Coexistence of uncountably many attracting sets for skew-products on the cylinder

TL;DR

This paper demonstrates that skew-product systems on the cylinder can have uncountably many attracting sets, each associated with an irrational rotation number, extending the understanding of complex attractor coexistence in quasiperiodic systems.

Contribution

It extends the existence of attracting sets to skew-products homotopic to the identity, showing uncountably many coexist for each irrational rotation number.

Findings

Uncountably many attracting sets coexist for different irrational rotation numbers.

Each attracting set corresponds to a quasiperiodically forced system with a specific rotation.

The results apply to skew-products homotopic to the identity on the cylinder.

Abstract

The aim of this paper is to show that the existence of attracting sets for quasiperiodically forced systems can be extended to appropriate skew-products on the cylinder, homotopic to the identity, in such a way that the general system will have (at least) one attracting set corresponding to every irrational rotation number \rho in the rotation interval of the base map. This attracting set is a copy of the attracting set of the system quasiperiodically forced by a (rigid) rotation of angle \rho. This shows the co-existence of uncountably many attracting sets, one for each irrational in the rotation interval of the basis.