Abstract
Let S i , i∈I, be a countable collection of Jordan curves in the extended complex plane \(\widehat{\mathbb{C}}\) 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 \(f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\) such that f(S i ) is a round circle for all i∈I. This implies that every Sierpiński carpet in \(\widehat{\mathbb{C}}\) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.
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The author was supported by NSF grants DMS 0244421, DMS 0456940, DMS 0652915, DMS 1058772, and DMS 1058283.
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Bonk, M. Uniformization of Sierpiński carpets in the plane. Invent. math. 186, 559–665 (2011). https://doi.org/10.1007/s00222-011-0325-8
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DOI: https://doi.org/10.1007/s00222-011-0325-8