  We prove that for a generic skew product with circle fiber over an Anosov
diffeomorphism the Milnor attractor (also called the likely limit set)
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.
