Valiron's construction in higher dimension.

Abstract. We consider holomorphic self-maps f of the unit ball B^N in C^N (N = 1, 2, 3, . . . ). In the one-dimensional case, when f 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 f, and therefore, in this case, the dynamical properties of f are well understood. In this paper, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on f at its Denjoy-Wolff point. As a result, we construct a semiconjugation h, which maps the ball into the right half-plane of C, and solves the functional equation h(f(z))= rh(z), where r > 1 is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of f.