On Thurston’s theorem on rational mappings, exponential maps, and limits of quadratic differentials

One of the most important theorems in complex dynamics is
Thurston’s characterization theorem of rational maps, which gives a
precise criterion whether postcritically finite (topological)
branched covers of the 2-sphere are realized as rational mappings; if
not, there is a well-defined topological-combinatorial obstruction.
We extend this theorem into the world of transcendental mappings, by
proving an analogous theorem on exponential maps. The proof uses a
limit theorem of quadratic differentials on Riemann surfaces which
might be of interest in its own right.