Julia sets appear quasiconformally in the Mandelbrot set
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2018
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| _version_ | 1866913194391371776 |
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| author | Kawahira, Tomoki Kisaka, Masashi |
| author_facet | Kawahira, Tomoki Kisaka, Masashi |
| contents | In this paper we prove the following: Take any "small Mandelbrot set" and zoom in a neighborhood of a parabolic or Misiurewicz parameter in it, then we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a parabolic or Misiurewicz Julia set. Furthermore, zoom in its middle part, then we can see a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears. A similar nested structure exists in the Julia set for any parameter in the "smaller Mandelbrot set". We can also find images of a Julia sets by quasiconformal maps with dilatation arbitrarily close to 1. This answers a question by Adrian Douady. All the parameters belonging to these images are semihyperbolic and this leads to the fact that the set of semihyperbolic but non-Misiurewicz and non-hyperbolic parameters is dense with Hausdorff dimension 2 in the boundary of the Mandelbrot set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_00176 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Julia sets appear quasiconformally in the Mandelbrot set Kawahira, Tomoki Kisaka, Masashi Dynamical Systems Primary 37F46, Secondary 37F25, 37F31 In this paper we prove the following: Take any "small Mandelbrot set" and zoom in a neighborhood of a parabolic or Misiurewicz parameter in it, then we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a parabolic or Misiurewicz Julia set. Furthermore, zoom in its middle part, then we can see a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears. A similar nested structure exists in the Julia set for any parameter in the "smaller Mandelbrot set". We can also find images of a Julia sets by quasiconformal maps with dilatation arbitrarily close to 1. This answers a question by Adrian Douady. All the parameters belonging to these images are semihyperbolic and this leads to the fact that the set of semihyperbolic but non-Misiurewicz and non-hyperbolic parameters is dense with Hausdorff dimension 2 in the boundary of the Mandelbrot set. |
| title | Julia sets appear quasiconformally in the Mandelbrot set |
| topic | Dynamical Systems Primary 37F46, Secondary 37F25, 37F31 |
| url | https://arxiv.org/abs/1804.00176 |