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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.14610 |
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| _version_ | 1866915142504022016 |
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| author | Bishop, Christopher J. Eremenko, Alexandre Lazebnik, Kirill |
| author_facet | Bishop, Christopher J. Eremenko, Alexandre Lazebnik, Kirill |
| contents | A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert's lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge's theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Shapes of Rational Lemniscates Bishop, Christopher J. Eremenko, Alexandre Lazebnik, Kirill Complex Variables Dynamical Systems 30C10, 30C62, 30E10, 41A20 A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert's lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge's theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps. |
| title | On the Shapes of Rational Lemniscates |
| topic | Complex Variables Dynamical Systems 30C10, 30C62, 30E10, 41A20 |
| url | https://arxiv.org/abs/2407.14610 |