Geometrically finite and semi-rational branched coverings of the two-sphere
HTML articles powered by AMS MathViewer
- by Guizhen Cui and Yunping Jiang PDF
- Trans. Amer. Math. Soc. 363 (2011), 2701-2714 Request permission
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.References
- D. Brown, Spider theory to explore parameter spaces. Cornell University Ph.D. thesis, 2001. Stony Brook thesis preprint server.
- Guizhen Cui, Yunping Jiang, and Dennis Sullivan, Dynamics of geometrically finite rational maps. Manuscript, 1994.
- 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, [2003], pp. 1–14. MR 2504307
- 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, [2003], pp. 15–29. MR 2504308
- G. Cui and L. Tan, A characterization of hyperbolic rational maps. Preprint.
- 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, DOI 10.1007/BF02392534
- 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, DOI 10.1090/S0894-0347-08-00609-7
- Ricardo Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 1, 1–11. MR 1224298, DOI 10.1007/BF01231694
- Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
- John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
- G. Zhang, Topological models of polynomials of simple Siegel disk type, CUNY Graduate Center Ph.D. thesis, 2002.
- Gaofei Zhang and Yunping Jiang, Combinatorial characterization of sub-hyperbolic rational maps, Adv. Math. 221 (2009), no. 6, 1990–2018. MR 2522834, DOI 10.1016/j.aim.2009.03.009
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
- MR Author ID: 238389
- Email: Yunping.Jiang@qc.cuny.edu
- Received by editor(s): June 2, 2009
- Received by editor(s) in revised form: September 18, 2009
- Published electronically: December 10, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 2701-2714
- MSC (2010): Primary 37F20, 37F10, 30D05
- DOI: https://doi.org/10.1090/S0002-9947-2010-05211-0
- MathSciNet review: 2763733