Automorphisms of ${\mathbb{C}}^m$ with bounded wandering domains

Abstract

We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of ${\mathbb{C}}^m$, in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded regular open sets $\Omega \subset {\mathbb{C}}^m$ whose closure is polynomially convex. Our result gives in particular the first example of a bounded Fatou component with a smooth boundary in the category of holomorphic automorphisms.

Minor revision. Accepted in Annali di Matematica Pura ed Applicata