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Conformal Geometry and Dynamics

ISSN 1088-4173



Extending rational maps

Author: Gaven J. Martin
Journal: Conform. Geom. Dyn. 8 (2004), 158-166
MSC (2000): Primary 30C60, 30C65, 30F40, 30D50
Published electronically: November 16, 2004
MathSciNet review: 2122524
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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

Keywords: Rational mapping, quasiconformal, quasiregular, extension
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)
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.