Milnor Attractors of Skew Products with the Fiber a Circle

TL;DR

This paper investigates the dynamics of generic skew products with circle fibers over Anosov diffeomorphisms, establishing conditions under which their Milnor attractors are Lyapunov stable and coincide with statistical attractors, with implications for transitivity.

Contribution

It proves that for a broad class of partially hyperbolic skew products with circle fibers, the Milnor attractor equals the statistical attractor and is Lyapunov stable, extending understanding of their long-term behavior.

Findings

Milnor attractor coincides with the statistical attractor in generic cases

The attractor is either of zero Lebesgue measure or the entire phase space

Such skew products are either transitive or have a zero measure non-wandering set

Abstract

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.