Borel complexity of the family of attractors for weak IFSs (Q2116379)

It is known that the family IFS\(^d\) of attractors for iterated function systems acting on a space \([0,1]^d\) is a meager \(F_\sigma\) subset of the space \(K([0,1]^d)\) of compact subsets of \([0,1]^d\) equipped with the Hausdorff metric, see [\textit{E. D'Aniello} and \textit{T. H. Steele}, J. Fractal Geom. 3, No. 2, 95--117 (2016; Zbl 1345.28014)]. In the reviewed article, the authors consider an analogous problem for weak iterated function systems. Let us recall basic definitions.\par Let \((X,d)\) be a metric space. A finite set of functions \(\{ s_1,\ldots,s_k\}\) acting on \(X\) is called a weak iterated function system if for each \(i\le k\) and every pair of distinct points \(x_0,x_1\in X\) the inequality \(d(s_i(x_0),s_i(x_1))< d(x_0,x_1)\) holds. A compact set \(A\subset X\) for which \(A=\bigcup_{i\le k}s_i[A]\) is called an attractor for \(\{ s_1,\ldots, s_k\}\). \par Let wIFS\(^d\) denote the set of all attractors for weak iterated function systems acting on \([0, 1]^d\), i.e., all compact sets \(A\subset [0,1]^d\) for which there is a weak iterated function system \(\{ s_1,\ldots, s_k\}\) acting on \([0,1]^d\) with \(A=\bigcup_{i\le k}s_i[A]\). \par The authors show that:\par (i) for \(d=1\), wIFS\(^1\) is a \(G_{\delta\sigma}\)-hard analytic subset of the space \(K([0,1])\); \par (ii) for \(d>1\), wIFS\(^d\) is a \(G_{\delta,\sigma}\)-hard subset of \(K([0,1]^d)\). \par Therefore, for any \(d\in\mathbb{N}\), wIFS is not \(F_{\sigma,\delta}\) subset of \(K([0,1])^d\), which means that wIFS\(^d\) is more complicated object than IFS\(^d\). An open problem remains whether wIFS\(^d\) is a Borel set.