  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 $\v$.
