A dichotomy for Fatou components of polynomial skew products
A dichotomy for Fatou components of polynomial skew products
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多项式偏斜积的 Fatou 分量的二分法
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发表时间：
2010
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通讯作者：
R. Roeder
中科院分区：
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作者：
R. Roeder
我们考虑形式为f(z，w)=(p(Z)，q(z，w))的多项式映射是Cp^2的全纯映射。Mattias Jonsson在(Math.AN.，1999)这种多项式斜积的连通性的概念类似于一元多项式映射的Julia集的连通性。我们证明了如下二分法：如果f是公理-A多项式斜积，且f是连通的，则f的每个Fatou分支都同胚于一个开球；否则，f的某个Fatou分支有无限个第一同调。
We consider polynomial maps of the form f(z,w) = (p(z),q(z,w)) that extend as holomorphic maps of CP^2. Mattias Jonsson introduces in (Math. Ann., 1999) a notion of connectedness for such polynomial skew products that is analogous to connectivity for the Julia set of a polynomial map in one-variable. We prove the following dichotomy: if f is an Axiom-A polynomial skew product, and f is connected, then every Fatou component of f is homeomorphic to an open ball; otherwise, some Fatou component of f has infinitely generated first homology.