Thurston's pullback map, invariant covers, and the global dynamics on curves
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| Format: | Preprint |
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2024
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| _version_ | 1866909374690099200 |
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| author | Bonk, Mario Hlushchanka, Mikhail Lodge, Russell |
| author_facet | Bonk, Mario Hlushchanka, Mikhail Lodge, Russell |
| contents | We consider rational maps $f$ on the Riemann sphere $\widehat {\mathbb{C}}$ with an $f$-invariant set $P\subset \widehat {\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the corresponding Thurston pullback map $σ_f$ on the completion $\overline{\mathcal{T}_P}$ of the associated Teichmüller space $\mathcal{T}_P$ with respect to the Weil-Petersson metric is easy to understand when $\overline{\mathcal{T}_P}$ admits a cover by sets with good combinatorial and dynamical properties. In particular, the map $f$ has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if $P$ contains all critical points of $f$ and each point in $P$ is periodic, then such a cover of $\overline{\mathcal{T}_P}$ can be obtained from a $σ_f$-invariant tessellation by ideal hyperbolic triangles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_00732 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Thurston's pullback map, invariant covers, and the global dynamics on curves Bonk, Mario Hlushchanka, Mikhail Lodge, Russell Dynamical Systems Complex Variables 37F10, 37F20 We consider rational maps $f$ on the Riemann sphere $\widehat {\mathbb{C}}$ with an $f$-invariant set $P\subset \widehat {\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the corresponding Thurston pullback map $σ_f$ on the completion $\overline{\mathcal{T}_P}$ of the associated Teichmüller space $\mathcal{T}_P$ with respect to the Weil-Petersson metric is easy to understand when $\overline{\mathcal{T}_P}$ admits a cover by sets with good combinatorial and dynamical properties. In particular, the map $f$ has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if $P$ contains all critical points of $f$ and each point in $P$ is periodic, then such a cover of $\overline{\mathcal{T}_P}$ can be obtained from a $σ_f$-invariant tessellation by ideal hyperbolic triangles. |
| title | Thurston's pullback map, invariant covers, and the global dynamics on curves |
| topic | Dynamical Systems Complex Variables 37F10, 37F20 |
| url | https://arxiv.org/abs/2411.00732 |