Schwarz reflections and the Tricorn
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866916878458290176 |
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| author | Lee, Seung-Yeop Lyubich, Mikhail Makarov, Nikolai G. Mukherjee, Sabyasachi |
| author_facet | Lee, Seung-Yeop Lyubich, Mikhail Makarov, Nikolai G. Mukherjee, Sabyasachi |
| contents | We continue our exploration of the family $\mathcal{S}$ of Schwarz reflection maps with respect to the cardioid and a circle which was initiated in our earlier work. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family and those of the basilica limb of the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials. We also show that every geometrically finite map in $\mathcal{S}$ arises as a conformal mating of a unique geometrically finite quadratic anti-holomorphic polynomial and a reflection map arising from the ideal triangle group. We then follow up with a combinatorial mating description for the periodically repelling maps in $\mathcal{S}$. Finally, we show that the locally connected topological model of the connectedness locus of $\mathcal{S}$ is naturally homeomorphic to such a model of the basilica limb of the Tricorn. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_01573 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Schwarz reflections and the Tricorn Lee, Seung-Yeop Lyubich, Mikhail Makarov, Nikolai G. Mukherjee, Sabyasachi Dynamical Systems Complex Variables 37F10, 37F20, 37F31, 37F32, 37F46, 30C45, 30D05, 30D40, 82B21 We continue our exploration of the family $\mathcal{S}$ of Schwarz reflection maps with respect to the cardioid and a circle which was initiated in our earlier work. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family and those of the basilica limb of the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials. We also show that every geometrically finite map in $\mathcal{S}$ arises as a conformal mating of a unique geometrically finite quadratic anti-holomorphic polynomial and a reflection map arising from the ideal triangle group. We then follow up with a combinatorial mating description for the periodically repelling maps in $\mathcal{S}$. Finally, we show that the locally connected topological model of the connectedness locus of $\mathcal{S}$ is naturally homeomorphic to such a model of the basilica limb of the Tricorn. |
| title | Schwarz reflections and the Tricorn |
| topic | Dynamical Systems Complex Variables 37F10, 37F20, 37F31, 37F32, 37F46, 30C45, 30D05, 30D40, 82B21 |
| url | https://arxiv.org/abs/1812.01573 |