On Connected Preimages of Simply-Connected Domains Under Entire Functions

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 [Bak70] implicitly gave a positive answer to this question, in order to prove that a transcendental entire function cannot have two disjoint completely invariant domains.It was recently observed by Julien Duval that there is a flaw in Baker's argument (which has also been used in later generalisations and extensions of Baker's result). 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 the proof of [Bak70], and a summary of other papers that are affected.

Untitled

“…However we can construct meromorphic functions with doubly periodic Julia and Fatou sets with toral bands that are Baker domains using maps satisfying (1.2) and (1.3). These are a variation of similar simply periodic maps discussed in [4,6,12,29], and Fatou gave examples of entire functions with Baker domains of the form z + 1 + 1/e z .…”

Geometrically finite transcendental entire functions

“…By assumption $$U \cap S(f)$$ is compact, and it is well‐known that $$U$$ is simply connected. Hence, by [42, Corollary 8.5], $$S(f) \subset U$$. In particular, $$f$$ is strongly geometrically finite.…”

Transcendental Julia Sets of Minimal Hausdorff Dimension

“…We will now record some useful lemmas about branched coverings that are topological in nature. The following are Propositions 3.1 and 3.2 of [RGS19].…”