2004/02/23 by Tobias Jaeger, Tobias H. Jaeger, Jaeger, Tobias H.
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS #msc:37C60 #msc:37C70
paper · pdf · doi:10.48550/arxiv.math/0402360
arxiv created 2004/02/23 · arxiv updated 2009/12/01
The existence of non-continuous invariant graphs (or strange non-chaotic attractors) in quasiperiodically forced systems has generated great interest, but there are still very few rigorous results about the properties of these objects. In particular, it is not known whether the topological closure of such graphs is typically a filled-in set, i.e consists of a single interval on every fibre, or not. We give a positive answer to this question for the class of so-called pinched skew products, where non-continuous invariant graphs occur generically, provided the rotation number on the base is diophantine and the system satisfies some additional conditions. For typical parameter families these conditions translate to a lower bound on the parameter. On the other hand, we also construct examples, where the non-continuous invariant graphs contain isolated points, such that their topological closure cannot be filled in.