On connected preimages of simply-connected domains under entire functions

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 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 entire 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 Baker's paper, and a summary of other papers that are affected.