Every closed subset of the plane containing 0 and at least one other point is realized as the singular-value set of a locally univalent entire function made from nested exponentials.
Heins: The set of asymptotic values of an entire function , Tolfte Skandinaviska Matematik- erkongressen (Lund, Sweden, 1953), Proceedings of the Scan dinavian Math
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Entire functions with prescribed singular values
Every closed subset of the plane containing 0 and at least one other point is realized as the singular-value set of a locally univalent entire function made from nested exponentials.