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 | References | Similar Articles | Additional Information

Abstract: We investigate when a rational endomorphism of the Riemann sphere extends to a mapping of the upper half-space 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 , 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

Email:
martin@math.auckland.ac.nz

DOI:
https://doi.org/10.1090/S1088-4173-04-00115-8

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.