Invariant escaping Fatou components with two rank 1 limit functions for automorphisms of $\mathbb{C}^2$

TL;DR

This paper constructs specific automorphisms of complex two-dimensional space with invariant escaping Fatou components having exactly two distinct rank 1 limit functions, and proves a growth lemma for certain automorphisms.

Contribution

It introduces new examples of automorphisms with invariant escaping Fatou components with two rank 1 limit functions and establishes a general growth lemma for related automorphisms.

Findings

Constructed automorphisms with invariant escaping Fatou components with two rank 1 limit functions.

Proved a growth lemma for orbits in invariant escaping Fatou components.

Demonstrated the existence of transcendental Hénon maps with specified Fatou component properties.

Abstract

We construct automorphisms of $\mathbb{C}^2$, and more precisely transcendental H\'enon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank 1. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):\mathbb{C}^2\rightarrow\mathbb{C}$ holomorphic.