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

TL;DR

This paper compares the sets of attractors generated by systems of contractions and weak contractions, showing their closures coincide and identifying differences in their attractor sets.

Contribution

It establishes that the closures of attractor families for contractions and weak contractions are equal and constructs examples illustrating their distinctions.

Findings

Closures of $L_n^d$ and $wL_n^d$ are equal.

There exist attractors in $ar{L}_2^d$ not realizable by weak IFS.

The set of attractors for systems with $n+1$ contractions is not contained in the closure of those with $n$ 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 $[0,1]^d$, - $wL_n^d$ 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 $\overline{L_n^d}=\overline{wL_n^d}$ and $L_{n+1}^d\setminus\overline{L_n^d}\neq\emptyset$, 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.