  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.
