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Conformal Geometry and Dynamics

ISSN 1088-4173



Spinning deformations of rational maps

Authors: Kevin M. Pilgrim and Tan Lei
Journal: Conform. Geom. Dyn. 8 (2004), 52-86
MSC (2000): Primary 37F30, 37F10; Secondary 30F60, 32G15
Published electronically: March 24, 2004
MathSciNet review: 2060378
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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.

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Additional Information

Kevin M. Pilgrim
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47401

Tan Lei
Affiliation: Unité CNRS-UPRESA 8088, Département de Mathématiques, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

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 Cergy-Pontoise
Article copyright: © Copyright 2004 American Mathematical Society