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Article

Milnor Attractors of Skew Products with the Fiber a Circle

Okunev A.

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.