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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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Extending rational maps
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by Gaven J. Martin PDF
Conform. Geom. Dyn. 8 (2004), 158-166 Request permission

Abstract:

We investigate when a rational endomorphism of the Riemann sphere $\overline {\mathbb {C}}$ extends to a mapping of the upper half-space ${\mathbb H}^3$ which is rational with respect to some measurable conformal structure. Such an extension has the property that it and all its iterates have uniformly bounded distortion. Such maps are called uniformly quasiregular. We show that, in the space of rational mappings of degree $d$, such an extension is possible in the structurally stable component where there is a single (attracting) component of the Fatou set and the Julia set is a Cantor set. We show that generally outside of this set no such extension is possible. In particular, polynomials can never admit such an extension.
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Additional Information
  • Gaven J. Martin
  • Affiliation: Department of Mathematics, University of Auckland and Massey University, Auckland, New Zealand
  • MR Author ID: 120465
  • Email: martin@math.auckland.ac.nz
  • Received by editor(s): April 15, 2002
  • Received by editor(s) in revised form: February 1, 2003
  • Published electronically: November 16, 2004
  • Additional Notes: Research supported in part by grants from the Australian Research Council, the Marsden Fund and Royal Society (NZ) and Institute Mittag-Leffler (Sweden)
  • © Copyright 2004 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 8 (2004), 158-166
  • MSC (2000): Primary 30C60, 30C65, 30F40, 30D50
  • DOI: https://doi.org/10.1090/S1088-4173-04-00115-8
  • MathSciNet review: 2122524