Topology of Fatou components for endomorphisms of $\mathbb{C}\mathbb{P}^k$: linking with the Green's current

Suzanne Lynch Hruska ¹   ; Roland K. W. Roeder ²

¹ Department of Mathematical Sciences University of Wisconsin Milwaukee PO Box 413 Milwaukee, WI 53201, U.S.A.
² IUPUI Department of Mathematical Sciences LD Building, Room 270 402 North Blackford Street Indianapolis, IN 46202-3216, U.S.A.

Fundamenta Mathematicae, Tome 210 (2010) no. 1, pp. 73-98

Résumé

Little is known about the global topology of the Fatou set $U(f)$ for holomorphic endomorphisms $f: \mathbb{C}\mathbb{P}^k \rightarrow \mathbb{C}\mathbb{P}^k$, when $k >1$. Classical theory describes $U(f)$ as the complement in $ \mathbb{C}\mathbb{P}^k$ of the support of a dynamically defined closed positive $(1,1)$ current. Given any closed positive $(1,1)$ current $S$ on $ \mathbb{C}\mathbb{P}^k$, we give a definition of linking number between closed loops in $\mathbb{C}\mathbb{P}^k \setminus \mathop{\rm supp} S$ and the current $S$. It has the property that if ${\rm lk}(\gamma,S) \neq 0$, then $\gamma$ represents a non-trivial homology element in $H_1( \mathbb{C}\mathbb{P}^k \setminus \mathop{\rm supp} S)$.As an application, we use these linking numbers to establish that many classes of endomorphisms of $\mathbb{C}\mathbb{P}^2$ 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 $\mathbb{C}\mathbb{P}^2$ 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 $\mathbb{C}\mathbb{P}^2$ 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.

Détail

DOI : 10.4064/fm210-1-4

Keywords: little known about global topology fatou set holomorphic endomorphisms mathbb mathbb rightarrow mathbb mathbb classical theory describes complement mathbb mathbb support dynamically defined closed positive current given closed positive current mathbb mathbb definition linking number between closed loops mathbb mathbb setminus mathop supp current has property gamma neq gamma represents non trivial homology element mathbb mathbb setminus mathop supp application these linking numbers establish many classes endomorphisms mathbb mathbb have fatou components infinitely generated first homology example prove fatou set has infinitely generated first homology polynomial endomorphism mathbb mathbb which restriction line infinity hyperbolic has disconnected julia set addition polynomial skew product mathbb mathbb has fatou set infinitely generated first homology vertical julia set disconnected conclude section concrete examples questions further study