Topology of Fatou Components for Endomorphisms of Cp: Linking with the Green’s Current

Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms f : CP → CP, when k > 1. Classical theory describes U(f) as the complement in CP of the support of a dynamically-defined closed positive (1, 1) current. Given any closed positive (1, 1) current S on CP, we give a definition of linking number between closed loops in CP \ suppS and the current S. It has the property that if lk(γ, S) 6= 0, then γ represents a non-trivial homology element in H1(CP k \ suppS). As an application, we use these linking numbers to establish that many classes of endomorphisms of CP 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 CP 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 CP 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.