Abstract
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 \(a\in \mathbb{R }\). We also show the absence of wandering domains for certain non-real entire functions for which \(S(f)\) is bounded and \(f^n|_{S(f)}\rightarrow \infty \) 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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Notes
Since this paper was first submitted, Bishop has announced the construction of such a function. Here the orbit of the wandering domain contains (infinitely many) critical values; hence Question 1.1 remains open.
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Acknowledgments
We would like to thank Walter Bergweiler, Jörn Peter and Phil Rippon for interesting discussions and helpful feedback. We also thank the referee for comments that have led to some improvement in the paper.
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H. Mihaljević-Brandt and L. Rempe-Gillen were supported by EPSRC grant EP/E017886/1. L. Rempe-Gillen was supported by EPSRC Fellowship EP/E052851/1. H. Mihaljević-Brandt and L. Rempe-Gillen gratefully acknowledge support received through the European CODY network.
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Mihaljević-Brandt, H., Rempe-Gillen, L. Absence of wandering domains for some real entire functions with bounded singular sets. Math. Ann. 357, 1577–1604 (2013). https://doi.org/10.1007/s00208-013-0936-z
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DOI: https://doi.org/10.1007/s00208-013-0936-z