Infinite-dimensional Thurston theory and transcendental dynamics I: infinite-legged spiders

We develop techniques that lay out a basis for generalizations of Thurston’s famous Topological Characterization of Rational Functions for an infinite set of marked points and branched coverings of infinite degree. Analogously to the classical theorem we consider Thurston’s $\sigma $-map acting on a Teichmüller space which is this time infinite-dimensional – and this leads to a completely different theory compared to the classical setting.

We demonstrate our techniques by giving an alternative proof of the result by Markus Förster about the classification of exponential functions with the escaping singular value.