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

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map of the polydisc which does not admit fixed points in . We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic and the Abel equation for a parabolic nonzero-step . This is done by studying the canonical Kobayashi hyperbolic semi-model of 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. [ABSTRACT FROM AUTHOR]