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Abstract

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.