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On the limit functions of iterates in wandering domains. (English) Zbl 0793.30022

Let \(f\) be a transcendental entire function and denote by \(F\) the set where the iterates \(f^ n\) of \(f\) form a normal family. A component \(U\) of \(F\) is called wandering if \(f^ n(U)\cap f^ m(U)=\emptyset\) for \(n\neq m\). Denote by \(\text{sing}(f^{-1})\) the set of singularities of the inverse function of \(f\) and define \(E=\bigcup^ \infty_{n=0} f^ n(\text{sing}(f^{-1}))\). It follows from a result of I. N. Baker [Ann. Acad. Sci. Fenn., Ser. A 467, 3-11 (1970; Zbl 0197.05302)] that if \(U\) is a wandering component of \(F\), then all finite limit functions of \(f^ n|_ U\) are contained in the closure of \(E\). In this paper, it is shown that they are in fact in the derived set of \(E\), answering a question of Baker. This result is used to obtain some new classes of entire functions which do not have wandering domains.
Reviewer: W.Bergweiler

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets

Citations:

Zbl 0197.05302