Borel complexity of the family of attractors for weak IFSs

TL;DR

This paper investigates the Borel complexity of the set of attractors for weak iterated function systems, revealing it is highly complex and differs significantly from classical IFS attractors, especially in one dimension.

Contribution

It establishes the Borel complexity classification of weak IFS attractors, showing they are $G_{\delta ext{-}\sigma}$-hard and analytic in one dimension, highlighting their intricate topological nature.

Findings

wIFS$^d$ is $G_{ ext{ extdelta} ext{ extsigma}}$-hard in the hyperspace.

wIFS$^d$ is not $F_{ ext{ extsigma} ext{ extdelta}}$, unlike classical IFS attractors.

wIFS$^1$ is an analytic subset of the hyperspace.

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 $G_{\delta\sigma}$-hard in $K([0,1]^d)$, for all $d\in\mathbb{N}$. In particular, wIFS$^d$ is not $F_{\sigma\delta}$ (in contrast to the family IFS$^d$ of attractors for classical iterated function systems acting on $[0,1]^d$, which is $F_{\sigma}$). Moreover, we show that in the one-dimensional case, wIFS$^1$ is an analytic subset of $K([0,1])$.