet  zmath  isi  elib  scopus
    13. Gonchenko S.V., Gonchenko A.S., Ovsyannikov I.I., Turaev D.V., “Examples of Lorenz-Like Attractors in Henon-Like Maps”, Math. Model. Nat. Phenom., 8:5 (2013), 48–70  crossref  mathscinet  zmath  isi  scopus
    14. Diaz L.J., Gelfert K., “Porcupine-Like Horseshoes: Topological and Ergodic Aspects”, Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics, 54, eds. Ibanez S., DelRio J., Pumarino A., Rodriguez J., Springer-Verlag Berlin, 2013, 199–219  crossref  mathscinet  zmath  isi  scopus
    15. 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
    16. Tatiana Golenishcheva-Kutuzova, Anton Gorodetski, Victor Kleptsyn, Denis Volk, “Translation numbers define generators of $F_k^+\to\mathrm{Homeo}_+(\mathbb S^1)$”, Mosc. Math. J., 14:2 (2014), 291–308  mathnet  mathscinet
    17. V. Kleptsyn, D. Volk, “Physical measures for nonlinear random walks on interval”, Mosc. Math. J., 14:2 (2014), 339–365  mathnet  crossref  mathscinet
    18. Ilyashenko Yu., Romaskevich O., “Sternberg Linearization Theorem for Skew Products”, J. Dyn. Control Syst., 22:3 (2016), 595–614  crossref  mathscinet  zmath  isi  elib  scopus
    19. 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
    20. Gharaei M. Homburg A.J., “Random interval diffeomorphisms”, Discret. Contin. Dyn. Syst.-Ser. S, 10:2 (2017), 241–272  crossref  mathscinet  zmath  isi  scopus
    21. 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
    22. 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
    23. 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
    24. 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
    25. 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
    26. Tian X., Wang Sh., Wang X., “Intermediate Lyapunov Exponents For Systems With Periodic Orbit Gluing Property”, Discret. Contin. Dyn. Syst., 39:2 (2019), 1019–1032  crossref  zmath  isi
    27. Diaz L.J. Gelfert K. Rams M., “Entropy Spectrum of Lyapunov Exponents For Nonhyperbolic Step Skew-Products and Elliptic Cocycles”, Commun. Math. Phys., 367:2 (2019), 351–416  crossref  mathscinet  zmath  isi  scopus
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    29. 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
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