Valiron's construction in higher dimension

TL;DR

This paper extends Valiron's classical semi-linearization technique from one-dimensional holomorphic self-maps of the disk to higher-dimensional unit balls, under weak assumptions at the Denjoy-Wolff point, constructing a semi-conjugation that simplifies the dynamics.

Contribution

It generalizes Valiron's construction to higher dimensions, providing a semi-conjugation for holomorphic self-maps of the unit ball with weak boundary assumptions.

Findings

Constructed a semi-conjugation $\sigma$ mapping the ball to the right half-plane.

Solved the functional equation $\sigma\circ \\v=\\lambda \\sigma$ with $\\lambda>1$.

Extended classical one-dimensional results to higher dimensions.

Abstract

We consider holomorphic self-maps $\v$ of the unit ball $\B^N$ in $\C^N$ ($N=1,2,3,...$). In the one-dimensional case, when $\v$ has no fixed points in $\D\defeq \B^1$ 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 $\v$ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation $\sigma$, which maps the ball into the right half plane of $\C$, and solves the functional equation $\sigma\circ \v=\lambda \sigma$, where $\lambda>1$ is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of $\v$.