We consider holomorphic self-maps $\varphi$ of the unit ball
$\mathbb B^N$ in $\mathbb C^N$ ($N=1,2,3,\dots$). In the one-dimensional case, when
$\varphi$ has no fixed points in $\mathbb D\defeq \mathbb B^1$ 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 $\varphi$ at its
Denjoy-Wolff point. As a result, we construct a semi-conjugation
$\sigma$, which maps the ball into the right half-plane of $\mathbb C$, and
solves the functional equation $\sigma\circ \varphi=\lambda \sigma$,
where $\lambda > 1$ is the (inverse of the) boundary dilation
coefficient at the Denjoy-Wolff point of $\varphi$.