Bounded Geometry and Characterization of Some Transcendental Entire and Meromorphic Maps

TL;DR

This paper characterizes certain transcendental entire and meromorphic maps with finite post-singular sets using bounded geometry, providing a combinatorial criterion for their equivalence to holomorphic maps of the same type.

Contribution

It introduces a new combinatorial characterization of post-singularly finite transcendental maps based on bounded geometry, extending understanding of their structure.

Findings

Post-singularly finite maps are equivalent to holomorphic maps if and only if they have bounded geometry.

Provides a classification framework for exponential and universal covering maps with finite post-singular sets.

Establishes conditions under which topological models correspond to holomorphic maps of the same class.

Abstract

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 $P e^{Q}$, 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 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.