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 $K([0,1])$.