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.
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, Trudy Mat. Inst. Steklova, 297, MAIK Nauka/Interperiodica, Moscow, 2017, 260–280; Proc. Steklov Inst. Math., 297 (2017), 235–253
\Bibitem{OkuShi17}
\by A.~V.~Okunev, I.~S.~Shilin
\paper On the attractors of step skew products over the Bernoulli shift
\inbook Order and chaos in dynamical systems
\bookinfo Collected papers. On the occasion of the 125th anniversary of the birth of Academician Dmitry Victorovich Anosov
\serial Trudy Mat. Inst. Steklova
\yr 2017
\vol 297
\pages 260--280
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\crossref{https://doi.org/10.1134/S0371968517020145}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2017
\vol 297
\pages 235--253
\crossref{https://doi.org/10.1134/S0081543817040149}
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Linking options:
https://www.mathnet.ru/eng/tm3804
https://doi.org/10.1134/S0371968517020145
https://www.mathnet.ru/eng/tm/v297/p260
This publication is cited in the following 3 articles:
M. Zaj, F. H. Ghane, “Non Hyperbolic Solenoidal Thick Bony Attractors”, Qual. Theory Dyn. Syst., 18:1 (2019), 35
Ivan Shilin, “Locally topologically generic diffeomorphisms with Lyapunov unstable Milnor attractors”, Mosc. Math. J., 17:3 (2017), 511–553
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