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Borel and Julia directions of meromorphic Schröder functions

Published online by Cambridge University Press:  22 June 2005

KATSUYA ISHIZAKI  and
NIRO YANAGIHARA
KATSUYA ISHIZAKI
Affiliation:
Department of Mathematics, NIPPON Institute of Technology 4-1 Gakuendai Miyashiro, Minamisaitama, Saitama-ken, 345-8501 Japan. e-mail: [email protected]
NIRO YANAGIHARA
Affiliation:
Minami-Iwasaki 671–18, Ichihara-City, Chiba-ken, 290–0244 Japan. e-mail: [email protected]
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Abstract

Meromorphic solutions of the Schröder equation $f(sz)\,{=}\,R(f(z)),$ where $|s|\,{>}\,1$ and $R(w)$ is a rational function with $\deg[R]\,{\geq}\,2$, are studied. We will show that, if $\arg[s]\notin 2\pi {\mathbb Q}$, then $f(z)$ has any Borel direction, without exceptional values other than Picard values, which depend on $R(w)$. Further the case $\arg[s]\,{\in}\,2 \pi {\mathbb Q}$ is also considered. We investigate the relation between Julia directions of $f(z)$ and the Julia set of $R(w)$.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 139 , Issue 1 , July 2005 , pp. 139 - 147
Copyright
2005 Cambridge Philosophical Society

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