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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Spinning deformations of rational maps
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by Kevin M. Pilgrim and Tan Lei PDF
Conform. Geom. Dyn. 8 (2004), 52-86 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.
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Additional 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