We consider holomorphic self-maps of the unit ball in (). In the one-dimensional case, when has no fixed points in and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map , and therefore, in this case, the dynamical properties of are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on at its Denjoy-Wolff point. As a result, we construct a semi-conjugation , which maps the ball into the right half-plane of , and solves the functional equation , where is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of .
Cite this article
Filippo Bracci, Graziano Gentili, Pietro Poggi-Corradini, Valiron’s construction in higher dimension. Rev. Mat. Iberoam. 26 (2010), no. 1, pp. 57–76DOI 10.4171/RMI/593