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

Suzanne Lynch Hruska; Roland K. W. Roeder — 2010

Fundamenta Mathematicae

Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f : ℂ ℙ^{k} \to ℂ ℙ^{k}$ , when k >1. Classical theory describes U(f) as the complement in $ℂ ℙ^{k}$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂ ℙ^{k}$ , we give a definition of linking number between closed loops in $ℂ ℙ^{k} ∖ s u p p S$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H ₁ (ℂ ℙ^{k} ∖ s u p p S)$ . As an application, we use these linking...

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Topology of Fatou components for endomorphisms of ℂ ℙ k : linking with the Green’s current

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