Extending Rational Expanding Thurston Maps

TL;DR

We show that every expanding rational Thurston map $f$ on the Riemann sphere, i.e. with $J(f)=\\widehat{\\C}$, admits a uniformly quasi-regular extension $F$ to a neighborhood $\\Omega\subset\\mathbb{R}^3$ of $\\widehat{\\C}$. The construction embeds $\\widehat{\\C}$ as the boundary of a family of shrinking spheres $S_n$ in $\\mathbb{R}^3$ and defines maps $f_n$ by replacing holomorphic behavior near critical points with suitably scaled winding maps, yielding a sequence of spheres $S_n$ accumulating on the sphere and a map $F$ sending $S_n$ to $S_{n-1}$. A careful analysis using Koenig linearization, distortion bounds, and interpolation on prisms ensures that each iterate $F^n$ is $K$-quasi-regular on the corresponding domain $\\Omega_n$ with a distortion constant $K$ independent of $n$, and that the extension agrees with $f$ on $\\widehat{\\C}$. The argument first treats the special case where all critical values are fixed, then extends to the general case by addressing various critical-orbit configurations (multiple critical points in orbits, pre-fixed, and periodic critical values) with adjusted radii and prisms. The results provide a non-homeomorphic higher-dimensional analogue of classical quasi-conformal extension results and apply uniformly across iterates, with potential connections to Lattès maps and higher-dimensional dynamics.

Abstract

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 $Ω\subset \mathbb{R}^3$ of $\widehat{\mathbb{C}}$ to a quasi-regular map $F\colon Ω\to \mathbb{R}^3$. 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 $Ω_n$ of $\widehat{\mathbb{C}}= \mathbb{S}^2 \subset \mathbb{R}^3$, is uniformly quasi-regular. Here $Ω_n$ shrink to $\widehat{\mathbb{C}}$, meaning that $\bigcap Ω_n = \widehat{\mathbb{C}}$. 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.

Figures (4)

• Figure 4.0.1: This indicates the backwards iteration of a post-critical fixed point. The dashed arrows and domains are an example where we iterate backwards for a few iterates with a conformal branch, then we take a pre-image containing the critical point and then again conformal branches of the inverse after passing the critical point.
• Figure 5.0.1: This image shows the approximating spheres $S_1, S_2, S_3$ and $S_4$ as well as the extension $F$ schematically.
• Figure 4.13.1: This illustrates the spheres $S_1, \dots, S_4$ in the case where the approximating spheres are monotone and in the case where the spheres are not monotone and $S_1'\cap S_2'\neq \emptyset$. The degree of the map $f'$ giving rise to $S_1', \dots, S_4'$ is 2.
• Figure 5.11.1: We illustrate the extension around critical points in defined by the maps $\alpha_1$, $\alpha_2$ and $\beta$.

Theorems & Definitions (106)

• Theorem 1.1
• Definition 3.1
• Proposition 3.2
• Definition 3.3
• Remark
• Theorem 3.4
• Definition 3.5
• Definition 3.6
• Definition 3.7
• Definition 3.8