Canonical Thurston obstructions for sub-hyperbolic semi-rational branched coverings

Chen, Tao; Jiang, Yunping

doi:10.1090/S1088-4173-2013-00250-6

Canonical Thurston obstructions for sub-hyperbolic semi-rational branched coverings
HTML articles powered by AMS MathViewer

by Tao Chen and Yunping Jiang PDF

Conform. Geom. Dyn. 17 (2013), 6-25 Request permission

Abstract:

We prove that the canonical Thurston obstruction for a sub-hyper- bolic semi-rational branched covering exists if the branched covering is not CLH-equivalent to a rational map.

References

Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
G. Cui, Y. Jiang and D. Sullivan, On geometrically finite branched covering 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
Guizhen Cui and Yunping Jiang, Geometrically finite and semi-rational branched coverings of the two-sphere, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2701–2714. MR 2763733, DOI 10.1090/S0002-9947-2010-05211-0
Guizhen Cui and Lei Tan, A characterization of hyperbolic rational maps, Invent. Math. 183 (2011), no. 3, 451–516. MR 2772086, DOI 10.1007/s00222-010-0281-8
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
Clifford J. Earle and Sudeb Mitra, Variation of moduli under holomorphic motions, In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) Contemp. Math., vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 39–67. MR 1759669, DOI 10.1090/conm/256/03996
Frederick P. Gardiner, Yunping Jiang, and Zhe Wang, Holomorphic motions and related topics, Geometry of Riemann surfaces, London Math. Soc. Lecture Note Ser., vol. 368, Cambridge Univ. Press, Cambridge, 2010, pp. 156–193. MR 2665009
F. R. Gantmacher, The theory of matrices. Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. Translated from the Russian by K. A. Hirsch; Reprint of the 1959 translation. MR 1657129
Y. Jiang, A framework towards understanding the characterization of holomorphic maps, to appear in Frontiers in Complex Dynamics, Princeton, 2013.
Yunping Jiang, Sudeb Mitra, and Zhe Wang, Liftings of holomorphic maps into Teichmüller spaces, Kodai Math. J. 32 (2009), no. 3, 547–563. MR 2582017, DOI 10.2996/kmj/1257948895
Gregory Stephen Lieb, Holomorphic motions and Teichmuller space, ProQuest LLC, Ann Arbor, MI, 1989. Thesis (Ph.D.)–Cornell University. MR 2638376
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
Kevin M. Pilgrim, Canonical Thurston obstructions, Adv. Math. 158 (2001), no. 2, 154–168. MR 1822682, DOI 10.1006/aima.2000.1971
W. Thurston, The combinatorics of iterated rational maps. Preprint Princeton University, Princeton, N.J. 1983.
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