Attractors and Semi-Attractors of IFS

Abstract

It is well known that a finite set of contractive self maps on a metric space, called an iterated function system (IFS), admit a nonempty compact invariant set called the attractor of the IFS. It is also well known that the chaos game converges to "draw" the attractor. We examine generalized notions of IFSs, attractors and the convergence of the chaos game to these generalized attractors. We focus on IFSs whose Hutchinson operator is a lower semi continuous multifunction, this includes infinite and possibly discontinuous IFS. In this case we develop several characterizations of smallest/minimal nonempty closed sub-invariant sets of the IFS. Under the same assumptions, we then give some necessary conditions for the chaos game to converge. Then, under the assumption that the set of all finite compositions of functions in the IFS are equicontinuous and certain compactness assumptions, we establish that the chaos game converges.