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

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 Teichmüller 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 Förster about the classification of exponential functions with the escaping singular value.

33 pages, 2 figures, version 3, formula (2.1) corrected, typos corrected, some explanations added or improved, Acknowledgements section updated

Cited by in corpus (3)

• Infinite-dimensional Thurston theory and transcendental dynamics II: classification of entire functions with escaping singular orbits
• Infinite-dimensional Thurston theory and transcendental dynamics IV: dependence on parameters and escape on (pre-)periodic rays
• Infinite-dimensional Thurston theory and transcendental dynamics III: entire functions with escaping singular orbits in the degenerate case