Extending Rational Expanding Thurston Maps

Abstract

We consider postcritically finite rational maps $f:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ whose Julia set is the whole Riemann sphere $\hat{\mathbb{C}}$. We call such a map an expanding rational Thurston map. Identifying $\hat{\mathbb{C}}$ with the unit sphere ${\mathbb{S}}^2$ in ${\mathbb{R}}^3$, we show that $f$ may be extended on a neighborhood $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$ of $\hat{\mathbb{C}}$ to a quasi-regular map $F:\hat{I}\text{c◯}\to {\mathbb{R}}^3$. 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 $\hat{I}{\text{c◯}}_n$ of $\hat{\mathbb{C}}={\mathbb{S}}^2\subset {\mathbb{R}}^3$, is uniformly quasi-regular. Here $\hat{I}{\text{c◯}}_n$ shrink to $\hat{\mathbb{C}}$, meaning that $\bigcap \hat{I}{\text{c◯}}_n=\hat{\mathbb{C}}$. 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 $F:{\mathbb{R}}^3\to {\mathbb{R}}^3$ due to Ahlfors.

59 pages, 4 figures