Finite and infinite degree Thurston maps with extra marked points
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916428440928256 |
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| author | Prochorov, Nikolai |
| author_facet | Prochorov, Nikolai |
| contents | We investigate the family of marked Thurston maps that are defined everywhere on the topological sphere $S^2$, potentially excluding at most countable closed set of essential singularities. We show that when an unmarked Thurston map $f$ is realized by a postsingularly finite holomorphic map, the marked Thurston map $(f, A)$, where $A \subset S^2$ is the corresponding finite marked set, admits such a realization if and only if it has no degenerate Levy cycle. To obtain this result, we analyze the associated pullback map $σ_{f, A}$ defined on the Teichmüller space $\mathcal{T}_A$ and demonstrate that some of its iterates admit well-behaved invariant complex sub-manifolds within $\mathcal{T}_A$. By applying powerful machinery of one-dimensional complex dynamics and hyperbolic geometry, we gain a clear understanding of the behavior of the map $σ_{f, A}$ restricted to the corresponding invariant subset of the Teichmüller space $\mathcal{T}_A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06206 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite and infinite degree Thurston maps with extra marked points Prochorov, Nikolai Dynamical Systems Primary 37F20, Secondary 37F10, 37F15, 37F34 We investigate the family of marked Thurston maps that are defined everywhere on the topological sphere $S^2$, potentially excluding at most countable closed set of essential singularities. We show that when an unmarked Thurston map $f$ is realized by a postsingularly finite holomorphic map, the marked Thurston map $(f, A)$, where $A \subset S^2$ is the corresponding finite marked set, admits such a realization if and only if it has no degenerate Levy cycle. To obtain this result, we analyze the associated pullback map $σ_{f, A}$ defined on the Teichmüller space $\mathcal{T}_A$ and demonstrate that some of its iterates admit well-behaved invariant complex sub-manifolds within $\mathcal{T}_A$. By applying powerful machinery of one-dimensional complex dynamics and hyperbolic geometry, we gain a clear understanding of the behavior of the map $σ_{f, A}$ restricted to the corresponding invariant subset of the Teichmüller space $\mathcal{T}_A$. |
| title | Finite and infinite degree Thurston maps with extra marked points |
| topic | Dynamical Systems Primary 37F20, Secondary 37F10, 37F15, 37F34 |
| url | https://arxiv.org/abs/2410.06206 |