Spinning deformations of rational maps
HTML articles powered by AMS MathViewer
- by Kevin M. Pilgrim and Tan Lei
- Conform. Geom. Dyn. 8 (2004), 52-86
- DOI: https://doi.org/10.1090/S1088-4173-04-00101-8
- Published electronically: March 24, 2004
- PDF | Request permission
Abstract:
We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as spinning. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges to infinity in the moduli space of rational maps of a fixed degree. When the limit exists, it has either a parabolic fixed point, or a pre-periodic critical point in the Julia set, depending on the combinatorics of the data defining the deformation. The proofs are soft and rely on two ingredients: the construction of a Riemann surface containing the closure of the family, and an analysis of the geometric limits of some simple dynamical systems. An interpretation in terms of Teichmüller theory is presented as well.References
- Joan S. Birman, The algebraic structure of surface mapping class groups, Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975) Academic Press, London, 1977, pp. 163–198. MR 0488019
- Jeffrey F. Brock, Iteration of mapping classes and limits of hyperbolic 3-manifolds, Invent. Math. 143 (2001), no. 3, 523–570. MR 1817644, DOI 10.1007/PL00005799 [Ché]Che Arnaud Chéritat, Recherche d’ensembles de Julia de mesure de Lebesgue positive. Ph.D. thesis, Université de Paris-Sud, 2001. [Cui]cui:limits Guizhen Cui, Geometrically finite rational maps with given combinatorics. Manuscript, 2000.
- Robert C. Gunning, Introduction to holomorphic functions of several variables. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory. MR 1052649
- Irwin Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), no. 3-4, 231–270. MR 611385, DOI 10.1007/BF02392465
- Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
- Curtis T. McMullen and Dennis P. Sullivan, Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), no. 2, 351–395. MR 1620850, DOI 10.1006/aima.1998.1726
- John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
- Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
- M. Rees, Components of degree two hyperbolic rational maps, Invent. Math. 100 (1990), no. 2, 357–382. MR 1047139, DOI 10.1007/BF01231191
- Lei Tan, On pinching deformations of rational maps, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 3, 353–370 (English, with English and French summaries). MR 1914001, DOI 10.1016/S0012-9593(02)01092-3
Bibliographic Information
- Kevin M. Pilgrim
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47401
- MR Author ID: 614176
- Email: pilgrim@indiana.edu
- Tan Lei
- Affiliation: Unité CNRS-UPRESA 8088, Département de Mathématiques, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
- Email: tanlei@math.u-cergy.fr
- Received by editor(s): May 13, 2003
- Received by editor(s) in revised form: February 18, 2004
- Published electronically: March 24, 2004
- Additional Notes: The first author was supported in part by NSF Grant No. DMS 9996070 and the Université de Cergy-Pontoise
- © Copyright 2004 American Mathematical Society
- Journal: Conform. Geom. Dyn. 8 (2004), 52-86
- MSC (2000): Primary 37F30, 37F10; Secondary 30F60, 32G15
- DOI: https://doi.org/10.1090/S1088-4173-04-00101-8
- MathSciNet review: 2060378