an:06428137 Zbl 1338.30016 Bishop, Christopher J. Constructing entire functions by quasiconformal folding EN Acta Math. 214, No. 1, 1-60 (2015). 00342569 2015
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30C65 30D15 30D05 quasiconformal maps; entire functions; Speiser class; Eremenko-Lyubich class; bounded singular set; finite singular set; wandering domains; area conjecture For a finite plane tree $$T$$, a polynomial $$p$$ is associated with only two critical values $$\pm 1$$ such that $$T_p = p^{-1}([-1,1])$$ is a plane tree which is equivalent to $$T$$. To an infinite plane tree $$T$$, there corresponds a certain entire function $$f$$. In order to explain this in more detail, let us consider the singular set $$S(f)$$ of $$f$$ which is the closure of the critical values and finite asymptotic values of $$f$$. The Speiser class $$\mathcal{S}$$ is the set of transcendental entire functions with a finite singular set. Furthermore, let $$\mathcal{S}_n \subset \mathcal{S}$$ be those functions with at most $$n$$ singular values and $$S_{p,q}$$ be those functions with $$p$$ critical values and $$q$$ finite asymptotic values. Finally, let $$\mathcal{B}$$ denote the Eremenko-Lyubich class of transcendental entire functions with bounded (not necessarily finite) singular set. Now, for an infinite plane tree $$T$$ with certain mild geometric conditions the author develops a method to construct a corresponding entire function $$f$$ in the class $$\mathcal{S}_{2,0}$$ with the only critical values $$\pm 1$$ such that $$T_f = f^{-1}([-1,1])$$ is a plane tree which approximates $$T$$ in a precise way. His method uses quasiconformal mappings and the measurable Riemann mapping theorem. Furthermore, he applies his method to solve a number of open problems, e.g., the area conjecture of Eremenko and Lyubich and the existence of a function in $$\mathcal{B}$$ whose Fatou set has a wandering domain. Rainer Br??ck (Dortmund)