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

2015/09/14 by Leandro Arosio, Pavel Gumenyuk, Arosio, Leandro +1
Mathematics · #39B12 #39B32 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 32H50 #Secondary: 37F99 #math.CV #math.DS #msc:32H50 #msc:37F99 #msc:39B12 #msc:39B32

paper · pdf · doi:10.48550/arxiv.1509.04169

A few references are added

arxiv created 2016/02/11 · arxiv updated 2016/02/15

Abstract

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map f: Δ^N → Δ^N of the polydisc which does not admit fixed points in Δ^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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