Jordan and Julia
The author considers the problem: which is the Julia set \(J(f)\) of a rational function \(f\) a quasiconformal Jordan arc or curve? In the case when \(J(f)\) is a Jordan arc (for which necessary and sufficient conditions are known) it is shown that \(J(f)\) is always a quasiconformal arc. If the Julia set is a Jordan curve then the complementary Fatou set \(F(f)\) consists of two components \(U\) and \(V\). In the case when \(U\) and \(V\) are attracting then D. Sullivan has proved that \(J\) is a quasicircle. The author shows that the only other case which gives rise to a quasicircle is when \(U\) and \(V\) form the Leau flower of the same parabolic fixed point. The proofs depend on a lemma about fusing critical points in a parabolic basis using quasiconformal surgery.
scientific article; zbMATH DE number 1031547