  We give a systematic account of iterated function systems (IFS) of weak
contractions of different types (Browder, Rakotch, topological). We show that
the existence of attractors and asymptotically stable invariant measures, and
the validity of the random iteration algorithm ("chaos game"), can be obtained
rather easily for weakly contractive systems. We show that the class of
attractors of weakly contractive IFSs is essentially wider than the class of
classical IFSs' fractals. On the other hand, we show that, in reasonable
spaces, a typical compact set is not an attractor of any weakly contractive
IFS. We explore the possibilities and restrictions to break the contractivity
barrier by employing several tools from fixed point theory: geometry of balls,
average contractions, remetrization technique, ordered sets, and measures of
noncompactness. From these considerations it follows that while the existence
of invariant sets and invariant measures can be assured rather easily for
general iterated function systems under mild conditions, to establish the
existence of attractors and unique invariant measures is a substantially more
difficult problem. This explains the central role of contractive systems in the
theory of IFSs.
