Valiron’s construction in higher dimension

Abstract

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