Spinning deformations of rational maps
Authors:
Kevin M. Pilgrim and Tan Lei
Journal:
Conform. Geom. Dyn. 8 (2004), 5286
MSC (2000):
Primary 37F30, 37F10; Secondary 30F60, 32G15
Published electronically:
March 24, 2004
MathSciNet review:
2060378
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We analyze a real oneparameter 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 preperiodic 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.
 [Bir]
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
(58 #7596)
 [Bro]
Jeffrey
F. Brock, Iteration of mapping classes and limits of hyperbolic
3manifolds, Invent. Math. 143 (2001), no. 3,
523–570. MR 1817644
(2002d:30052), http://dx.doi.org/10.1007/PL00005799
 [Ché]
Arnaud Chéritat, Recherche d'ensembles de Julia de mesure de Lebesgue positive. Ph.D. thesis, Université de ParisSud, 2001.
 [Cui]
Guizhen Cui, Geometrically finite rational maps with given combinatorics. Manuscript, 2000.
 [Gun]
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
(92b:32001a)
 [Kra]
Irwin
Kra, On the NielsenThurstonBers type of some selfmaps of Riemann
surfaces, Acta Math. 146 (1981), no. 34,
231–270. MR
611385 (82m:32019), http://dx.doi.org/10.1007/BF02392465
 [McM2]
Curtis
T. McMullen, Complex dynamics and renormalization, Annals of
Mathematics Studies, vol. 135, Princeton University Press, Princeton,
NJ, 1994. MR
1312365 (96b:58097)
 [MS]
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
(99e:58145), http://dx.doi.org/10.1006/aima.1998.1726
 [Mil]
John
Milnor, Dynamics in one complex variable, Friedr. Vieweg &
Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
(2002i:37057)
 [Nag]
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 WileyInterscience Publication.
MR 927291
(89f:32040)
 [Rees]
M.
Rees, Components of degree two hyperbolic rational maps,
Invent. Math. 100 (1990), no. 2, 357–382. MR 1047139
(91b:58187), http://dx.doi.org/10.1007/BF01231191
 [Tan]
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
(2003c:37061), http://dx.doi.org/10.1016/S00129593(02)010923
 [Bir]
 Joan S. Birman, The algebraic structure of surface mapping class groups, in Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975), pp. 163198. London, Academic Press, 1977. MR 58:7596
 [Bro]
 Jeffrey F. Brock, Iteration of mapping classes and limits of hyperbolic threemanifolds. Invent. Math. 143 (2001), 523570. MR 2002d:30052
 [Ché]
 Arnaud Chéritat, Recherche d'ensembles de Julia de mesure de Lebesgue positive. Ph.D. thesis, Université de ParisSud, 2001.
 [Cui]
 Guizhen Cui, Geometrically finite rational maps with given combinatorics. Manuscript, 2000.
 [Gun]
 Robert C. Gunning, Introduction to holomorphic functions of several variables. Vol. I. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory. MR 92b:32001a
 [Kra]
 Irwin Kra, On the NielsenThurstonBers type of some selfmaps of Riemann surfaces, Acta Math. 146 (1981), 231270. MR 82m:32019
 [McM2]
 Curtis T. McMullen, Complex dynamics and renormalization, Princeton University Press, Princeton, NJ, 1994. MR 96b:58097
 [MS]
 Curtis T. McMullen and Dennis P. Sullivan, Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), 351395. MR 99e:58145
 [Mil]
 John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 2002i:37057
 [Nag]
 Subhashis Nag, The complex analytic theory of Teichmüller spaces, John Wiley & Sons Inc., New York, 1988. A WileyInterscience Publication. MR 89f:32040
 [Rees]
 Mary Rees, Components of degree two hyperbolic rational maps. Invent. Math. 100 (1990), 357382. MR 91b:58187
 [Tan]
 Tan Lei, On pinching deformations of rational maps, Ann. Sci. École Norm. Sup. 35 (2002), 353370. MR 2003c:37061
Similar Articles
Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society
with MSC (2000):
37F30,
37F10,
30F60,
32G15
Retrieve articles in all journals
with MSC (2000):
37F30,
37F10,
30F60,
32G15
Additional Information
Kevin M. Pilgrim
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47401
Email:
pilgrim@indiana.edu
Tan Lei
Affiliation:
Unité CNRSUPRESA 8088, Département de Mathématiques, Université de CergyPontoise, 2 Avenue Adolphe Chauvin, 95302 CergyPontoise cedex, France
Email:
tanlei@math.ucergy.fr
DOI:
http://dx.doi.org/10.1090/S1088417304001018
PII:
S 10884173(04)001018
Keywords:
Holomorphic dynamics,
quasiconformal deformation,
modular group
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 CergyPontoise
Article copyright:
© Copyright 2004
American Mathematical Society
