Abstract
We investigate when a rational endomorphism of the Riemann sphereC¯\overline {\mathbb {C}}extends to a mapping of the upper half-spaceH3{\mathbb H}^3which 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 calleduniformly quasiregular. We show that, in the space of rational mappings of degreedd, 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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