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

TL;DR

This paper extends Thurston's topological characterization to infinite-dimensional settings, developing new techniques for infinite-degree branched coverings and applying them to classify exponential functions with escaping singular values.

Contribution

It introduces a framework for infinite-dimensional Thurston theory and provides an alternative proof for classifying exponential functions with escaping singular values.

Findings

Established an infinite-dimensional Thurston theory framework.

Provided an alternative proof for exponential function classification.

Connected classical and infinite-dimensional Thurston theories.

Abstract

We develop techniques that lay out a basis for generalizations of the famous Thurston's 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 the Thurston's $\sigma$-map acting on a Teichm\"uller space which is this time infinite-dimensional -- and this leads to a completely different theory comparing to the classical setting. We demonstrate our techniques by giving an alternative proof of the result by Markus F\"orster about the classification of exponential functions with the escaping singular value.