Valiron’s construction in higher dimension

Abstract

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 ,a nd solves the functional equation σ ◦ ϕ = λσ ,w here λ> 1i s the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of ϕ.