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

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Funktsional'nyi Analiz i ego Prilozheniya, 2007, Volume 41, Issue 4, Pages 30–45
DOI: https://doi.org/10.4213/faa2877 (Mi faa2877)

This article is cited in 26 scientific papers (total in 26 papers)

Stability of Existence of Nonhyperbolic Measures for

C^{1}

$C^1$ -Diffeomorphisms

V. A. Kleptsyn^abcd, M. B. Nalsky^ab

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

Full-text PDF (275 kB) Citations (26)

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

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.

Received: 10.04.2006

English version:
Functional Analysis and Its Applications, 2007, Volume 41, Issue 4, Pages 271–283
DOI: https://doi.org/10.1007/s10688-007-0025-8

Bibliographic databases:

Document Type: Article

UDC: 519.987.5+517.938.5

Language: Russian

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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Linking options:

https://www.mathnet.ru/eng/faa2877

https://doi.org/10.4213/faa2877

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Erratum

Letter to the Editors
V. A. Kleptsyn, M. B. Nalsky
Funktsional. Anal. i Prilozhen., 2005, 39:2, 95

This publication is cited in the following 26 articles:

Abbas Fakhari, Maryam Khalaj, “Connectedness of the Set of Central Lyapunov Exponents”, J Dyn Control Syst, 30:4 (2024)
Martha Łącka, “Non-hyperbolic ergodic measures with the full support and positive entropy”, Monatsh Math, 200:1 (2023), 163
Ali Tahzibi, Jinhua Zhang, “Disintegrations of non‐hyperbolic ergodic measures along the center foliation of DA maps”, Bulletin of London Math Soc, 55:3 (2023), 1404
L. J. Díaz, K. Gelfert, M. Rams, “Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles”, Commun. Math. Phys., 394:1 (2022), 73
Barrientos P.G. Malicet D., “Extremal Exponents of Random Products of Conservative Diffeomorphisms”, Math. Z., 296:3-4 (2020), 1185–1207
Wang X. Zhang J., “Ergodic Measures With Multi-Zero Lyapunov Exponents Inside Homoclinic Classes”, J. Dyn. Differ. Equ., 32:2 (2020), 631–664
Bonatti Ch. Zhang J., “Periodic Measures and Partially Hyperbolic Homoclinic Classes”, Trans. Am. Math. Soc., 372:2 (2019), 755–802
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
Bonatti Ch. Zhang J., “On the Existence of Non-Hyperbolic Ergodic Measures as the Limit of Periodic Measures”, Ergod. Theory Dyn. Syst., 39:11 (2019), 2932–2967
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
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
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
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
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
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
Bochi J. Bonatti Ch. Diaz L.J., “Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures”, Commun. Math. Phys., 344:3 (2016), 751–795
Ali Tahzibi, Andrey Gogolev, “Center Lyapunov exponents in partially hyperbolic dynamics”, JMD, 8:3/4 (2015), 549
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
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
Bonatti C., Grines V., Pécou E., “Non-hyperbolic ergodic measures with large support”, Nonlinearity, 23:3 (2010), 687–710

Citing articles in Google Scholar: Russian citations, English citations
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Functional Analysis and Its Applications

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