Proves an if-and-only-if quasiconformal characterization of Schottky sets on the sphere that applies to Sierpiński carpets and gaskets and generalizes Bonk's carpet result without uniform relative separation.
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Complete quasisymmetric classification and rigidity proof for Julia sets of postcritically finite McMullen maps, establishing first rigid examples in carpet, necklace, and cluster classes.
No local quasiconformal map exists between Sierpiński carpet limit sets of convex-cocompact Kleinian groups and Julia sets of postcritically finite rational maps.
A criterion and constructions are given for Cantor bubble Julia sets in rational maps with attracting or parabolic fixed points, including high-period cycles, Hausdorff dimension two, and a quasisymmetric equivalence condition to round bubbles.
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Quasiconformal characterization of Schottky sets
Proves an if-and-only-if quasiconformal characterization of Schottky sets on the sphere that applies to Sierpiński carpets and gaskets and generalizes Bonk's carpet result without uniform relative separation.