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

TL;DR

This work analyzes and compares attractor sets for traditional iterated function systems (IFS) and weak iterated function systems (wIFS) on $[0,1]^d$ by examining the closures of the families $L_n^d$ and $wL_n^d$ in the hyperspace $K([0,1]^d)$ under the Hausdorff metric. It proves that $\\overline{L_n^d}=\\overline{wL_n^d}$ for all $n,d$ and that new attractors appear at each level, i.e., $L_{n+1}^d\\setminus\\overline{L_n^d}\\neq\\emptyset$, while also producing a compact set in $\\overline{L_2^d}$ that is not an attractor for any wIFS. A key construction shows $wL_2^d\\setminus IFS^d\\neq\\emptyset$, and an explicit $X\\in\\overline{L^1_2}$ is given which is not a wIFS attractor, highlighting qualitative differences between the two attractor families. The results offer a precise boundary between IFS and wIFS attractors and provide a diagram illustrating the relationships among the various sets.

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.

Figures (1)

• Figure :

Theorems & Definitions (12)

• Lemma 2.1
• Remark
• Lemma 2.2
• proof
• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof