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- W2967079169 abstract "We introduce a new class of entire functions $mathcal{E}$ which consists of all $F_0inmathcal{O}(mathbb{C})$ for which there exists a sequence $(F_n)in mathcal{O}(mathbb{C})$ and a sequence $(lambda_n)inmathbb{C}$ satisfying $F_n(z)=lambda_{n+1}e^{F_{n+1}(z)}$ for all $ngeq 0$. 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 $Vsubset mathbb{C}$ 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 $Vsubsetmathbb{C}$ 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." @default.
- W2967079169 created "2019-08-22" @default.
- W2967079169 creator A5022706711 @default.
- W2967079169 date "2019-08-16" @default.
- W2967079169 modified "2023-09-27" @default.
- W2967079169 title "Entire functions with prescribed singular values" @default.
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