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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The rational maps $z \mapsto 1 + 1/\omega z^d $ have no Herman rings


Authors: Rodrigo Bamón and Juan Bobenrieth
Journal: Proc. Amer. Math. Soc. 127 (1999), 633-636
MSC (1991): Primary 58F23, 30D05.
MathSciNet review: 1469397
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04566-9
PII: S 0002-9939(99)04566-9
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
Article copyright: © Copyright 1999 American Mathematical Society