Seminar

Department of Mathematics

Monday, April 7th, 2008 at 3PM

BA6183, 40 St. George Street

DYNAMICAL SYSTEMS SEMINAR -- Linking with the Green's Current for holomorphic endomorphisms of CP^2.

Roland Roeder

U Toronto

Little is known about the structure of the Fatou Set (domain of normality) for holomorphic maps F: CP^2 -> CP^2. However, it's complement (the Julia set J) is precisely the support of a closed current T called the Green's Current. Since T represents a cohomology class, knowledge of the geometry of T can be used to study the first homology of the Fatou set, be means of linking numbers.

In this talk, I will explain how to make a well-defined linking number within CP^2 between a closed loop \gamma in the Fatou set and the Green's Current T. Then, I will show how this linking number can be used in a simple test case for which the geometry of T is well understood: polynomial skew products F(z,w) = (p(z),q(z,w)). This results in an (often satisfied) sufficient condition for Axiom-A polynomial skew products to have Fatou set with infinitely generated first homology.

This is joint work with Suzanne Hruska.