[Paper Review] Basins of attraction of automorphisms in C^3

Berit Stensønes, Liz Vivas|arXiv (Cornell University)|Nov 11, 2011

Advanced Differential Equations and Dynamical Systems1 references3 citations

TL;DR

This paper constructs holomorphic automorphisms on $\mathbb{C}^3$ tangent to the identity at the origin whose basins of attraction are not simply connected, specifically showing that such basins can be biholomorphic to $\mathbb{C}^* \times \mathbb{C}^2$. The construction uses a finite composition of shears and overshears to define a map whose dynamics are governed by a projected map on $\mathbb{C}^2$, where Hakim's theory on non-degenerate characteristic directions ensures the basin is biholomorphic to $\mathbb{C}^2$, lifting to the desired structure in $\mathbb{C}^3$. The key contribution is the first explicit example of a non-simply connected basin in higher-dimensional complex dynamics.

ABSTRACT

In this paper we shall give examples of maps and automorphisms with regions of attraction that are not simply connected.

Motivation & Objective

• To construct holomorphic automorphisms in $\mathbb{C}^k$ for $k \geq 3$ with basins of attraction that are not simply connected.
• To extend the known one-dimensional result—where basins are always simply connected—to higher dimensions, where such topology is no longer guaranteed.
• To demonstrate that basins of attraction for automorphisms tangent to the identity in $\mathbb{C}^3$ can be biholomorphic to $\mathbb{C}^* \times \mathbb{C}^2$, a non-simply connected domain.
• To provide a constructive method using shears and overshears to generate such automorphisms with controlled dynamical behavior.

Proposed method

• Define the map $F: \mathbb{C}^3 \to \mathbb{C}^3$ as a finite composition of shears and overshears: $F = \phi_3 \circ \phi_2^{-1} \circ \phi_1^{-1} \circ \phi_2 \circ \phi_1$, where each $\phi_i$ is a holomorphic automorphism of $\mathbb{C}^3$.
• Use the shears $\phi_1(z,t,w) = (z,t,w - zt)$, $\phi_2(z,t,w) = (ze^{aw}, te^{bw}, w)$, and $\phi_3(z,t,w) = (z,t,we^{-(a+b+c)zt} + (a+b)z^2t^2)$ to construct $F$ with a specific Taylor expansion involving $\zeta = zt$.
• Show that $F$ is tangent to the identity at the origin and preserves the axes $zt=0$, ensuring that $F(z,0,w) = (z,0,w)$ and $F(0,t,w) = (0,t,w)$.
• Define a projection $\pi: \mathbb{C}^3 \to \mathbb{C}^2$ by $\pi(z,t,w) = (zt, w)$, which makes the diagram with $G: \mathbb{C}^2 \to \mathbb{C}^2$ commute, where $G$ is a map on $\zeta = zt$ with a non-degenerate characteristic direction.
• Apply Hakim's theorem: if a non-degenerate characteristic direction has directors with positive real part, the associated basin is biholomorphic to $\mathbb{C}^2$, which holds for $G$ when $c > 2a$.
• Use the biholomorphism $\phi: \tilde{\Omega} \to \mathbb{C}^2$ for the basin $\tilde{\Omega}$ of $G$ to define $\Psi(z,t,w) = (z, \phi(zt,w))$, proving $\Omega$ is biholomorphic to $\mathbb{C}^* \times \mathbb{C}^2$.

Experimental results

Research questions

• RQ1 Can basins of attraction for holomorphic automorphisms in $\mathbb{C}^k$ with $k \geq 3$ fail to be simply connected, unlike in $\mathbb{C}^1$?
• RQ2 What dynamical conditions allow a basin of attraction in $\mathbb{C}^3$ to be biholomorphic to $\mathbb{C}^* \times \mathbb{C}^2$?
• RQ3 How can one construct an automorphism tangent to the identity in $\mathbb{C}^3$ such that its basin has nontrivial topology?
• RQ4 Is it possible to extend such constructions to higher dimensions $k \geq 3$ to obtain basins biholomorphic to $(\mathbb{C}^*)^{k-2} \times \mathbb{C}^2$?

Key findings

• The basin of attraction $\Omega$ for the constructed automorphism $F$ in $\mathbb{C}^3$ is biholomorphic to $\mathbb{C}^* \times \mathbb{C}^2$, a non-simply connected domain.
• The map $F$ is tangent to the identity at the origin and is constructed as a finite composition of shears and overshears, ensuring it is a global holomorphic automorphism.
• The projection $\pi: \mathbb{C}^3 \to \mathbb{C}^2$, $\pi(z,t,w) = (zt,w)$, induces a map $G$ on $\mathbb{C}^2$ whose basin $\tilde{\Omega}$ is biholomorphic to $\mathbb{C}^2$ via Hakim's theorem.
• The map $G$ has a non-degenerate characteristic direction $(1,0)$ with a director $\frac{c - 2a}{2a} > 0$ when $c > 2a$, satisfying the conditions for $\tilde{\Omega} \cong \mathbb{C}^2$, which lifts to the full basin $\Omega \cong \mathbb{C}^* \times \mathbb{C}^2$.
• The construction generalizes to $\mathbb{C}^k$ for $k \geq 3$, yielding basins biholomorphic to $(\mathbb{C}^*)^{k-2} \times \mathbb{C}^2$ by extending the shearing structure to $k-1$ variables with $\zeta = \prod_{i=1}^{k-1} z_i$.
• The set of characteristic directions for the automorphism is $k$-dimensional, and the maps are $0$-dicritical, meaning the set of singular directions is $0$-dimensional, consistent with Abate and Tovena's classification.