Uniformization of Sierpiński carpets in the plane

2010/09/30 by Mario Bonk
Mathematics · #math.CV #msc:30A24

paper · pdf · doi:10.1007/s00222-011-0325-8

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

arxiv created 2011/03/07 · arxiv updated 2015/05/20

Abstract

Let S_i, i∈ 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∈ I. This implies that every Sierpiński carpet in \oC whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.

Cited by

• Uniformization of cofat domains on metric two-spheres