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Journal of the American Mathematical Society

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

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Exponential Thurston maps and limits of quadratic differentials
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by John Hubbard, Dierk Schleicher and Mitsuhiro Shishikura PDF
J. Amer. Math. Soc. 22 (2009), 77-117 Request permission

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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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
  • Email: jhh8@cornell.edu
  • Dierk Schleicher
  • Affiliation: School of Engineering and Science, Jacobs University Bremen, Postfach 750 561, D-28725 Bremen, Germany
  • MR Author ID: 359328
  • Email: dierk@jacobs-university.de
  • Mitsuhiro Shishikura
  • Affiliation: Department of Mathematics, Faculty of Sciences, Kyoto University, Kyoto 606-8502, Japan
  • Email: mitsu@math.kyoto-u.ac.jp
  • 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: https://doi.org/10.1090/S0894-0347-08-00609-7
  • MathSciNet review: 2449055