The linear fractional model on the ball

Abstract

Given a holomorphic self-map $\phi$ of the ball of ${\mathbb{C}}^N$, we study whether there exists a map $\sigma$ and a linear fractional transformation $A$ such that $\sigma \circ \phi =A\circ \sigma$. This is an important result when $N=1$ with a great number of applications. We extend this result to the multi-dimensional setting for a large class of maps. Applications to commuting holomorphic self-maps are given.