On McMullen-like mappings
We introduce a generalization of the McMullen family f_(z) = z^n /zd^. In 1988 C. McMullen showed that the Julia set of f_ is a Cantor set of circles if and only if 1/n 1/d < 1 and the simple critical values of f_ belong to the trap door. We generalize this behavior and we define a McMullen-like...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2015 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:169434 |
| Acceso en línea: | https://ddd.uab.cat/record/169434 |
| Access Level: | acceso abierto |
| Palabra clave: | Complex dynamics Julia sets McMullen family Rational maps |
| Sumario: | We introduce a generalization of the McMullen family f_(z) = z^n /zd^. In 1988 C. McMullen showed that the Julia set of f_ is a Cantor set of circles if and only if 1/n 1/d < 1 and the simple critical values of f_ belong to the trap door. We generalize this behavior and we define a McMullen-like mapping as a rational map f associated to a hyperbolic postcritically finite polynomial P and a pole data D where we encode, basically, the location of every pole of f and the local degree at each pole. In the McMullen family the polynomial P is z z^n and the pole data D is the pole located at the origin that maps to infinity with local degree d. As in the McMullen family f_ we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial P and the pole data D. We prove that the arithmetic condition is necessary using the theory of Thurston's obstructions, and sufficient by quasiconformal surgery. |
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