  We study parameter families of quasiperiodically forced (qpf) circle maps
with Diophantine frequency. Under certain C1-open conditions concerning their
geometry, we prove that these families exhibit nonuniformly hyperbolic
behaviour, often referred to as the existence of strange nonchaotic attractors,
on parameter sets of positive measure. This provides a nonlinear version of
results by Young on quasiperiodic SL (2;R)-cocycles and complements previous
results in this direction which hold for sets of frequencies of positive
measure, but did not allow for an explicit characterisation of these
frequencies. As an application, we study a qpf version of the Arnold circle map
and show that the Arnold tongue corresponding to rotation number 1/2 collapses
on an open set of parameters. The proof requires to perform a parameter
exclusion with respect to some twist parameter and is based on the multiscale
analysis of the dynamics on certain dynamically defined critical sets. A
crucial ingredient is to obtain good control on the parameter-dependence of the
critical sets. Apart from the presented results, we believe that this step will
be important for obtaining further information on the behaviour of parameter
families like the qpf Arnold circle map.
