Mating parabolic rational maps with Hecke groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911539184795648 |
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| author | Bullett, Shaun Lomonaco, Luna Lyubich, Mikhail Mukherjee, Sabyasachi |
| author_facet | Bullett, Shaun Lomonaco, Luna Lyubich, Mikhail Mukherjee, Sabyasachi |
| contents | We prove that any degree $d$ rational map having a parabolic fixed point of multiplier $1$ with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group $H_{d+1}$, with the mating realized by an algebraic correspondence. This confirms the parabolic version of a conjecture on mateability between rational maps and Hecke groups made in \cite{BF1}. The proof is in two steps. The first is the construction of a pinched polynomial-like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial-like map to an algebraic correspondence via a suitable branched covering. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_14780 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mating parabolic rational maps with Hecke groups Bullett, Shaun Lomonaco, Luna Lyubich, Mikhail Mukherjee, Sabyasachi Dynamical Systems Complex Variables Group Theory 30C10, 30F35, 37F05, 37F10, 37F31, 37F32 (Primary), 30F10, 37C85 (Secondary) We prove that any degree $d$ rational map having a parabolic fixed point of multiplier $1$ with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group $H_{d+1}$, with the mating realized by an algebraic correspondence. This confirms the parabolic version of a conjecture on mateability between rational maps and Hecke groups made in \cite{BF1}. The proof is in two steps. The first is the construction of a pinched polynomial-like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial-like map to an algebraic correspondence via a suitable branched covering. |
| title | Mating parabolic rational maps with Hecke groups |
| topic | Dynamical Systems Complex Variables Group Theory 30C10, 30F35, 37F05, 37F10, 37F31, 37F32 (Primary), 30F10, 37C85 (Secondary) |
| url | https://arxiv.org/abs/2407.14780 |