Entire functions with prescribed singular values

Abstract

We introduce a new class of entire functions $\mathcal{E}$ which consists of all $F_0\in\mathcal{O}(\mathbb{C})$ for which there exists a sequence $(F_n)\in \mathcal{O}(\mathbb{C})$ and a sequence $(\lambda_n)\in\mathbb{C}$ satisfying $F_n(z)=\lambda_{n+1}e^{F_{n+1}(z)}$ for all $n\geq 0$. This new class is closed under the composition and its is dense in the space of all non-vanishing entire functions. We prove that every closed set $V\subset \mathbb{C}$ containing the origin and at least one more point is the set of singular values of some locally univalent function in $\mathcal{E}$, hence this new class has non-trivial intersection with both the Speiser class and the Eremenko-Lyubich class of entire functions. As a consequence we provide a new proof of an old result by Heins which states that every closed set $V\subset\mathbb{C}$ is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally we show that the class $\mathcal{E}$ contains functions with an empty Fatou set and also functions whose Fatou set is non-empty.