Dynamical approximations of postsingularly finite entire maps
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866907767440146432 |
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| author | Mukundan, Malavika Prochorov, Nikolai Reinke, Bernhard |
| author_facet | Mukundan, Malavika Prochorov, Nikolai Reinke, Bernhard |
| contents | We prove that every postsingularly finite entire map $g$ can be approximated by a sequence of postcritically finite complex polynomials $(g_n)$ such that their postsingular dynamics $g|P_g$ and $g_n|P_{g_n}$ are conjugate for every $n \in \mathbb{N}$. To establish this result, we introduce the notion of combinatorial convergence for sequences of entire Thurston maps defined on the topological plane $\mathbb{R}^2$ and having the same marked set $A$. We prove that if such a sequence $(f_n)$ converges combinatorially to a Thurston map $f$, then the sequence of Thurston pullback maps $(σ_{f_n})$ converges to $σ_f$ locally uniformly on the Teichmüller space $\mathrm{Teich}(\mathbb{R}^2, A)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_17793 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dynamical approximations of postsingularly finite entire maps Mukundan, Malavika Prochorov, Nikolai Reinke, Bernhard Dynamical Systems 37F20 (primary), 37F10, 37F34 (secondary) We prove that every postsingularly finite entire map $g$ can be approximated by a sequence of postcritically finite complex polynomials $(g_n)$ such that their postsingular dynamics $g|P_g$ and $g_n|P_{g_n}$ are conjugate for every $n \in \mathbb{N}$. To establish this result, we introduce the notion of combinatorial convergence for sequences of entire Thurston maps defined on the topological plane $\mathbb{R}^2$ and having the same marked set $A$. We prove that if such a sequence $(f_n)$ converges combinatorially to a Thurston map $f$, then the sequence of Thurston pullback maps $(σ_{f_n})$ converges to $σ_f$ locally uniformly on the Teichmüller space $\mathrm{Teich}(\mathbb{R}^2, A)$. |
| title | Dynamical approximations of postsingularly finite entire maps |
| topic | Dynamical Systems 37F20 (primary), 37F10, 37F34 (secondary) |
| url | https://arxiv.org/abs/2305.17793 |