Meromorphic functions with two completely invariant domains

Details Meromorphic functions with two completely invariant domains Meromorphic functions with two completely invariant domains

2003-10-31

arxiv.org/abs/math/0310495v2

in the book: Ph. Rippon and G. Stallard, eds, Transcendental Dynamics and Complex Analysis, p. 74-89 (LMS Lect. Notes Ser., 34

Mathematics

Complex Variables

14 pages, 6 figures, Examples 3 and 4 have been replaced by a simpler example, minor corrections made

Scientific paper

We show that if a meromorphic function has two completely invariant Fatou components and only finitely many critical and asymptotic values, then its Julia set is a Jordan curve. However, even if both domains are attracting basins, the Julia set need not be a quasicircle. We also show that all critical and asymptotic values are contained in the two completely invariant components. This need not be the case for functions with infinitely many critical and asymptotic values.

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