The Parabolic Mandelbrot Set
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866914748384149504 |
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| author | Petersen, Carsten Lunde Roesch, Pascale |
| author_facet | Petersen, Carsten Lunde Roesch, Pascale |
| contents | We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus $M_1$ of the family of quadratic rational maps tangent to the identity at $\infty$. We prove that this locus in homeomorphic to the Mandelbrot set $M$ and that the homeomorphism is unique, provided it identifies maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this homeomorphism from $M$ to $M_1$ is nowhere Hölder on the boundary and so can not have even locally a quasi-conformal extension to complements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_09407 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Parabolic Mandelbrot Set Petersen, Carsten Lunde Roesch, Pascale Dynamical Systems 37F46 (Primary) 30D05, 37F31 (Secondary) We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus $M_1$ of the family of quadratic rational maps tangent to the identity at $\infty$. We prove that this locus in homeomorphic to the Mandelbrot set $M$ and that the homeomorphism is unique, provided it identifies maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this homeomorphism from $M$ to $M_1$ is nowhere Hölder on the boundary and so can not have even locally a quasi-conformal extension to complements. |
| title | The Parabolic Mandelbrot Set |
| topic | Dynamical Systems 37F46 (Primary) 30D05, 37F31 (Secondary) |
| url | https://arxiv.org/abs/2107.09407 |