Uniformization of Sierpiński carpets in the plane

Details Uniformization of Sierpiński carpets in the plane Uniformization of Sierpiński carpets in the plane

2010-09-21

arxiv.org/abs/1009.4094v2

Mathematics

Complex Variables

Revised version. 89 pages. To appear in Invent. math

Scientific paper

Let $S_i$, $i\in I$, be a countable collection of Jordan curves in the extended complex plane $\Sph$ 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 $f\: \Sph\ra \Sph$ such that $f(S_i)$ is a round circle for all $i\in I$. This implies that every Sierpi\'nski carpet in $\oC$ whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpi\'nski carpet by a quasisymmetric map.

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