Extending rational maps

Martin, Gaven

doi:10.1090/S1088-4173-04-00115-8

Extending rational maps
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by Gaven J. Martin

Conform. Geom. Dyn. 8 (2004), 158-166

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

Published electronically: November 16, 2004

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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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