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Residual Julia Sets of Meromorphic Functions
Published online by Cambridge University Press: 03 July 2006
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
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 141 , Issue 1 , July 2006 , pp. 113 - 126
- Copyright
- 2006 Cambridge Philosophical Society
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