Classification of critically fixed anti-Thurston maps
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866913569821425664 |
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| author | Geyer, Lukas Hlushchanka, Mikhail |
| author_facet | Geyer, Lukas Hlushchanka, Mikhail |
| contents | We provide a complete combinatorial classification of critically fixed anti-Thurston maps, i.e., orientation-reversing branched covers of the 2-sphere that fix every critical point. The first step in the proof, and an interesting result in its own right, is a combinatorial classification of critically fixed anti-rational maps as "Schottky maps" associated to certain plane graphs. Both of these classification results heavily rely on an orientation-reversing version of Thurstons's theory, including the canonical decomposition of anti-Thurston maps, which we develop in this paper. Lastly, we give some applications to the global curve attractor and twisting problems, as well as to anti-rational maps with symmetries and to critically fixed anti-polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2006_10788 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Classification of critically fixed anti-Thurston maps Geyer, Lukas Hlushchanka, Mikhail Dynamical Systems 37F20, 37F10 (Primary) 30D05 (Secondary) We provide a complete combinatorial classification of critically fixed anti-Thurston maps, i.e., orientation-reversing branched covers of the 2-sphere that fix every critical point. The first step in the proof, and an interesting result in its own right, is a combinatorial classification of critically fixed anti-rational maps as "Schottky maps" associated to certain plane graphs. Both of these classification results heavily rely on an orientation-reversing version of Thurstons's theory, including the canonical decomposition of anti-Thurston maps, which we develop in this paper. Lastly, we give some applications to the global curve attractor and twisting problems, as well as to anti-rational maps with symmetries and to critically fixed anti-polynomials. |
| title | Classification of critically fixed anti-Thurston maps |
| topic | Dynamical Systems 37F20, 37F10 (Primary) 30D05 (Secondary) |
| url | https://arxiv.org/abs/2006.10788 |