Uniformization of Sierpiński carpets in the plane

By M. Bonk. 2010

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Let Si, 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(Si) 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c2e6c080b855af20806ca418b778391f7a8945ad

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Quasisymmetric rigidity of square Sierpinski carpets (2011, methodology): “sm of Sthat sends one family to the other family. The proof of Corollary 5.2 relies on some previous work. In Section 4 we collect certain uniformization and rigidity results that were established in [Bo1] and [BKM], and derive some consequences. Among these results is Proposition 4.9 which gives an”