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 [Formula: see text] of the polydisc which does not admit fixed points in [Formula: see text]. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic [Formula: see text] and the Abel equation for a parabolic nonzero-step [Formula: see text]. This is done by studying the canonical Kobayashi hyperbolic semi-model of [Formula: see text] 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.