Non-removability of Sierpiński spaces
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- by Dimitrios Ntalampekos and Jang-Mei Wu PDF
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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.References
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Additional Information
- Dimitrios Ntalampekos
- Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794
- MR Author ID: 1259365
- Email: dimitrios.ntalampekos@stonybrook.edu
- Jang-Mei Wu
- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61822
- MR Author ID: 184770
- Email: jmwu@illinois.edu
- Received by editor(s): December 14, 2018
- Received by editor(s) in revised form: April 8, 2019
- Published electronically: July 10, 2019
- Additional Notes: The research of the second named author was partially supported by Simons Foundation Collaboration Grant $\# 353435$.
- Communicated by: Jeremy Tyson
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 203-212
- MSC (2010): Primary 30C65, 57N15; Secondary 54C99
- DOI: https://doi.org/10.1090/proc/14698
- MathSciNet review: 4042843