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On the attractors of step skew products over the Bernoulli shift

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Abstract

We study the statistical and Milnor attractors of step skew products over the Bernoulli shift. In the case when the fiber is a circle, we prove that for a topologically generic step skew product the statistical and Milnor attractors coincide and are Lyapunov stable. To this end we study some properties of the projection of the attractor onto the fiber, which might be of independent interest. In the case when the fiber is a segment, we give a description of the Milnor attractor as the closure of the union of graphs of finitely many almost everywhere defined functions from the base of the skew product to the fiber.

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References

  1. V. I. Arnold, V. S. Afraimovich, Yu. S. Il’yashenko, and L. P. Shil’nikov, “Bifurcation theory,” Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 5, 5–218 (1986); Engl. transl. in Dynamical Systems V (Springer, Berlin,1994), Encycl. Math. Sci. 5, pp. 1–205.

    MathSciNet  Google Scholar 

  2. C. Bonatti, L. J. Díaz, and M. Viana, Dynamics beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective (Springer, Berlin, 2005).

    MATH  Google Scholar 

  3. A. Bonifant and J. Milnor, “Schwarzian derivatives and cylinder maps,” in Holomorphic Dynamics and Renormalization (Am. Math. Soc., Providence, RI, 2008), Fields Inst. Commun. 53, pp. 1–21.

    Google Scholar 

  4. B. Deroin, “Propriétés ergodiques des groupes de difféomorphismes du cercle par rapport à la mesure de Lebesgue,” Preprint (2008), http://www.math.ens.fr/~deroin/Publications/coursneuchatel.pdf

    Google Scholar 

  5. H. Furstenberg, “Noncommuting random products,” Trans. Am. Math. Soc. 108, 377–428 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Gorodetski and Yu. Ilyashenko, “Minimal and strange attractors,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6 (6), 1177–1183 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. S. Gorodetski and Yu. S. Ilyashenko, “Certain new robust properties of invariant sets and attractors of dynamical systems,” Funkts. Anal. Prilozh. 33 (2), 16–30 (1999) [Funct. Anal. Appl. 33, 95–105 (1999)].

    Article  Google Scholar 

  8. Yu. Ilyashenko, “Thick attractors of step skew products,” Regul. Chaotic Dyn. 15 (2–3), 328–334 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  9. Yu. Ilyashenko, “Thick attractors of boundary preserving diffeomorphisms,” Indag. Math., New Ser. 22 (3–4), 257–314 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  10. Yu. S. Ilyashenko, V. A. Kleptsyn, and P. Saltykov, “Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins,” J. Fixed Point Theory Appl. 3 (2), 449–463 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Kan, “Open sets of diffeomorphisms having two attractors, each with everywhere dense basin,” Bull. Am. Math. Soc. 31, 68–74 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge Univ. Press, Cambridge, 1997).

    MATH  Google Scholar 

  13. V. A. Kleptsyn and M. B. Nalski, “Persistence of nonhyperbolic measures for C1-diffeomorphisms,” Funkts. Anal. Prilozh. 41 (4), 30–45 (2007) [Funct. Anal. Appl. 41, 271–283 (2007)].

    Article  Google Scholar 

  14. V. Kleptsyn and D. Volk, “Physical measures for nonlinear random walks on interval,” Moscow Math. J. 14 (2), 339–365 (2014).

    MathSciNet  MATH  Google Scholar 

  15. V. Kleptsyn and D. Volk, “Nonwandering sets of interval skew products,” Nonlinearity 27 (7), 1595–1601 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  16. Yu. G. Kudryashov, “Bony attractors and magical billiards,” Cand. Sci. (Phys.–Math.) Dissertation (Moscow State Univ., Moscow, 2011).

    Google Scholar 

  17. K. Kuratowski, Topology II (Academic, New York, 1968).

    Google Scholar 

  18. J. Milnor, “On the concept of attractor,” Commun. Math. Phys. 99, 177–195 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. S. Minkov and A. V. Okunev, “Omega-limit sets of generic points of partially hyperbolic diffeomorphisms,” Funkts. Anal. Prilozh. 50 (1), 59–66 (2016) [Funct. Anal. Appl. 50, 48–53 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  20. C. A. Morales and M. J. Pacifico, “Mixing attractors for 3-flows,” Nonlinearity 14 (2), 359–378 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Okunev, “Milnor attractors of skew products with the fiber a circle,” J. Dyn. Control Syst. 23 (2), 421–433 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  22. R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd ed. (Springer, Berlin, 1993), Lect. Notes Math. 1364.

    MATH  Google Scholar 

  23. B. Santiago, “The semicontinuity lemma,” Preprint (2012), http://www.im.ufrj.br/~bruno_santiago/Semicontinuity.pdf

    Google Scholar 

  24. I. Shilin, “Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors,” arXiv: 1604.02437 [math.DS].

  25. M. Viana and J. Yang, “Physical measures and absolute continuity for one-dimensional center direction,” Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30 (5), 845–877 (2013).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. National Research University “Higher School of Economics,”, ul. Myasnitskaya 20, Moscow, 101000, Russia

    A. V. Okunev

  2. Moscow Center for Continuous Mathematical Education, Bol’shoi Vlas’evskii per. 11, Moscow, 119002, Russia

    I. S. Shilin

Authors
  1. A. V. Okunev
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  2. I. S. Shilin
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Correspondence to A. V. Okunev.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 297, pp. 260–280.

To the memory of Dmitry Anosov

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Okunev, A.V., Shilin, I.S. On the attractors of step skew products over the Bernoulli shift. Proc. Steklov Inst. Math. 297, 235–253 (2017). https://doi.org/10.1134/S0081543817040149

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  • DOI: https://doi.org/10.1134/S0081543817040149

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