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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.18015 |
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| _version_ | 1866911222862970880 |
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| author | Meyer, Daniel Münch, Julia |
| author_facet | Meyer, Daniel Münch, Julia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18015 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extending Rational Expanding Thurston Maps Meyer, Daniel Münch, Julia Complex Variables Dynamical Systems 30C65, 37F31 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. |
| title | Extending Rational Expanding Thurston Maps |
| topic | Complex Variables Dynamical Systems 30C65, 37F31 |
| url | https://arxiv.org/abs/2510.18015 |