Borel complexity of the family of attractors for weak IFSs

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 $K([0,1])$.