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

Abstract

For n, d ∈ N we consider the families:• L d n of attractors for iterated function systems (IFS) consisting of n contractions acting on [0, 1] d ,• wL d n 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 in the Hausdorff metric. In particular, we show thatWhat is more, we construct a compact set belonging to L d 2 which is not an attractor for any wIFS. We present a diagram summarizing our considerations.