Valiron and Abel equations for holomorphic self-maps of the polydisc

TL;DR

This paper extends classical Valiron and Abel equations to holomorphic self-maps of the polydisc without fixed points, introducing hyperbolicity and parabolicity concepts, and providing solutions and normal forms for these equations.

Contribution

It generalizes classical one-variable results to the polydisc setting, introducing new notions and solving the Valiron and Abel equations for fixed point free maps.

Findings

Solved the Valiron equation for hyperbolic maps.

Solved the Abel equation for parabolic non-zero step maps.

Described the space of all solutions for the Valiron equation.

Abstract

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map $f: \Delta^N \to \Delta^N$ of the polydisc which does not admit fixed points in $\Delta^N$. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic $f$ and the Abel equation for a parabolic nonzero-step $f$. This is done by studying the canonical Kobayashi hyperbolic semi-model of $f$ and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation we also describe the space of all solutions.