Valiron’s construction in higher dimension

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 renormalizationprocedure due to Valiron which allows to semi-linearize the map $\phi$, andtherefore, in this case, the dynamical properties of $\phi$ are wellunderstood. In what follows, we generalize the classical Valiron constructionto higher dimensions under some weak assumptions on $\v$ at its Denjoy-Wolffpoint. As a result, we construct a semi-conjugation $\sigma$, which maps theball 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 $\v$.