[Paper Review] Complex Hénon maps and discrete groups

Raluca Tanase|arXiv (Cornell University)|Mar 12, 2015

Mathematical Dynamics and Fractals7 references3 citations

TL;DR

This paper extends the group action on the escaping set of complex Hénon maps to the boundary $J^+$, representing $J^+$ as a quotient of $\mathbb{S}^1 \times \mathbb{C}$ by a discrete group $\Gamma \cong \mathbb{Z}[1/2]/\mathbb{Z}$, establishing a semi-conjugacy between the Hénon map and a model dynamics on the quotient manifold. For small perturbations of hyperbolic or semi-parabolic polynomials, $J^+$ is shown to be a topological manifold or a quotient of such a manifold, with explicit equivalence relations and fractal invariants encoded by a cocycle $\alpha$. The key contribution is a two-dimensional analogue of the Carathéodory loop for Hénon maps.

ABSTRACT

Consider the standard family of complex Hénon maps $H(x,y) = (p(x) - ay, x)$, where $p$ is a quadratic polynomial and $a$ is a complex parameter. Let $U^{+}$ be the set of points that escape to infinity under forward iterations. The analytic structure of the escaping set $U^{+}$ is well understood from previous work of J. Hubbard and R. Oberste-Vorth as a quotient of $(\mathbb{C}-\overline{\mathbb{D}}) imes\mathbb{C}$ by a discrete group of automorphisms $Γ$ isomorphic to $\mathbb{Z}[1/2]/\mathbb{Z}$. On the other hand, the boundary $J^{+}$ of $U^{+}$ is a complicated fractal object on which the Hénon map behaves chaotically. We show how to extend the group action to $\mathbb{S}^1 imes\mathbb{C}$, in order to represent the set $J^{+}$ as a quotient of $\mathbb{S}^1 imes \mathbb{C}/\,Γ$ by an equivalence relation. We analyze this extension for Hénon maps that are small perturbations of hyperbolic polynomials with connected Julia sets or polynomials with a parabolic fixed point.

Motivation & Objective

• To extend the discrete group action from the escaping set $U^+$ to its boundary $J^+$, enabling a topological and dynamical description of $J^+$.
• To characterize the structure of $J^+$ as a quotient of $\mathbb{S}^1 \times \mathbb{C}$ by a group $\Gamma \cong \mathbb{Z}[1/2]/\mathbb{Z}$, generalizing the known structure of $U^+$.
• To establish a semi-conjugacy between the Hénon map on $J^+$ and a model dynamics on the quotient manifold $\mathcal{M} = \mathbb{S}^1 \times \mathbb{C}/\Gamma$.
• To analyze the behavior of the cocycle $\alpha: \mathbb{S}^1 \to \mathbb{C}^*$, which encodes dynamical invariants and is shown to be a full invariant for small Jacobian perturbations of parabolic polynomials.
• To prove that $J^+$ is a topological manifold or a quotient of one under specific dynamical conditions, using group actions and laminar structures.

Proposed method

• Extends the known group action on $U^+ \cong (\mathbb{C} \setminus \overline{\mathbb{D}}) \times \mathbb{C} / \Gamma$ to $\mathbb{S}^1 \times \mathbb{C}$, where $\Gamma \cong \mathbb{Z}[1/2]/\mathbb{Z}$, to include the boundary $J^+$.
• Uses the stable multiplier condition and a trivialization of the lamination of $U^+ \cup J^+$ to analyze the extension of the group action to the circle boundary.
• Introduces the cocycle $\alpha: \mathbb{S}^1 \to \mathbb{C}^*$, which captures the dynamics and is shown to be a fractal set via a custom plotting algorithm.
• Applies results from Lyubich and Robertson on critical locus characterization to analyze degeneracy and invariants of $\alpha$.
• Employs growth estimates for group elements $\Gamma_{p,a}$ to prove proper discontinuity and absence of fixed points in the action on $\mathbb{S}^1 \times \mathbb{C}$.
• Constructs a semi-conjugacy from the quotient manifold $\mathcal{M} = \mathbb{S}^1 \times \mathbb{C}/\Gamma$ to $J^+$, with explicit equivalence relations in the case of hyperbolic and semi-parabolic maps.

Experimental results

Research questions

• RQ1 How can the discrete group action on the escaping set $U^+$ be extended to the boundary $J^+$ of the escaping set in complex Hénon maps?
• RQ2 What is the topological structure of $J^+$ when the Hénon map is a small perturbation of a hyperbolic polynomial with a connected Julia set?
• RQ3 Can the dynamics on $J^+$ be semi-conjugate to a model dynamics on a quotient manifold $\mathcal{M} = \mathbb{S}^1 \times \mathbb{C}/\Gamma$?
• RQ4 What role does the cocycle $\alpha: \mathbb{S}^1 \to \mathbb{C}^*$ play in encoding the dynamics and invariants of the Hénon family?
• RQ5 How does the group $\Gamma_{c,a}$ act on $\mathbb{S}^1 \times \mathbb{C}$ for small Jacobian perturbations of semi-parabolic Hénon maps?

Key findings

• The group $\Gamma_{c,a}$ acts properly discontinuously and without fixed points on $\mathbb{S}^1 \times \mathbb{C}$ for small perturbations of hyperbolic polynomials with connected Julia sets, ensuring $\mathcal{M} = \mathbb{S}^1 \times \mathbb{C}/\Gamma$ is a topological manifold.
• For Hénon maps near the interior of the main cardioid (i.e., with an attracting fixed point), $J^+$ is itself a topological manifold, and the semi-conjugacy becomes an actual conjugacy.
• In general, $J^+$ is a quotient of the manifold $\mathcal{M}$ by an explicit equivalence relation, as formalized in Theorem 9.11.
• The cocycle $\alpha$ is shown to be a full invariant of the family $\mathcal{P}_\lambda$ for small Jacobian, with $\alpha(1) = \mu$, the eigenvalue of the semi-parabolic fixed point.
• The image of $\alpha$ is a fractal set, and its structure is visualized via a custom plotting algorithm developed in Section 7.
• For semi-parabolic Hénon maps with $0 < |a| < a_0$, the boundary $\partial\mathcal{C}_0$ of the primary critical component is homeomorphic to the Julia set $J_p$ of the parabolic polynomial $p$, and Theorem 10.4 extends the group action and quotient structure to this case.