Valiron's construction in higher dimension

Abstract

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