Fatou–Bieberbach Domains: A New Construction and a Theme on the Runge Property

Abstract

Fatou–Bieberbach domains are a phenomenon specific to several complex variables. Techniques for producing such domains are limited and fundamental questions about containment between two Fatou–Bieberbach are still being raised. We show that given a countable collection of Runge Fatou–Bieberbach domains with a ball in common and a common point omitted, there exists a Runge Fatou–Bieberbach domain that contains the union. Additionally, we provide a new construction for Fatou–Bieberbach domains modelled on the attracting basin, using right-side composition instead of the prototypical left-side composition. We use this construction to show that there exists a strictly decreasing family of Fatou–Bieberbach domains whose intersection contains a Fatou–Bieberbach domain. Additionally, we provide a generalized condition for constructing attracting basins from a sequence of automorphisms.