Topology of Fatou components for endomorphisms of CP^k: Linking with the Green's Current
by
Roland Roeder
IUPUI
Coauthors: Suzanne Hruska

Little is known about the global topology of Fatou components for holomorphic endomorphisms f: CP^k → CP^k. These Fatou components are precisely the complement of the support of a dynamically defined closed positive (1, 1) current known as the Green's current.

Motivated by dynamics, we develop a well-defined linking number between a closed loop g and a closed (1, 1) current T in CP^k. It allows us to observe that for certain types of holomorphic endomorphisms of CP^k, the Fatou set has infinitely generated first homology. These linking numbers may also be useful in other applications from several complex variables.

Date received: December 20, 2009