On Connected Preimages of Simply-Connected Domains Under Entire Functions



Rempe-Gillen, Lasse and Sixsmith, Dave ORCID: 0000-0002-3543-6969
(2019) On Connected Preimages of Simply-Connected Domains Under Entire Functions. Geometric and Functional Analysis, 29 (5). 1579 - 1615.

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Abstract

Let f be a transcendental entire function, and let U,V⊂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⊂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)=ez+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.

Item Type: Article
Additional Information: 35 pages, 8 figures. V3: Accepted manuscript, to appear in Geometric and Functional Analysis. Revisions and corrections were made throughout; in particular the auxiliary Propositions 3.1 and 7.2 were incorrectly stated in V2. These have been corrected; further discussion and a figure have been added in Section 7. The statement of Proposition 8.6 has also been strengthened
Uncontrolled Keywords: Primary 30D20, Secondary 30D05, 37F10, 30D30
Depositing User: Symplectic Admin
Date Deposited: 12 Sep 2018 10:19
Last Modified: 28 Oct 2021 14:10
DOI: 10.1007/s00039-019-00488-2
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3025460