Fatou-Bieberbach Domains

TL;DR

This paper constructs multiple disjoint Fatou-Bieberbach domains in complex Euclidean spaces with dense unions, explores their boundary properties, and relates them to automorphism dynamics, advancing understanding of complex domain structures.

Contribution

It demonstrates the existence of multiple disjoint Fatou-Bieberbach domains with dense unions, constructs domains with specific subvariety intersections, and links Runge domains to automorphism basins.

Findings

Existence of m disjoint FB domains with dense union in a9^k.

Construction of FB domains containing arbitrary countable subvarieties.

Runge FB domains as attracting basins for automorphism sequences.

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 $\CC^k$.