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

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.

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