LAUSR

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 $$^d$$ d of attractors for weak iterated function systems acting on $$[0,1]^d$$ [ 0 , 1 ] d (as a subset of the hyperspace $$K([0,1]^d)$$ K ( [ 0 , 1 ] d ) of all compact subsets of $$[0,1]^d$$ [ 0 , 1 ] d equipped with the Hausdorff metric). We prove that wIFS $$^d$$ d is $$G_{\delta\sigma}$$ G δ σ -hard in $$K([0,1]^d)$$ K ( [ 0 , 1 ] d ) , for all $${d\in\mathbb{N}}$$ d ∈ N . In particular,wIFS $$^d$$ d is not $$F_{\sigma\delta}$$ F σ δ (in contrast to the family IFS $$^d$$ d of attractors for classical iterated function systems acting on $$[0,1]^d$$ [ 0 , 1 ] d , which is $$F_{\sigma}$$ F σ ). Moreover, we show that in the one-dimensional case, wIFS $$^1$$ 1 is an analytic subset of $$K([0,1])$$ K ( [ 0 , 1 ] ) .