Transitive sets for quasi-periodically forced monotone maps

Abstract

There has been considerable interest in recent years in quasi-periodically forced systems, partly due to the fact that these commonly exhibit strange non-chaotic attractors. Relatively little is known rigorously about such systems. In this paper we concentrate on investigating the structure of the simplest possible invariant sets for a particular class of quasi-periodically forced maps, namely those that are monotone in each fibre. Due to the quasi-periodic nature of the forcing, periodic orbits cannot occur, and their role is played by various types of invariant graph. Any compact invariant set is bounded by two invariant graphs, which are respectively upper and lower semi-continuous. If the set is minimal, these two graphs intersect on a residual set, on which both are continuous. Any transitive set Ω contains either one or two minimal sets, which must be the closure of one or the other of the boundaries of Ω. If Ω contains only one minimal set, then again its upper and lower boundaries intersect on a residual set. This case contains the original example of Grebogi et al. and its generalizations by Keller and by Glendinning. If Ω contains two minimal sets, then its upper and lower boundaries cannot intersect, though as far as we are aware, there is no known example where the existence of such a transitive set has been proven rigorously.