Valiron’s Construction in Higher Dimension

We consider holomorphic self-maps φ of the unit ball B in C (N = 1, 2, 3, ...). In the one-dimensional case, when φ has no fixed points in D := B 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.