Fatou-Bieberbach Domains

Abstract

We show that for any $m\in\NN\cup\{\infty\}$ there exist $m$ disjoint FB domains whose union is dense in $\CC^k$. In fact we show that any point not in the union is a boundary point for all the domains. We construct FB domains that contains arbitrary countable collections of subvarieties of $\CC^k$, and we construct FB domains that intersect elements of countable collections of affine subspaces of $\CC^k$ in connected proper subsets. Moreover, we show that any Runge FB domain is the attracting basin for a sequence of automorphisms of $\CC^k$, although not necessarily if you only allow iteration of one automorphism. We also show that an increasing sequence of Runge $\CC^k$'s is a $\CC^k$.