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 δσ -hard in K([0, 1] d ), for all d ∈ 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]).