Extending Rational Expanding Thurston Maps

We consider postcritically finite rational maps $f\colon \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ whose Julia set is the whole Riemann sphere $\widehat{\mathbb{C}}$. We call such a map an expanding rational Thurston map. Identifying $\widehat{\mathbb{C}}$ with the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$, we show that $f$ may be extended on a neighborhood $\Omega\subset \mathbb{R}^3$ of $\widehat{\mathbb{C}}$ to a quasi-regular map $F\colon \Omega \to \mathbb{R}^3$. In fact, $F$ is uniformly quasi-regular in the following sense. The sequence of iterates $F^n$, each of which is defined on a neighborhood $\Omega_n$ of $\widehat{\mathbb{C}}= \mathbb{S}^2 \subset \mathbb{R}^3$, is uniformly quasi-regular. Here $\Omega_n$ shrink to $\widehat{\mathbb{C}}$, meaning that $\bigcap \Omega_n = \widehat{\mathbb{C}}$. This result may be viewed as a non-homeomorphic version of the extension of a quasi-conformal mapping $f:\mathbb{R}^2\to \mathbb{R}^2$ to a quasi-conformal mapping $F\colon \mathbb{R}^3 \to \mathbb{R}^3$ due to Ahlfors.