Transcendental Julia Sets of Minimal Hausdorff Dimension
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866918085482512384 |
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| author | Burkart, Jack Lazebnik, Kirill |
| author_facet | Burkart, Jack Lazebnik, Kirill |
| contents | We show the existence of transcendental entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of $f$ has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of $f$, answering a question of Rippon and Stallard (arXiv:1703.11001). Our proof relies on a quasiconformal-surgery approach developed in arXiv:2101.04219. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_05001 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Transcendental Julia Sets of Minimal Hausdorff Dimension Burkart, Jack Lazebnik, Kirill Complex Variables Dynamical Systems 37F10, 30D05, 37F35 We show the existence of transcendental entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of $f$ has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of $f$, answering a question of Rippon and Stallard (arXiv:1703.11001). Our proof relies on a quasiconformal-surgery approach developed in arXiv:2101.04219. |
| title | Transcendental Julia Sets of Minimal Hausdorff Dimension |
| topic | Complex Variables Dynamical Systems 37F10, 30D05, 37F35 |
| url | https://arxiv.org/abs/2109.05001 |