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Iteration of meromorphic functions. (English) Zbl 0791.30018
The iteration theory for meromorphic transcendental functions develops along the lines of the rational case. However, there is a variety of phenomena that do not occur in the rational case. The author of the present paper focuses his interest on two of them, namely Baker domains and wandering domains. He gives a survey on known results, but the article also includes new results. In addition, the reader will find several open questions which are subject of actual research work and a comprehensive bibliography. After recalling definitions and basic properties of Fatou sets and Julia sets, the author explains a method to prove existence theorems for (repelling) period points of given minimal period by use of Nevanlinna’s theory and Ahlfors’ five-island-theorem. Then he deals with the connectivity of the Fatou set and components of the Fatou set, especially he scetches one possibility to prove non- wandering domains to be simply connected. Next, he collects several examples and various classifications of meromorphic functions having (or having not) Baker domains or wandering domains. At the end, further topics are investigated. Among them there are: completely invariant components of the Fatou set, entire transcendental functions having Cantor bouquets as Julia sets and Newton’s method for meromorphic functions.
Reviewer: H.Kriete (Aachen)

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
65H05 Numerical computation of solutions to single equations
37B99 Topological dynamics
30D30 Meromorphic functions of one complex variable, general theory
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