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

TL;DR

This paper explores the topology of Fatou components for holomorphic endomorphisms of complex projective spaces, linking them with Green's currents to show many have infinitely generated first homology.

Contribution

It introduces a linking number concept for currents and loops, revealing new topological properties of Fatou sets in higher-dimensional complex dynamics.

Findings

Fatou components often have infinitely generated first homology.

Hyperbolic restrictions lead to disconnected Julia sets and complex Fatou topology.

Vertical Julia set disconnections imply infinite homology in Fatou sets.

Abstract

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.