Geometrically finite and semi-rational branched coverings of the two-sphere

Authors:
Guizhen Cui and Yunping Jiang

Journal:
Trans. Amer. Math. Soc. **363** (2011), 2701-2714

MSC (2010):
Primary 37F20, 37F10, 30D05

Published electronically:
December 10, 2010

MathSciNet review:
2763733

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Abstract | References | Similar Articles | Additional Information

Abstract: In 1982, Thurston gave a necessary and sufficient condition for a critically finite branched covering of the two-sphere to itself to be combinatorially equivalent to a rational map. We discuss extending this result to geometrically finite rational maps. We give an example to show that Thurston's original condition is not sufficient. This example is topologically pathological near accumulation points of the postcritical set. We give two conditions forbidding such pathology, show that they are equivalent, and (in a sequel to the present paper) will show that Thurston's condition together with this tameness is both necessary and sufficient to characterize geometrically finite rational maps.

**1.**D. Brown,

Spider theory to explore parameter spaces.

Cornell University Ph.D. thesis, 2001.

Stony Brook thesis preprint server.**2.**

Guizhen Cui, Yunping Jiang, and Dennis Sullivan,

Dynamics of geometrically finite rational maps. Manuscript, 1994.**3.**Guizhen Cui, Yunping Jiang, and Dennis Sullivan,*On geometrically finite branched coverings. I. Locally combinatorial attracting*, Complex dynamics and related topics: lectures from the Morningside Center of Mathematics, New Stud. Adv. Math., vol. 5, Int. Press, Somerville, MA, 20??, pp. 1–14. MR**2504307****4.**Guizhen Cui, Yunping Jiang, and Dennis Sullivan,*On geometrically finite branched coverings. II. Realization of rational maps*, Complex dynamics and related topics: lectures from the Morningside Center of Mathematics, New Stud. Adv. Math., vol. 5, Int. Press, Somerville, MA, 20??, pp. 15–29. MR**2504308****5.**G. Cui and L. Tan,

A characterization of hyperbolic rational maps.

Preprint.**6.**Adrien Douady and John H. Hubbard,*A proof of Thurston’s topological characterization of rational functions*, Acta Math.**171**(1993), no. 2, 263–297. MR**1251582**, 10.1007/BF02392534**7.**John Hubbard, Dierk Schleicher, and Mitsuhiro Shishikura,*Exponential Thurston maps and limits of quadratic differentials*, J. Amer. Math. Soc.**22**(2009), no. 1, 77–117. MR**2449055**, 10.1090/S0894-0347-08-00609-7**8.**Ricardo Mañé,*On a theorem of Fatou*, Bol. Soc. Brasil. Mat. (N.S.)**24**(1993), no. 1, 1–11. MR**1224298**, 10.1007/BF01231694**9.**Curtis T. McMullen,*Complex dynamics and renormalization*, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR**1312365****10.**John Milnor,*Dynamics in one complex variable*, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR**1721240****11.**G. Zhang,

Topological models of polynomials of simple Siegel disk type,

CUNY Graduate Center Ph.D. thesis, 2002.**12.**Gaofei Zhang and Yunping Jiang,*Combinatorial characterization of sub-hyperbolic rational maps*, Adv. Math.**221**(2009), no. 6, 1990–2018. MR**2522834**, 10.1016/j.aim.2009.03.009

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Additional Information

**Guizhen Cui**

Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

Email:
gzcui@math.ac.cn

**Yunping Jiang**

Affiliation:
Department of Mathematics, Queens College of CUNY, Flushing, New York 11367 – and – Department of Mathematics, CUNY Graduate Center, New York, New York 10016

Email:
Yunping.Jiang@qc.cuny.edu

DOI:
https://doi.org/10.1090/S0002-9947-2010-05211-0

Keywords:
Geometrically finite branched covering,
semi-rational branched covering,
sub-hyperbolic semi-rational branched covering

Received by editor(s):
June 2, 2009

Received by editor(s) in revised form:
September 18, 2009

Published electronically:
December 10, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.