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

Filippo Bracci, Jasmin Raissy+1 more|arXiv (Cornell University)|Mar 24, 2017

Advanced Differential Equations and Dynamical Systems14 references3 citations

TL;DR

This paper constructs holomorphic automorphisms of $\mathbb{C}^k$ for $k \geq 2$ with an invariant, non-recurrent attracting Fatou component biholomorphic to $\mathbb{C} \times (\mathbb{C}^*)^{k-1}$, using a combination of local Fatou coordinates, fiber bundle extensions, and Kobayashi metric estimates. The key result establishes the first known example of a non-simply connected attracting Fatou component in complex dynamics, resolving a long-standing open question on Runge embeddings of $\mathbb{C} \times \mathbb{C}^*$ in $\mathbb{C}^2$.

ABSTRACT

We prove the existence of automorphisms of $\mathbb C^k$, $k\ge 2$, having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb C imes (\mathbb C^\ast)^{k-1}$ 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 $\mathbb C imes (\mathbb C^\ast)^{k-1}$ in $\mathbb C^k$.

Motivation & Objective

• Establish the existence of holomorphic automorphisms of $\mathbb{C}^k$ ($k \geq 2$) with an invariant, non-recurrent, attracting Fatou component biholomorphic to $\mathbb{C} \times (\mathbb{C}^*)^{k-1}$.
• Resolve the long-standing open question of whether $\mathbb{C} \times \mathbb{C}^*$ admits a Runge embedding in $\mathbb{C}^2$ by constructing such an embedding as a Fatou component.
• Show that the constructed Fatou component avoids $k$ analytic discs intersecting transversally at the attracting fixed point.
• Prove that the global Fatou component is biholomorphic to $\mathbb{C} \times (\mathbb{C}^*)^{k-1}$ via a fiber bundle structure induced by Fatou coordinates.
• Establish that the component is a full Fatou component by ruling out the existence of larger invariant domains using Kobayashi metric estimates.

Proposed method

• Construct a local basin of attraction $B$ in $\mathbb{C}^k$ using a holomorphic germ with a fixed point at the origin and eigenvalues satisfying Pöschel's admissibility condition.
• Define a local Fatou coordinate $\psi$ on $B$ such that $\psi \circ F = \psi + 1$, ensuring uniform convergence of iterates to the origin.
• Extend the Fatou coordinate $\psi$ to a global map $g_1$ on the global basin $\Omega = \bigcup_{n \geq 0} F^{-n}(B)$, establishing it as a holomorphic fiber bundle map with fiber $ (\mathbb{C}^*)^{k-1} $.
• Define $k-1$ additional local coordinates $\sigma_j$ satisfying functional equations $\sigma_j \circ F = \lambda_j \cdots \lambda_k \cdot e^{-\frac{k-j+1}{k\psi}} \sigma_j$, which are extended globally over $\Omega_0 \subset \Omega$.
• Use the transition functions of the fiber bundle, which lie in $\mathrm{GL}_{k-1}(\mathbb{C})$, to conclude that $\Omega$ is biholomorphic to $\mathbb{C} \times (\mathbb{C}^*)^{k-1}$ via [7, Corollary 8.3.3].
• Apply Pöschel's results and detailed estimates of the Kobayashi metric to show that the Fatou component $V$ containing $\Omega$ equals $\Omega$, proving it is a full Fatou component.

Experimental results

Research questions

• RQ1 Does there exist a holomorphic automorphism of $\mathbb{C}^k$ ($k \geq 2$) with a non-recurrent, attracting Fatou component biholomorphic to $\mathbb{C} \times (\mathbb{C}^*)^{k-1}$?
• RQ2 Can $\mathbb{C} \times \mathbb{C}^*$ be realized as a Runge domain in $\mathbb{C}^2$ via a Fatou component of an automorphism?
• RQ3 Is it possible to construct such a Fatou component that avoids $k$ analytic discs intersecting transversally at the attracting fixed point?
• RQ4 Can the global basin of attraction of a holomorphic automorphism be shown to be biholomorphic to $\mathbb{C} \times (\mathbb{C}^*)^{k-1}$ using fiber bundle structures from Fatou coordinates?
• RQ5 Can the Kobayashi metric be used to prove that the global basin is the full Fatou component and not part of a larger invariant domain?

Key findings

• The paper constructs, for each $k \geq 2$, a holomorphic automorphism of $\mathbb{C}^k$ with an invariant, non-recurrent, attracting Fatou component biholomorphic to $\mathbb{C} \times (\mathbb{C}^*)^{k-1}$, providing the first such example in higher dimensions.
• Corollary 0.2 establishes that $\mathbb{C} \times (\mathbb{C}^*)^{k-1}$ admits a Runge embedding in $\mathbb{C}^k$, resolving a long-standing open problem in complex analysis.
• The constructed Fatou component avoids $k$ analytic discs intersecting transversally at the attracting fixed point, which is a key topological obstruction to biholomorphy to $\mathbb{C}^k$.
• Using the functional equation $\psi \circ F = \psi + 1$ and extension techniques, the global basin $\Omega$ is shown to be a holomorphic fiber bundle over $\mathbb{C}$ with fiber $ (\mathbb{C}^*)^{k-1} $, implying $\Omega \cong \mathbb{C} \times (\mathbb{C}^*)^{k-1} $.
• By applying Pöschel’s admissibility condition and detailed estimates of the Kobayashi metric, the authors prove that the Fatou component $V$ equals $\Omega$, confirming it is a full Fatou component.
• The cohomology group $H^{k-1}(\Omega, \mathbb{C})$ is nontrivial, showing that the Fatou component has the highest possible non-vanishing cohomological degree allowed for a Runge domain in $\mathbb{C}^k$, consistent with Serre’s vanishing theorem.