Automorphisms of $\mathbb C^k$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb C\times (\mathbb C^\ast)^{k-1}$

Filippo Bracci,Berit Stensønes,Jasmin Raissy

doi:10.4171/jems/1019

Abstract

We prove the existence of automorphisms of $\mathbb C^k$, $k\ge 2$, having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb C\times (\mathbb C^\ast)^{k-1}$ which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. Such a Fatou component also avoids k analytic discs intersecting transversally at the fixed point. As a corollary, we obtain a Runge copy of $\mathbb C\times (\mathbb C^\ast)^{k-1}$ in $\mathbb C^k$.

Highlights

Let F be a holomorphic endomorphism of Ck, k ≥ 2
In the study of the dynamics of F, that is of the behavior of its iterates, a natural dichotomy is given by the division of the space into the Fatou set and the Julia set
A Fatou component Ω for a map F is called invariant if F (Ω) = Ω

Summary

Introduction

Let F be a holomorphic endomorphism of Ck, k ≥ 2. We call an invariant Fatou component Ω for a map F attracting if there exists a point p ∈ Ω with limn→∞ F n(z) = p for all z ∈ Ω. Every non-recurrent invariant attracting Fatou component Ω of a polynomial automorphism of C2 is biholomorphic to C2. There exist holomorphic automorphisms of Ck having an invariant, non-recurrent, attracting Fatou component biholomorphic to C × (C∗)k−1. This shows that there exist (non polynomial) automorphisms of C2 having a non- connected attracting non-recurrent Fatou component. Our construction shows that the invariant non-recurrent attracting Fatou component biholomorphic to C × (C∗)k−1 avoids k analytic discs which intersect transversally at the fixed point. We have to introduce a completely new argument, which is based on Poschel’s results in [13] and detailed estimates for the Kobayashi metric on certain domains

Outline of the proof in dimension 2

The local basin of attraction B

Local Fatou coordinates on B

The topology of the global basin Ω

Findings

The global basin Ω and the Fatou component containing B