The rational maps $z\mapsto 1+1/\omega z^d$ have no Herman rings
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- by Rodrigo Bamón and Juan Bobenrieth PDF
- Proc. Amer. Math. Soc. 127 (1999), 633-636 Request permission
Abstract:
We prove that for every $d \in \mathbb {N}, d \geq 2$, the rational maps in the family $\{ z \mapsto 1 + 1/\omega z^d : \omega \in \textbf {C} \setminus \{0 \} \}$ have no Herman rings. From this we conclude a dynamical characterization for the parameters in the Mandelbrot set of these families. Further, we show that hyperbolic maps are dense in this family if and only if the set of parameters for which the Julia set is the whole sphere has no interior.References
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Additional Information
- Rodrigo Bamón
- Affiliation: Departamento de Matemática, Facultad de Ciencias, Universidad de Chile, Casilla 653 Santiago, Chile
- Email: rbamon@abello.dic.uchile.cl
- Juan Bobenrieth
- Affiliation: Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
- Email: jbobenri@zeus.dci.ubiobio.cl
- Received by editor(s): March 5, 1997
- Received by editor(s) in revised form: May 29, 1997
- Additional Notes: The first author was partially supported by Fondecyt, Proyecto 1960848.
- Communicated by: Mary Rees
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 633-636
- MSC (1991): Primary 58F23, 30D05
- DOI: https://doi.org/10.1090/S0002-9939-99-04566-9
- MathSciNet review: 1469397