CP^k 自同态的 Fatou 分支拓扑：与 Green 电流的环绕
Topology of Fatou components for endomorphisms of CP^k: Linking with the Green's Current
AI总结：
本文通过定义闭正 (1,1) 电流与闭回路之间的环绕数，研究 CP^k 全纯自同态 Fatou 集的整体拓扑，并证明 CP^2 中多类自同态的 Fatou 分支具有无限生成的一阶同调。
摘要：
当 k>1 时，关于全纯自同态 f: CP^k → CP^k 的 Fatou 集 U(f) 的整体拓扑，目前所知甚少。经典理论将 U(f) 描述为 CP^k 中一个动力学定义的闭正 (1,1) 电流的支集的补集。给定 CP^k 上任意闭正 (1,1) 电流 S，我们给出了 CP^k \ supp S 中闭回路与该电流 S 之间的环绕数定义。该定义具有如下性质：若 lk(γ,S) ≠ 0，则 γ 在 H_1(CP^k \ supp S) 中代表一个非平凡的同调元素。作为应用，我们利用这些环绕数证明 CP^2 的许多自同态类具有一阶同调无限生成的 Fatou 分支。例如，我们证明：对于任意 CP^2 的多项式自同态，只要其在无穷远直线上的限制是双曲的且 Julia 集不连通，则其 Fatou 集的一阶同调无限生成。此外，我们还证明：若某个垂直 Julia 集不连通，则 CP^2 的多项式斜积的 Fatou 集具有无限生成的一阶同调。最后，我们以一节具体例子和进一步研究的问题作为结尾。
英文摘要：
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 $lk(γ,S) \neq 0$, then $γ$ 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.
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