Abstract
In 1884, Koenigs showed that when $\varphi$ is an analytic self-map of the unit disk fixing the origin, with $0\lt |\varphi'(0)|\lt 1$, then Schroeder's functional equation, $f\circ\varphi=\varphi'(0)f$, can be solved for a unique analytic function $f$ in the disk with $f'(0)=1$. Here we consider a natural analogue of Schroeder's equation in the unit ball of ${\Bbb C}^N$ for $N\gt 1$, namely, $f\circ\varphi=\varphi'(0)f$ where $\varphi$ is an analytic self-map of the unit ball fixing the origin and $f$ is to be a ${\Bbb C}^N$-valued analytic map on the ball. Under some natural hypotheses on $\varphi$, we give necessary and sufficient conditions for the existence of a solution $f$ satisfying $f'(0)=I$ and then describe all analytic solutions in the ball. We also discuss various phenomena which may occur in the several variable setting that do not occur when $N=1$.
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