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Bounded geometry and characterization of some transcendental maps. (English) Zbl 1379.37089

Summary: We define two classes of topological infinite degree covering maps modeled on two families of transcendental holomorphic maps. The first, which we call exponential maps of type \((p,q)\), involves branched covers, and is modeled on transcendental entire maps of the form \(Pe^Q\), where \(P\) and \(Q\) are polynomials of degrees \(p\) and \(q\). The second is the class of universal covering maps from the plane to the sphere with two removed points modeled on transcendental meromorphic maps with two asymptotic values. The problem we address is how to give a combinatorial characterization of the holomorphic maps contained in these classes, whose post-singular sets are finite. The main results in this paper are that a post-singularly finite topological exponential map of type \((0,1)\) or a certain post-singularly finite topological exponential map of type \((p,1)\) or a post-singularly finite universal covering map from the plane to the sphere with two points removed is combinatorially equivalent to a holomorphic same type map if and only if this map has bounded geometry.

MSC:

37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30F30 Differentials on Riemann surfaces
30D30 Meromorphic functions of one complex variable (general theory)
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