## 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