Critically fixed Thurston maps: classification, recognition, and twisting
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911399786053632 |
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| author | Hlushchanka, Mikhail Prochorov, Nikolai |
| author_facet | Hlushchanka, Mikhail Prochorov, Nikolai |
| contents | An orientation-preserving branched covering map $f\colon S^2 \to S^2$ is called a critically fixed Thurston map if $f$ fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between Möbius conjugacy classes of critically fixed rational maps and isomorphism classes of planar embedded connected graphs. In the paper, we generalize this result to the whole family of critically fixed Thurston maps. Namely, we show that each critically fixed Thurston map $f$ is obtained by applying the blow-up operation, introduced by Kevin Pilgrim and Tan Lei, to a pair $(G,φ)$, where $G$ is a planar embedded graph in $S^2$ without isolated vertices and $φ$ is an orientation-preserving homeomorphism of $S^2$ that fixes each vertex of $G$. This result allows us to provide a classification of combinatorial equivalence classes of critically fixed Thurston maps. We also develop an algorithm that reconstructs (up to isotopy) the pair $(G,φ)$ associated with a critically fixed Thurston map $f$. Finally, we solve some special instances of the twisting problem for the family of critically fixed Thurston maps obtained by blowing up pairs $(G, \mathrm{id}_{S^2})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_14759 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Critically fixed Thurston maps: classification, recognition, and twisting Hlushchanka, Mikhail Prochorov, Nikolai Dynamical Systems 37F20, 37F10 An orientation-preserving branched covering map $f\colon S^2 \to S^2$ is called a critically fixed Thurston map if $f$ fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between Möbius conjugacy classes of critically fixed rational maps and isomorphism classes of planar embedded connected graphs. In the paper, we generalize this result to the whole family of critically fixed Thurston maps. Namely, we show that each critically fixed Thurston map $f$ is obtained by applying the blow-up operation, introduced by Kevin Pilgrim and Tan Lei, to a pair $(G,φ)$, where $G$ is a planar embedded graph in $S^2$ without isolated vertices and $φ$ is an orientation-preserving homeomorphism of $S^2$ that fixes each vertex of $G$. This result allows us to provide a classification of combinatorial equivalence classes of critically fixed Thurston maps. We also develop an algorithm that reconstructs (up to isotopy) the pair $(G,φ)$ associated with a critically fixed Thurston map $f$. Finally, we solve some special instances of the twisting problem for the family of critically fixed Thurston maps obtained by blowing up pairs $(G, \mathrm{id}_{S^2})$. |
| title | Critically fixed Thurston maps: classification, recognition, and twisting |
| topic | Dynamical Systems 37F20, 37F10 |
| url | https://arxiv.org/abs/2212.14759 |