turkmath.org

The Fatou set of a holomorphic endomorphism of a complex manifold is the largest open set where the family iterates of the map form a normal family, and a Fatou component is a connected component of the Fatou set. We investigate the description of Fatou components for polynomial skew-products in two complex variables. The non-existence of wandering domains near a super-attracting invariant fiber was shown by Lilov, and the geometrically-attracting case was studied by Peters, Vivas and Smit. In collaboration with Astorg, Buff, Dujardin and Peters we proved that wandering domains can exist near a parabolic invariant fiber. I will present these results and, if time allows it I will also report on some recent results, obtained in collaboration with Peters, on the remaining case, namely the dynamics near an elliptic invariant fiber.