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

Abstract

For $n,d\in \mathbb{N}$ we consider the families: - $L_n^d$ of attractors for iterated function systems (IFS) consisting of $n$ contractions acting on $[0,1{]}^d$, - $w{L}_n^d$ 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 that $\overline{L_n^d}=\overline{w{L}_n^d}$ and $L_{n+1}^d\setminus \overline{L_n^d}\ne \varnothing$, for all $n,d\in \mathbb{N}$. What is more, we construct a compact set belonging to $\overline{L_2^d}$ which is not an attractor for any wIFS. We present a diagram summarizing our considerations.