Comparison of the sets of attractors for systems of contractions and weak contractions

Abstract

For n,d∈N we consider the families:•Lnd of attractors for iterated function systems (IFS) consisting of n contractions acting on [0,1]d,•wLnd of attractors for weak iterated function systems (wIFS) consisting of n weak contractions acting on [0,1]d.We study closures of the above families as subsets of the hyperspace K([0,1]d) of all compact subsets of [0,1]d equipped with the Hausdorff–Pompeiu metric. In particular, we show that Lnd¯=wLnd¯ and Ln+1d∖Lnd¯≠∅, for all n,d∈N. What is more, we construct a compact set belonging to L2d¯ which is not an attractor for any wIFS. We present a diagram summarizing our considerations.