Attractors for classes of iterated function systems

We consider various classes of iterated function systems, such as those comprised of similarities and those satisfying the open set condition, and weakly contractive systems. Necessary properties are developed for a compact set to be an attractor for the various classes of iterated function systems. Since the Hausdorff metric space is complete, we are able to develop several results concerning typical compact sets and their invariance with respect to iterated function systems