Non-removability of Sierpiński spaces

Abstract

We prove that all Sierpiński spaces in S n {\mathbb {S}}^n , n ≥ 2 n\geq 2 , are non-removable for (quasi)conformal maps, generalizing the result of the first named author [Non-removability of Sierpiński carpets, preprint, 2018]. More precisely, we show that for any Sierpiński space X ⊂ S n X\subset \mathbb {S}^n there exists a homeomorphism f : S n → S n f\colon \mathbb {S}^n\to \mathbb {S}^n , conformal in S n ∖ X \mathbb {S}^n\setminus X , that maps X X to a set of positive measure and is not globally (quasi)conformal. This is the first class of examples of non-removable sets in higher dimensions.