Abstract
We show that for any m ∈ ℕ ∪ {∞}, there exist m disjoint FB (Fatou–Bieberbach) domains whose union is dense in ℂk. 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 ℂk, and we construct FB domains that intersect elements of countable collections of affine subspaces of ℂk in connected proper subsets. Moreover, we show that any Runge FB domain is the attracting basin for a sequence of automorphisms of ℂk, although not necessarily if you only allow iteration of one automorphism. We also show that an increasing sequence of Runge ℂk's is a ℂk.
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