Non-removability of Sierpiński spaces

Ntalampekos, Dimitrios; Wu, Jang-Mei

doi:10.1090/proc/14698

Non-removability of Sierpiński spaces
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by Dimitrios Ntalampekos and Jang-Mei Wu PDF

Proc. Amer. Math. Soc. 148 (2020), 203-212 Request permission

Abstract:

We prove that all Sierpiński spaces in ${\mathbb {S}}^n$, $n\geq 2$, 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 $X\subset \mathbb {S}^n$ there exists a homeomorphism $f\colon \mathbb {S}^n\to \mathbb {S}^n$, conformal in $\mathbb {S}^n\setminus X$, 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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