Finite and infinite degree Thurston maps with extra marked points

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 Teichmüller 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 Teichmüller space ${\mathcal{T}}_A$.