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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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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
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".


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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Bibliographic 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
  • Email:
  • Dierk Schleicher
  • Affiliation: School of Engineering and Science, Jacobs University Bremen, Postfach 750 561, D-28725 Bremen, Germany
  • MR Author ID: 359328
  • Email:
  • Mitsuhiro Shishikura
  • Affiliation: Department of Mathematics, Faculty of Sciences, Kyoto University, Kyoto 606-8502, Japan
  • Email:
  • Received by editor(s): March 28, 2006
  • Published electronically: June 3, 2009
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 22 (2009), 77-117
  • MSC (2000): Primary 30F30; Secondary 30F60, 32G15, 37F20, 37F30
  • DOI:
  • MathSciNet review: 2449055