Conformally Natural extensions revisited

In this note we revisit the notion of conformal barycenter of a measure on ${\text{SS}}^n$ as defined by Douady and Earle in Acta Math. Vol 157, 1986. The aim is to extend rational maps from the Riemann sphere $\text{Cbar}\text{isom}{\text{SS}}^2$ to the (hyperbolic) three ball ${\text{BB}}^3$ and thus to ${\text{SS}}^3$ by reflection. The construction which was pioneered by Douady and Earle in the case of homeomorphisms actually gives extensions for more general maps such as entire transcendental maps on $\text{C}\subset \text{Cbar}$. And it works for maps in any dimension.