Invariant Nonrecurrent Fatou Components of Automorphisms of C 2

Let Ω be an invariant nonrecurrent Fatou component associated with the automorphism F : C → C. Assume that all of the limit maps of {F|Ω} are constant. We prove the following theorem. If there is more than one such limit map then there are uncountably many. The images of these limit maps form a closed set in the boundary of Ω containing no isolated points. Additionally there cannot be more than one limit map unless the derivative of F along a specific subset of the curve of fixed points of F has eigenvalues 1 and e, with θ non-Diophantine. We also examine the case where the limit maps are not all constant. The image of a nonconstant limit map is an immersed variety in the boundary of Ω. We show that any two such immersed varieties intersect either trivially or in a set that is open in their intrinsic topologies. We present some examples of maps with invariant nonrecurrent Fatou components.