Finite and infinite degree Thurston maps with a small postsingular set

TL;DR

This paper extends Thurston map theory to include maps with a single essential singularity, providing a characterization theorem for maps with four postsingular values and analyzing their Teichmüller space dynamics.

Contribution

It introduces a new class of Thurston maps with singularities and establishes an analog of Thurston's theorem for these maps, expanding the theoretical framework.

Findings

Established a Thurston-type characterization theorem for maps with a singularity

Analyzed pullback maps on Teichmüller space for these Thurston maps

Derived properties of Hurwitz classes related to the maps

Abstract

We develop the theory of Thurston maps that are defined everywhere on the topological sphere $S^2$ with a possible exception of a single essential singularity. We establish an analog of the celebrated W. Thurston's characterization theorem for a broad class of such Thurston maps having four postsingular values. To achieve this, we analyze the corresponding pullback maps defined on the one-complex dimensional Teichm\"uller space. This analysis also allows us to derive various properties of Hurwitz classes of the corresponding Thurston maps.