Non-removability of Sierpinski spaces

2018/12/31 by Dimitrios Ntalampekos, Jang-Mei Wu
Mathematics · #math.MG #math.CV #math.GN #msc:30C65 #msc:57N15 #msc:54C99

paper · pdf · doi:10.1090/proc/14698

published as Proc. Amer. Math. Soc. 148 (2020), 203-212 · 10 pages

arxiv created 2019/05/20 · arxiv updated 2020/07/24

Abstract

We prove that all Sierpiński spaces in \mathbbSⁿ, n≥ 2, are non-removable for (quasi)conformal maps, generalizing the result of the first named author arXiv:1809.05605. More precisely, we show that for any Sierpiński space X⊂ \mathbbSⁿ there exists a homeomorphism f\colon \mathbbSⁿ→ \mathbbSⁿ, conformal in \mathbbSⁿ∖ X, that maps 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.

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