Uniformization of Sierpi\'nski carpets in the plane

TL;DR

This paper proves that Sierpiński carpets with uniformly relatively separated uniform quasicircle boundaries can be mapped to round carpets via quasiconformal maps, generalizing uniformization results for complex fractal structures.

Contribution

It establishes a uniformization theorem for Sierpiński carpets with specific geometric conditions, extending the class of fractals that can be conformally mapped to round carpets.

Findings

Existence of quasiconformal maps transforming carpets to round ones.

Characterization of carpets with uniformly relatively separated uniform quasicircle boundaries.

Extension of uniformization results to a broader class of fractals.

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 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.