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Constructing entire functions by quasiconformal folding. (English) Zbl 1338.30016
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.

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 30D15 Special classes of entire functions of one complex variable and growth estimates 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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