Valiron's construction in higher dimension

We consider holomorphic self-maps \v of the unit ball ${\text{B}}^N$ in ${\text{C}}^N$ ($N=1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}$). In the one-dimensional case, when \v has no fixed points in $\text{D}\text{defeq}{\text{B}}^1$ and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map $\varphi$, and therefore, in this case, the dynamical properties of $\varphi$ 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 $\text{C}$, and solves the functional equation $\sigma \circ \check{=}\lambda \sigma$, where $\lambda >1$ is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of \v.