Valiron's construction in higher dimension

Abstract

We consider holomorphic self-maps $\v$ of the unit ball $\B^N$ in $\C^N$ ($N=1,2,3,...$). In the one-dimensional case, when $\v$ has no fixed points in $\D\defeq \B^1$ and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map $\phi$, and therefore, in this case, the dynamical properties of $\phi$ are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on $\v$ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation $\sigma$, which maps the ball into the right half plane of $\C$, and solves the functional equation $\sigma\circ \v=\lambda \sigma$, where $\lambda>1$ is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of $\v$.