Abstract
Let f be a transcendental entire function, and let \({U,V \subset \mathbb{C}}\) be disjoint simply-connected domains. Must one of \({f^{-1}(U)}\) and \({f^{-1}(V)}\) be disconnected? In 1970, Baker implicitly gave a positive answer to this question, in order to prove that a transcendental entire function cannot have two disjoint completely invariant domains. (A domain \({U\subset \mathbb{C}}\) is completely invariant under f if \({f^{-1}(U)=U}\).) It was recently observed by Julien Duval that Baker's argument, which has also been used in later generalisations and extensions of Baker's result, contains a flaw. We show that the answer to the above question is negative; so this flaw cannot be repaired. Indeed, for the function \({f(z)= e^z+z}\), there is a collection of infinitely many pairwise disjoint simply-connected domains, each with connected preimage. We also answer a long-standing question of Eremenko by giving an example of a transcendental meromorphic function, with infinitely many poles, which has the same property. Furthermore, we show that there exists a function f with the above properties such that additionally the set of singular values S(f) is bounded; in other words, f belongs to the Eremenko–Lyubich class. On the other hand, if S(f) is finite (or if certain additional hypotheses are imposed), many of the original results do hold. For the convenience of the research community, we also include a description of the error in Baker's proof, and a summary of other papers that are affected.
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Acknowledgements
We thank Alex Eremenko for making us aware of the error in Baker’s proof discovered by Duval, and for many interesting and useful discussions in the preparation of this article. We also thank Julien Duval for allowing us to include a description of his argument (see Figure 7), and for reading an early draft of our paper. Finally, we thank the referee for several helpful comments, suggestions, and corrections.
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The first author was partially supported by a Philip Leverhulme Prize.
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Rempe-Gillen, L., Sixsmith, D. On Connected Preimages of Simply-Connected Domains Under Entire Functions. Geom. Funct. Anal. 29, 1579–1615 (2019). https://doi.org/10.1007/s00039-019-00488-2
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DOI: https://doi.org/10.1007/s00039-019-00488-2