Uniformization of gasket Julia sets
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910716415442944 |
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| author | Luo, Yusheng Ntalampekos, Dimitrios |
| author_facet | Luo, Yusheng Ntalampekos, Dimitrios |
| contents | The object of the paper is to characterize gasket Julia sets of rational maps that can be uniformized by round gaskets. We restrict to rational maps without critical points on the Julia set. Under these conditions, we prove that a Julia set can be quasiconformally uniformized by a round gasket if and only if it is a fat gasket, i.e., boundaries of Fatou components intersect tangentially. We also prove that a Julia set can be uniformized by a round gasket with a David homeomorphism if and only if every Fatou component is a quasidisk; equivalently, there are no parabolic cycles of multiplicity 2. Our theorem applies to show that gasket Julia sets and limit sets of Kleinian groups can be locally quasiconformally homeomorphic, although globally this is conjectured to be false. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniformization of gasket Julia sets Luo, Yusheng Ntalampekos, Dimitrios Dynamical Systems Complex Variables Primary 37F10, 37F31, Secondary 30C62, 37F30 The object of the paper is to characterize gasket Julia sets of rational maps that can be uniformized by round gaskets. We restrict to rational maps without critical points on the Julia set. Under these conditions, we prove that a Julia set can be quasiconformally uniformized by a round gasket if and only if it is a fat gasket, i.e., boundaries of Fatou components intersect tangentially. We also prove that a Julia set can be uniformized by a round gasket with a David homeomorphism if and only if every Fatou component is a quasidisk; equivalently, there are no parabolic cycles of multiplicity 2. Our theorem applies to show that gasket Julia sets and limit sets of Kleinian groups can be locally quasiconformally homeomorphic, although globally this is conjectured to be false. |
| title | Uniformization of gasket Julia sets |
| topic | Dynamical Systems Complex Variables Primary 37F10, 37F31, Secondary 30C62, 37F30 |
| url | https://arxiv.org/abs/2411.17227 |