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Izv. RAN. Ser. Mat., 2006, Volume 70, Issue 6, Pages 19–44 (Mi izv678)  

This article is cited in 16 scientific papers (total in 16 papers)

The regularity of central leaves of partially hyperbolic sets and its applications

A. S. Gorodetski

California Institute of Technology

Abstract: We consider partially hyperbolic maps which are close to the direct product of a hyperbolic map and an identity map and prove that their central leaves depend Hölder continuously on the base point in the $C^r$-metric. We use this result to construct an open set of diffeomorphisms with rather unusual properties (they have transitive sets with periodic points of different indices and orbits with zero Lyapunov exponent). This paper concludes a series of joint papers with Yu. S. Ilyashenko.

DOI: https://doi.org/10.4213/im678

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English version:
Izvestiya: Mathematics, 2006, 70:6, 1093–1116

Bibliographic databases:

UDC: 517.938
MSC: 37D30, 37C20
Received: 11.07.2001

Citation: A. S. Gorodetski, “The regularity of central leaves of partially hyperbolic sets and its applications”, Izv. RAN. Ser. Mat., 70:6 (2006), 19–44; Izv. Math., 70:6 (2006), 1093–1116

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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
    2. Ilyashenko Yu. S., Kleptsyn V. A., Saltykov P., “Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins”, J. Fixed Point Theory Appl., 3:2 (2008), 449–463  crossref  mathscinet  zmath  isi  elib  scopus
    3. 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
    4. 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
    5. Ilyashenko Yu., Negut A., “Invisible parts of attractors”, Nonlinearity, 23:5 (2010), 1199–1219  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Ilyashenko Yu., “Thick attractors of boundary preserving diffeomorphisms”, Indag. Math. (N.S.), 22:3-4 (2011), 257–314  crossref  mathscinet  zmath  isi  scopus
    7. 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
    8. Yu Ilyashenko, A Negut, “Hölder properties of perturbed skew products and Fubini regained”, Nonlinearity, 25:8 (2012), 2377  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. N. A. Solodovnikov, “Boundary-preserving mappings of a manifold with intermingling basins of components of the attractor, one of which is open”, Trans. Moscow Math. Soc., 75 (2014), 69–76  mathnet  crossref  elib
    10. Barrientos P.G., Ki Yu., Raibekas A., “Symbolic Blender- Horseshoes and Applications”, Nonlinearity, 27:12 (2014), 2805–2839  crossref  mathscinet  zmath  isi  scopus
    11. Volk D., “Persistent Massive Attractors of Smooth Maps”, Ergod. Theory Dyn. Syst., 34:2 (2014), 693–704  crossref  mathscinet  zmath  isi  scopus
    12. Ali Tahzibi, Andrey Gogolev, “Center Lyapunov exponents in partially hyperbolic dynamics”, JMD, 8:3/4 (2015), 549  crossref  mathscinet  scopus
    13. J. Math. Sci. (N. Y.), 209:6 (2015), 979–987  mathnet  crossref
    14. 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
    15. 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, eds. Katok A., Pesin Y., Hertz F., Amer Mathematical Soc, 2017, 111–121  crossref  mathscinet  zmath  isi
    16. Ilyashenko Yu., Shilin I., “Attractors and Skew Products”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, eds. Katok A., Pesin Y., Hertz F., Amer Mathematical Soc, 2017, 155–175  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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