Uniformization of Sierpiński carpets in the plane

• DOI: 10.1007/s00222-011-0325-8
• Publication Date: 2011-04-07T23:54:29Z
• Authors: Mario Bonk
• Keywords: Disjoint sets; Uniformization (probability theory)

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 regions. If are uniform quasicircles and uniformly relatively separated, then there exists quasiconformal map $f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}$ such f(S ) is round circle for all i∈I. This implies every Sierpiński carpet whose peripheral circles separated can mapped to by quasisymmetric map.