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