Circle diffeomorphisms forced by expanding circle maps

TL;DR

This paper studies the dynamics of skew product maps where circle diffeomorphisms are driven by expanding circle maps, revealing robust mixing behavior and convergence properties through invariant attracting graphs.

Contribution

It introduces an open class of such systems that are topologically mixing and exhibit fiberwise convergence, constructed via invariant attracting graphs in the natural extension.

Findings

Systems are robustly topologically mixing.

Almost all points in fibers converge under iteration.

Invariant attracting graphs are constructed in the natural extension.

Abstract

We discuss dynamics of skew product maps defined by circle diffeomorphisms forced by expanding circle maps. We construct an open class of such systems that are robust topologically mixing and for which almost all points in the same fiber converge under iteration. This property follows from the construction of an invariant attracting graph in the natural extension, a skew product of circle diffeomorphisms forced by a solenoid homeomorphism.