A gasket Julia set is quasiconformally uniformizable by a round gasket exactly when its Fatou components meet tangentially, and David-uniformizable exactly when all Fatou components are quasidisks.
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Uniformization of gasket Julia sets
A gasket Julia set is quasiconformally uniformizable by a round gasket exactly when its Fatou components meet tangentially, and David-uniformizable exactly when all Fatou components are quasidisks.