Small and minimal attractors of an IFS

Abstract

Lasota and Myjak demonstrated that one can study attractors to an iterated function system (IFS) possibly containing discontinuous functions. We continue this line of thought by working with IFS with a lower semicontinuous Hutchinson-Barnsley operator and two new (but not significantly different) types of attractors, the small attractor (the smallest closed nonempty invariant set) and minimal attractors (a minimal closed nonempty invariant set). We characterize exactly when an IFS possesses a small attractor and provide several practically verifiable sufficient conditions for this. To study minimal attractors we create the notion of the weak basin; we show a minimal attractor behaves much like a small attractor on its weak basin and that the weak basin is the largest set in which a minimal attractor behaves like this. We then give a characterization of when the weak basin is open. Further, we show that, when the iterates of the Hutchinson-Barnsley operator of an IFS form an equicontinuous set of multifunctions, both the weak basin and the point wise basin (of a point wise attractor) is closed.