Absence of wandering domains for some real entire functions with bounded singular sets

Details Absence of wandering domains for some real entire functions with bounded singular sets Absence of wandering domains for some real entire functions with bounded singular sets

2011-03-31

arxiv.org/abs/1104.0034v1

Mathematics

Dynamical Systems

23 pages

Scientific paper

Let $f$ be a real entire function whose set $S(f)$ of singular values is real and bounded. We show that, if $f$ satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all maps of the form $z\mapsto \lambda \frac{\sinh(z)}{z} + a$ with $\lambda>0$ and $a\in\R$. We also show the absence of wandering domains for certain non-real entire functions for which $S(f)$ is bounded and $f^n|_{S(f)}\to\infty$ uniformly. As a special case of our theorem, we give a short, elementary and non-technical proof that the Julia set of the exponential map $f(z)=e^z$ is the entire complex plane. Furthermore, we apply similar methods to extend a result of Bergweiler, concerning Baker domains of entire functions and their relation to the postsingular set, to the case of meromorphic functions.

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