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Dynamics of functions meromorphic outside a small set

Published online by Cambridge University Press:  04 June 2001

I. N. BAKER,
P. DOMÍNGUEZ and
M. E. HERRING
I. N. BAKER
Affiliation:
Department of Mathematics, Imperial College of Science, Technology and Medicine, London SW7 2BZ, UK
P. DOMÍNGUEZ
Affiliation:
F.C. Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio, Col. San Manuel, C.U., Puebla, Pue. C.P. 72570, México
M. E. HERRING
Affiliation:
Department of Mathematics, Imperial College of Science, Technology and Medicine, London SW7 2BZ, UK
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Abstract

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The theory of Fatou and Julia is extended to include the dynamics of functions f which are meromorphic in \widehat{\mathbb{C}} outside a totally disconnected compact set E(f) at whose points the cluster set of f is \widehat{\mathbb{C}}. The Julia set is defined not only by the standard approach but is also characterized in terms of the set of points whose orbits approach a point of E(f). For the subclass where E(f) has a complement of class OAD and the inverse of f has a finite set of singular points it is shown that neither wandering components nor Baker domains occur in F(f)>. As an application, functions of a certain general class are shown to have a totally disconnected Julia set.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 21 , Issue 3 , June 2001, pp. 647 - 672
Copyright
2001 Cambridge University Press