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

Abstract

We construct automorphisms of ${\mathbb{C}}^2$, and more precisely transcendental Hénon 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 $g(z,w):{\mathbb{C}}^2\to \mathbb{C}$ holomorphic.

16 pages

References in corpus (3)

• Singular orbits and Baker domains
• Automorphisms of ${\mathbb{C}}^k$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb{C}\times ({\mathbb{C}}^{\ast }{)}^{k-1}$
• Automorphisms of ${\mathbb{C}}^2$ with parabolic cylinders