Topology of Fatou components for endomorphisms of CP^k: Linking with the Green's Current
Topology of Fatou components for endomorphisms of CP^k: Linking with the Green's Current
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CP^k 自同态的 Fatou 分量拓扑：与格林电流的联系
DOI：
10.4064/fm210-1-4
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
R. Roeder
中科院分区：
文献类型：
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作者：
Suzanne Lynch Hruska;R. Roeder
对于全纯自同态$f：\mathbb{Cp}^k\to\mathbb{Cp}^k$，当$k>；1$时，对Fatou集$U(F)$的全局拓扑知之甚少。经典理论将$U(F)$描述为动态定义的闭正$(1，1)$流的支撑度$\mathbb{Cp}^k$中的补集。给定$\mathbb{CP}^k$上的闭正$(1，1)$当前$S$，给出了$\mathbb{CP}^k\setminus\supp S$中的闭环与当前$S$之间的连通数的定义.它的性质是：如果$lk(\Gamma，S)\neq0$，则$\Gamma$表示$H_1(\mathbb{CP}^k\setminus\supp S)$中的一个非平凡同调元素。
作为应用，我们利用这些连接数建立了$\mathbb{Cp}^2$的许多类自同态具有无限生成的第一同调的Fatou分支。例如，我们证明了Fatou集对任意多项式自同态都有无限生成的第一同调，其中对无穷远处的直线的限制是双曲的，并且有不连通的Julia集。此外，我们还证明了如果某个垂直Julia集是不连通的，则$\mathbb{Cp}^2$的多项式斜积有Fatou集，并且具有无限生成的第一同调.然后，我们用一节具体的例子和问题来结束，以供进一步研究。
Little is known about the global topology of the Fatou set $U(f)$ for holomorphic endomorphisms $f: \mathbb{CP}^k \to \mathbb{CP}^k$, when $k >1$. Classical theory describes $U(f)$ as the complement in $ \mathbb{CP}^k$ of the support of a dynamically-defined closed positive $(1,1)$ current. Given any closed positive $(1,1)$ current $S$ on $ \mathbb{CP}^k$, we give a definition of linking number between closed loops in $\mathbb{CP}^k \setminus \supp S$ and the current $S$. It has the property that if $lk(\gamma,S) \neq 0$, then $\gamma$ represents a non-trivial homology element in $H_1(\mathbb{CP}^k \setminus \supp S)$.
As an application, we use these linking numbers to establish that many classes of endomorphisms of $\mathbb{CP}^2$ have Fatou components with infinitely generated first homology. For example, we prove that the Fatou set has infinitely generated first homology for any polynomial endomorphism of $\mathbb{CP}^2$ for which the restriction to the line at infinity is hyperbolic and has disconnected Julia set. In addition we show that a polynomial skew product of $\mathbb{CP}^2$ has Fatou set with infinitely generated first homology if some vertical Julia set is disconnected. We then conclude with a section of concrete examples and questions for further study.