Invariant Nonrecurrent Fatou Components of Automorphisms of $C^2$

TL;DR

This paper studies invariant nonrecurrent Fatou components of automorphisms in complex two-dimensional space, showing limitations on the number of constant limit maps and exploring cases with nonconstant limits, supported by examples.

Contribution

It provides new results on the uniqueness of constant limit maps for invariant Fatou components and explores cases with nonconstant limits, expanding understanding of complex dynamics in $\

Findings

Most invariant nonrecurrent Fatou components have at most one constant limit map.

Examples of Fatou components with nonconstant limit maps are provided.

Abstract

We examine invariant nonrecurrent Fatou components of automorphisms of $\mathbb{C}^2$ in the case where all limit maps are constant. We show that except in special cases there cannot be more than one such limit map. We also briefly examine such Fatou components where the limit maps may be nonconstant. Lastly we present a few examples of such Fatou components.