Abstract
For a generic skew product with the fiber a circle over an Anosov diffeomorphism, we prove that the Milnor attractor coincides with the statistical attractor, is Lyapunov stable, and either has zero Lebesgue measure or coincides with the whole phase space. As a consequence, we conclude that such skew product is either transitive or has non-wandering set of zero measure. The result is proved under the assumption that the fiber maps preserve the orientation of the circle, and the skew product is partially hyperbolic.
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Notes
This lemma is a slightly improved version of the following statement by S. Minkov: if the Milnor attractor has positive measure, the attractor of the inverse diffeomorphism either has positive measure or is Lyapunov unstable.
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Acknowledgments
The author is grateful to professor Yu.S. Ilyashenko for constant attention to this work and to I. Shilin and S. Minkov for useful discussions.
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The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2016-2017 (grant No 16-05-0066) and supported within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program
National Research University Higher School of Economics; supported by part by a grant of the Simons Foundation.
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Okunev, A. Milnor Attractors of Skew Products with the Fiber a Circle. J Dyn Control Syst 23, 421–433 (2017). https://doi.org/10.1007/s10883-016-9334-7
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DOI: https://doi.org/10.1007/s10883-016-9334-7