Topology of Fatou components for endomorphisms of CP^k: Linking with the Green's Current

Details 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

2008-10-09

arxiv.org/abs/0810.1673v3

Mathematics

Dynamical Systems

Revision 3: Added improvements to the discussion of skew products. 21 pages. Comments welcome

Scientific paper

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.

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