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Residual Julia Sets of Meromorphic Functions

Published online by Cambridge University Press:  03 July 2006

TUEN WAI NG
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. e-mail: ntw@maths.hku.hk
JIAN HUA ZHENG
Affiliation:
Department of Mathematical Science, Tsinghua University, Beijing 100084, People's Republic of China. e-mail: jzheng@math.tsinghua.edu.cn
YAN YU CHOI
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. e-mail: h0034138@graduate.hku.hk

Abstract

In this paper, we study the residual Julia sets of meromorphic functions. In fact, we prove that if a meromorphic function $f$ belongs to the class ${\bf S}$ and its Julia set is locally connected, then the residual Julia set of $f$ is empty if and only if its Fatou set $F(f)$ has a completely invariant component or consists of only two components. We also show that if $f$ is a meromorphic function which is not of the form $\alpha + (z\,-\,\alpha)^{-k}e^{g(z)}$, where $k$ is a natural number, $\alpha$ is a complex number and $g$ is an entire function, then $f$ has buried components provided that $f$ has no completely invariant components and its Julia set $J(f)$ is disconnected. Moreover, if $F(f)$ has an infinitely connected component, then the singleton buried components are dense in $J(f)$. This generalizes a result of Baker and Domínguez. Finally, we give some examples of meromorphic functions with buried points but without any buried components.

Type
Research Article
Copyright
2006 Cambridge Philosophical Society

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