Coexistence of uncountably many attracting sets for skew-products 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.