Borel and Julia directions of meromorphic Schröder functions

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 $R(w)$.