Finite and infinite degree Thurston maps with extra marked points

TL;DR

This paper characterizes when marked Thurston maps can be realized by holomorphic maps, linking the existence of such realizations to the absence of degenerate Levy cycles, using complex dynamics and hyperbolic geometry tools.

Contribution

It provides a criterion for realizing marked Thurston maps as holomorphic maps based on Levy cycle degeneracy, expanding understanding of Thurston map realizations.

Findings

Realization of marked Thurston maps depends on no degenerate Levy cycles.

Invariant complex sub-manifolds in Teichmüller space are key to understanding map behavior.

Application of complex dynamics and hyperbolic geometry techniques to Thurston maps.

Abstract

We investigate the family of marked Thurston maps that are defined everywhere on the topological sphere $S^2$, potentially excluding at most countable closed set of essential singularities. We show that when an unmarked Thurston map $f$ is realized by a postsingularly finite holomorphic map, the marked Thurston map $(f, A)$, where $A \subset S^2$ is the corresponding finite marked set, admits such a realization if and only if it has no degenerate Levy cycle. To obtain this result, we analyze the associated pullback map $\sigma_{f, A}$ defined on the Teichm\"uller space $\mathcal{T}_A$ and demonstrate that some of its iterates admit well-behaved invariant complex sub-manifolds within $\mathcal{T}_A$. By applying powerful machinery of one-dimensional complex dynamics and hyperbolic geometry, we gain a clear understanding of the behavior of the map $\sigma_{f, A}$ restricted to the corresponding invariant subset of the Teichm\"uller space $\mathcal{T}_A$.