We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map f: δN → δN of the polydisc which does not admit fixed points in δN. 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.
Valiron and Abel equations for holomorphic self-maps of the polydisc
Gumenyuk P.
2016-01-01
Abstract
We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map f: δN → δN of the polydisc which does not admit fixed points in δN. 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.File in questo prodotto:
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