CHAPTER 1 Introduction In the early 1980’s, Herman [1] and Grebogi et al. [2] independently discovered the existence of strange non-chaotic attractors (SNA’s) in quasiperiodically forced (qpf) systems. These objects combine a complicated geometry1 with non-chaotic dynamics, a combination which is rather unusual and has only been observed in a few very particular cases before (the most prominent example is the Feigenbaum map, see [3] for a discussion and further references). In quasiperiodically forced systems, however, they seem to occur quite frequently and even over whole inter- vals in parameter space [2, 4, 5]. As a novel phenomenon this evoked considerable interest in theoretical physics, and in the sequel a large number of numerical stud- ies explored the surprisingly rich dynamics of these relatively simple maps. In particular, the widespread existence of SNA’s was confirmed both numerically (see [6]–[19], just to give a selection) and even experimentally [21, 22, 23]. Further, it turned out that SNA play an important role in the bifurcations of invariant circles [5, 14, 18, 20]. The studied systems were either discrete time maps, such as the qpf logistic map [10, 13, 18] and the qpf Arnold circle map [5, 9, 12, 14], or skew product flows which are forced at two or more incommensurate frequencies. Especially the latter underline the significance of qpf systems for understanding real-world phenomena, as most of them were derived from models for different physical systems (e.g. quasiperiodically driven damped pendula and Josephson junctions [6, 7, 8] or Duﬃng oscillators [22]. Their Poincar´ e maps again give rise to discrete-time qpf systems, on which the present article will focus. However, despite all efforts there are still only very few mathematically rigorous results about the subject, with the only exception of qpf Schr¨odinger cocycles (see below). There are results concerning the regularity of invariant curves ([24], see also [25]), and there has been some progress in carrying over basic results from one-dimensional dynamics [26, 27, 28]. But so far, the two original examples in [1] and [2] remain the only ones for which the existence of SNA’s has been proved rigorously. In both cases, the arguments used were highly specific for the respective class of maps and did not allow for much further generalisation, nor did they give very much insight into the geometrical and structural properties of the attractors. The systems Herman studied in [1] were matrix cocycles, with quasiperiodic Schr¨ odinger cocycles as a special case. The linear structure of these systems and their intimate relation to Schr¨ odinger operators with quasiperiodic potential made it possible to use a fruitful blend of techniques from operator theory, dynamical 1 This means in particular that they are not a piecewise differentiable (or even continuous) sub-manifold of the phase space. 1

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