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arXiv · 1509.04169

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

Leandro Arosio·Pavel Gumenyuk
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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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Leandro Arosio, Pavel Gumenyuk. 2015-09-14. Valiron and Abel equations for holomorphic self-maps of the polydisc. https://arxiv.org/abs/1509.04169

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