Valiron and Abel equations for holomorphic self-maps of the polydisc
Valiron and Abel equations for holomorphic self-maps of the polydisc
复制标题
多圆盘全纯自映射的 Valiron 和 Abel 方程
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
P. Gumenyuk
中科院分区：
文献类型：
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作者：
Leandro Arosio;P. Gumenyuk
对于不允许在$\Delta^N$中存在不动点的多面体的全纯自映射$f: \Delta^N \到\Delta^N$，我们引入了双曲性和抛物性的概念。我们将两个经典的单变量结果推广到多盘上：我们解出了双曲型的Valiron方程和抛物线型的非零阶的Abel方程。这是通过研究$f$的正则Kobayashi双曲半模型和得到多面体自同构的正规形式来实现的。对于Valiron方程，我们也描述了所有解的空间。
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.