We prove that for a generic skew product with circle fiber over an Anosovdiffeomorphism the Milnor attractor (also called the likely limit set)coincides with the statistical attractor, is Lyapunov stable, and either haszero Lebesgue measure or coincides with the whole phase space. As a consequencewe conclude that such skew product is either transitive or has non-wanderingset of zero measure. The result is proved under the assumption that the fibermaps preserve the orientation of the circle, and the skew product is partiallyhyperbolic.
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