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Journal of the American Mathematical Society

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Exponential Thurston maps and limits of quadratic differentials

Authors: John Hubbard, Dierk Schleicher and Mitsuhiro Shishikura
Journal: J. Amer. Math. Soc. 22 (2009), 77-117
MSC (2000): Primary 30F30; Secondary 30F60, 32G15, 37F20, 37F30
Published electronically: June 3, 2009
Previous version: Original version posted July 9, 2008
Corrected version: Current version corrects publisher's introduction of inconsistent spelling of "Teichmüller".
MathSciNet review: 2449055
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Abstract | References | Similar Articles | Additional Information


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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Additional Information

John Hubbard
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853, and Centre de Mathématiques et d’Informatique, Université de Provence, 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France

Dierk Schleicher
Affiliation: School of Engineering and Science, Jacobs University Bremen, Postfach 750 561, D-28725 Bremen, Germany
MR Author ID: 359328

Mitsuhiro Shishikura
Affiliation: Department of Mathematics, Faculty of Sciences, Kyoto University, Kyoto 606-8502, Japan

Keywords: Quadratic differential, decomposition, limit model, iteration, exponential map, classification
Received by editor(s): March 28, 2006
Published electronically: June 3, 2009
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.