Exponential Thurston maps and limits of quadratic differentials

Hubbard, John¹ ; Schleicher, Dierk² ; Shishikura, Mitsuhiro³

¹ 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
² School of Engineering and Science, Jacobs University Bremen, Postfach 750 561, D-28725 Bremen, Germany
³ Department of Mathematics, Faculty of Sciences, Kyoto University, Kyoto 606-8502, Japan

Journal of the American Mathematical Society, Tome 22 (2009) no. 1, pp. 77-117

Résumé

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.

Détail

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

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