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

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 $lk(\gamma ,S)\ne 0$, 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.

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

References in corpus (3)

• Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents
• Henon mappings in the complex domain II: projective and inductive limits of polynomials
• Axiom A polynomial skew products of C^2 and their postcritical sets

Cited by in corpus (1)

• A dichotomy for Fatou components of polynomial skew products