Borel complexity of the family of attractors for weak IFSs

TL;DR

The authors study the descriptive complexity of the family of attractors for weak IFSs, comparing it to classical IFS attractors. They show that while IFS$^d$-attractors form an $F_{\sigma}$ set, the weak version wIFS$^d$ is $G_{\delta\sigma}$-hard (and not $F_{\sigma\delta}$) for all $d$, with wIFS$^1$ being analytic in $K([0,1])$. The core method is a continuous reduction from a canonical $F_{\sigma\delta}$-complete subset of $\mathcal{P}(\mathbb{N}^2)$ to wIFS$^1$ and then embedding this construction to higher dimensions via a product encoding, yielding $G_{\delta\sigma}$-completeness for all $d$. These results demonstrate that the family of weak IFS attractors is significantly more complex than the classical attractors, and the paper leaves open whether wIFS$^d$ is Borel for general $d$.

Abstract

This paper is an attempt to measure the difference between the family of iterated function systems attractors and a broader family, the set of attractors for weak iterated function systems. We discuss Borel complexity of the set wIFS$^d$ of attractors for weak iterated function systems acting on $[0,1]^d$ (as a subset of the hyperspace $K([0,1]^d)$ of all compact subsets of $[0,1]^d$ equipped in the Hausdorff metric). We prove that wIFS$^d$ is $G_{δσ}$-hard in $K([0,1]^d)$, for all $d\in\mathbb{N}$. In particular, wIFS$^d$ is not $F_{σδ}$ (in contrast to the family IFS$^d$ of attractors for classical iterated function systems acting on $[0,1]^d$, which is $F_σ$). Moreover, we show that in the one-dimensional case, wIFS$^1$ is an analytic subset of $K([0,1])$.

Theorems & Definitions (12)

• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Remark 1
• Remark 2
• Lemma 3.3
• proof
• Lemma 3.4
• proof