Quasiconformal characterization of Schottky sets

TL;DR

This paper characterizes when subsets of the 2-sphere are quasiconformally equivalent to Schottky sets, providing a new uniformization theorem applicable to Sierpiński carpets and gaskets without requiring uniform relative separation.

Contribution

It establishes a quasiconformal characterization of Schottky sets, extending uniformization results to gaskets and removing the need for uniform relative separation.

Findings

Provides a necessary and sufficient condition for quasiconformal equivalence to Schottky sets.

Applies to Sierpiński carpets and gaskets, yielding new uniformization results.

Contains Bonk's uniformization as a special case, broadening previous work.

Abstract

The complement of the union of a collection of disjoint open disks in the $2$-sphere is called a Schottky set. We prove that a subset $S$ of the $2$-sphere is quasiconformally equivalent to a Schottky set if and only if every pair of complementary components of $S$ can be mapped to a pair of open disks with a uniformly quasiconformal homeomorphism of the sphere. Our theorem applies to Sierpi\'nski carpets and gaskets, yielding for the first time a general quasiconformal uniformization result for gaskets. Moreover, it contains Bonk's uniformization result for carpets as a special case and does not rely on the condition of uniform relative separation that is used in relevant works.