Strange non-chaotic attractors in quasi-periodically forced circle maps: Diophantine forcing

Abstract

AbstractWe study parameter families of quasi-periodically forced (qpf) circle maps with Diophantine frequency. Under certain $\mathcal {C}^1$-open conditions concerning their geometry, we prove that these families exhibit non-uniformly hyperbolic behaviour, often referred to as the existence of strange non-chaotic attractors, on parameter sets of positive measure. This provides a nonlinear version of results by Young on quasi-periodic $\mathrm {SL}(2,\mathbb {R})$-cocycles and complements previous results in this direction which hold for sets of frequencies of positive measure, but did not allow for an explicit characterization 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.