Uniformization of Sierpiński carpets in the plane

Abstract

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 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.

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

References in corpus (1)

• Quasisymmetric parametrizations of two-dimensional metric spheres

Cited by in corpus (7)

• Modulus and Poincaré inequalities on non-self-similar Sierpinski carpets
• Local rigidity for hyperbolic groups with Sierpiński carpet boundaries
• Almost uniform domains and Poincaré inequalities
• Quasisymmetric geometry of the Julia sets of McMullen maps
• Quasisymmetric rigidity of Sierpinski carpets $F_{n,p}$
• Conformal uniformization of planar packings by disk packings
• Non-removability of Sierpinski spaces