Escaping Fatou components with disjoint hyperbolic limit sets

TL;DR

This paper constructs specific automorphisms of complex two-dimensional space exhibiting cycles of escaping Fatou components with two distinct hyperbolic limit sets at infinity, advancing understanding of complex dynamics.

Contribution

It introduces automorphisms with a cycle of escaping Fatou components having exactly two disjoint hyperbolic limit sets, a novel configuration in complex dynamics.

Findings

Automorphisms with a cycle of escaping Fatou components were constructed.

Each component has exactly two rank 1 limit functions.

Limit sets are disjoint hyperbolic subsets of the line at infinity.

Abstract

We construct automorphisms of $\mathbb{C}^2$ with a cycle of escaping Fatou components, on which there are exactly two limit functions, both of rank 1. On each such Fatou component, the limit sets for these limit functions are two disjoint hyperbolic subsets of the line at infinity.