Abstract
We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahlfors L. Conformal Invariants: Topics in Geometric Function Theory. New York: McGraw-Hill Book Company, 1973
Beardon A. Iteration of Rational Functions. Graduate Texts in Mathematics, vol. 132. Berlin: Springer-Verlag, 1991
Benson F, Margalit D. A Primer on Mapping Class Groups. Princeton: Princeton University Press, 2011
Bielefeld B, Douady A, McMullen C. Conformal Dynamics Problem List. ArXiv:math/9201271, 1990
Buff X, Cui G, Tan L. Teichmüller spaces and holomorphic dynamics. In: Handbook of Teichmüller Theory, vol. 4. Zürich: Eur Math Soc, 2014, 717–756
Cui G. Conjugacies between rational maps and extremal quasiconformal maps. Proc Amer Math Soc, 2001, 129: 1949–1953
Cui G, Jiang Y. Geometrically finite and semi-rational branched coverings of the two-sphere. Trans Amer Math Soc, 2011, 363: 2701–2714
Cui G, Peng W, Tan L. On a theorem of Rees-Shishikura. Fac Sci Toulouse Math (6), 2012, 21: 981–993
Cui G, Tan L. Distortion control of conjugacies between quadratic polynomials. Sci China Math, 2010, 53: 625–634
Cui G, Tan L. A characterization of hyperbolic rational maps. Invent Math, 2011, 183: 451–516
Douady A, Hubbard J. A proof of Thurston’s topological characterization of rational functions. Acta Math, 1993, 171: 263–297
Goldberg L, Milnor J. Fixed points of polynomial maps, II: Fixed point portraits. Ann Sci école Norm Sup (4), 1993, 26: 51–98
Haïssinsky P. Chirurgie parabolique. C R Math Acad Sci Paris, 1998, 327: 195–198
Haïssinsky P. Déformation J-équivalente de polynômes géométriquement finis. Fund Math, 2000, 163: 131–141
Haïssinsky P. Pincements de polynômes. Comment Math Helv, 2002, 77: 1–23
Haïssinsky P, Tan L. Convergence of pinching deformations and matings of geometrically finite polynomials. Fund Math, 2004, 181: 143–188
Jiang Y, Zhang G. Combinatorial characterization of sub-hyperbolic rational maps. Adv Math, 2009, 221: 1990–2018
Kawahira T. Semiconjugacies between the Julia sets of geometrically finite rational maps. Ergodic Theory Dynam Systems, 2003, 23: 1125–1152
Kawahira T. Semiconjugacies between the Julia sets of geometrically finite rational maps II. In: Dynamics on the Riemann Sphere. Zürich: Eur Math Soc, 2006, 131–138
Lehto O. Univalent Functions and Teichmüller Spaces. New York: Springer-Verlag, 1987
Lyubich M, Minsky Y. Laminations in holomorphic dynamics. J Differential Geom, 1997, 47: 17–94
Makienko P. Unbounded components in parameter space of rational maps. Conform Geom Dyn, 2000, 4: 1–21
Maskit B. Parabolic elements in Kleinian groups. Ann of Math (2), 1983, 117: 659–668
McMullen C. Cusps are dense. Ann of Math (2), 1991, 133: 217–247
McMullen C. Rational maps and Teichmüller space. In: Linear and Complex Analysis Problem Book 3. Lecture Notes in Mathematics, vol. 1574. Berlin: Springer-Verlag, 1994, 430–433
McMullen C. Complex Dynamics and Renormalization. Annals of Mathematics Studies, vol. 135. Princeton: Princeton University Press, 1994
McMullen C. Self-similarity of Siegel disks and the Hausdorff dimension of Julia sets. Acta Math, 1998, 180: 247–292
McMullen C. Hausdorff dimension and conformal dynamics, II: Geometrically finite rational maps. Comment Math Helv, 2000, 75: 535–593
McMullen C, Sullivan D. Quasiconformal homeomorphisms and dynamics III: The Teichmüller space of a holomorphic dynamical system. Adv Math, 1998, 135: 351–395
Milnor J. Dynamics in One Complex Variable, 3rd Ed. Annals of Mathematics Studies, vol. 160. Princeton: Princeton University Press, 2006
Petersen C, Meyer D. On the notions of mating. Ann Fac Sci Toulouse Math (6), 2012, 21: 839–876
Pommerenke C. Univalent Functions. Göttingen: Vandenhoeck and Ruprecht, 1975
Reich E, Strebel K. Extremal quasiconformal mapping with given boundary values. Bull Amer Math Soc, 1973, 79: 488–490
Shishikura M. On a theorem of Mary Rees for matings of polynomials. In: The Mandelbrot Set, Theme and Variations. London Mathematical Society Lecture Note Series, vol. 274. Cambridge: Cambridge University Press, 2000, 289–305
Strebel K. On quasiconformal mappings of open Riemann surfaces. Comment Math Helv, 1978, 53: 301–321
Tan L. Matings of quadratic polynomials. Ergodic Theory Dynam Systems, 1992, 12: 589–620
Tan L. On pinching deformations of rational maps. Ann Sci éc Norm Supér (4), 2002, 35: 353–370
Urbánski M. Rational functions with no recurrent critical points. Ergodic Theory Dynam Systems, 1994, 14: 1–29
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11125106).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the Memory of Professor Lei Tan
Rights and permissions
About this article
Cite this article
Cui, G., Tan, L. Hyperbolic-parabolic deformations of rational maps. Sci. China Math. 61, 2157–2220 (2018). https://doi.org/10.1007/s11425-018-9426-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-018-9426-4