Uniformization of Sierpiski carpets in the plane

Let S ^sub i^, iI, be a countable collection of Jordan curves in the extended complex plane (ProQuest: Formulae and/or non-USASCII text omitted; see image) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then t...

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Bibliographic Details
Published inInventiones mathematicae Vol. 186; no. 3; p. 559
Main Author Bonk, Mario
Format Journal Article
LanguageEnglish
Published Heidelberg Springer Nature B.V 01.12.2011
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Summary:Let S ^sub i^, iI, be a countable collection of Jordan curves in the extended complex plane (ProQuest: Formulae and/or non-USASCII text omitted; see image) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map (ProQuest: Formulae and/or non-USASCII text omitted; see image) such that f(S ^sub i^) is a round circle for all iI. This implies that every Sierpiski carpet in (ProQuest: Formulae and/or non-USASCII text omitted; see image) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiski carpet by a quasisymmetric map.[PUBLICATION ABSTRACT]
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-011-0325-8