Exponential Thurston maps and limits of quadratic differentials

Hubbard, John; Schleicher, Dierk; Shishikura, Mitsuhiro

doi:10.1090/S0894-0347-08-00609-7

Exponential Thurston maps and limits of quadratic differentials
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by John Hubbard, Dierk Schleicher and Mitsuhiro Shishikura

J. Amer. Math. Soc. 22 (2009), 77-117

DOI: https://doi.org/10.1090/S0894-0347-08-00609-7

Published electronically: June 3, 2009

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Previous version: Original version posted July 9, 2008
Corrected version: Current version corrects publisher's introduction of inconsistent spelling of "Teichmüller".

Abstract:

We give a topological characterization of postsingularly finite topological exponential maps, i.e., universal covers $g\colon \mathbb {C}\to \mathbb {C}\setminus \{0\}$ such that $0$ has a finite orbit. Such a map either is Thurston equivalent to a unique holomorphic exponential map $\lambda e^z$ or it has a topological obstruction called a degenerate Levy cycle. This is the first analog of Thurston’s topological characterization theorem of rational maps, as published by Douady and Hubbard, for the case of infinite degree.

One main tool is a theorem about the distribution of mass of an integrable quadratic differential with a given number of poles, providing an almost compact space of models for the entire mass of quadratic differentials. This theorem is given for arbitrary Riemann surfaces of finite type in a uniform way.

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