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 with 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 \(K([0,1])\).