Bounded Geometry and Characterization of Some Transcendental Entire and Meromorphic Maps

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), are branched covers and is modeled on transcendental entire maps of the form Pe, 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 to give a combinatorial characterization of the holomorphic maps contained in these classes whose postsingular 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.