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 Tr. Mat. Inst. Steklova, 2017, Volume 297, Pages 260–280 (Mi tm3804)

On the attractors of step skew products over the Bernoulli shift

A. V. Okuneva, I. S. Shilinb

a National Research University "Higher School of Economics", ul. Myasnitskaya 20, Moscow, 101000 Russia
b Moscow Center for Continuous Mathematical Education, Bol'shoi Vlas'evskii per. 11, Moscow, 119002 Russia

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.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00748-à This work was supported by the Russian Foundation for Basic Research, project no. 16-01-00748-a.

DOI: https://doi.org/10.1134/S0371968517020145

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English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 297, 235–253

Bibliographic databases:

UDC: 517.938

Citation: A. V. Okunev, I. S. Shilin, “On the attractors of step skew products over the Bernoulli shift”, Order and chaos in dynamical systems, Collected papers. On the occasion of the 125th anniversary of the birth of Academician Dmitry Victorovich Anosov, Tr. Mat. Inst. Steklova, 297, MAIK Nauka/Interperiodica, Moscow, 2017, 260–280; Proc. Steklov Inst. Math., 297 (2017), 235–253

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3804
• https://doi.org/10.1134/S0371968517020145
• http://mi.mathnet.ru/eng/tm/v297/p260

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This publication is cited in the following articles:
1. Ivan Shilin, “Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors”, Mosc. Math. J., 17:3 (2017), 511–553
2. 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
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