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