More on the Concept of Anti-integrability for Hénon Maps

TL;DR

The work extends the anti-integrability framework for Hénon maps to parameter scalings where $\lim_{a\to\infty} b/\sqrt{a}=\hat{r}>0$, provided the one-dimensional quadratic map $x\mapsto \frac{1}{\hat{r}}(1-x^2)$ is hyperbolic. By reformulating the AI limit as a small-parameter problem with $\epsilon=1/\sqrt{a}$ and $r=b/\sqrt{a}$, the authors prove a rigorous continuation from AI orbits to genuine orbits for all sufficiently small $\epsilon$, yielding a compact invariant set $\mathcal{A}_{\epsilon}$ on which the Henon map is topologically conjugate to a two-sided Markov shift on $\Sigma_{0,\hat{r}}$. They show $\mathbf{x}^*(\epsilon;\mathbf{x}^\dag)\to\mathbf{x}^\dag$ as $\epsilon\to0$ and $\mathcal{A}_{\epsilon}\to\mathcal{A}_{0}$ in the Hausdorff sense, linking AI-limit dynamics to hyperbolic behavior via quasi-hyperbolicity. The paper further explores hyperbolic plateaus across multiple geometric representations of parameter space (sphere, disc, semi-disc, and $(\epsilon,r)$-plane) and connects the continued dynamics to inverse-limit dynamics when $1/\hat{r}^2$ is hyperbolic. Overall, the results provide a robust framework for chaotic dynamics in Hénon-like maps under generalized parameter scaling and reveal rich structure in the associated hyperbolic plateaus.

Abstract

For the family of Hénon maps $(x,y)\mapsto (\sqrt{a}(1-x^2)-b y,x)$ of $\mathbb{R}^2$, the so-called anti-integrable (AI) limit concerns the limit $a\to\infty$ with fixed Jacobian $b$. At the AI limit, the dynamics reduces to a subshift of finite type. There is a one-to-one correspondence between sequences allowed by the subshift and the AI orbits. The theory of anti-integrability says that each AI orbit can be continued to becoming a genuine orbit of the Hénon map for $a$ sufficiently large (and fixed Jacobian). In this paper, we assume $b$ is a smooth function of $a$ and show that the theory can be extended to investigating the limit $\lim_{a\to\infty} b/\sqrt{a}=\hat{r}$ for any $\hat{r}>0$ provided that the one dimensional quadratic map $x\mapsto \displaystyle\frac{1}{\hat{r}}(1-x^2)$ is hyperbolic.

Figures (4)

• Figure 1: Hyperbolic plateaus on the sphere and its stereographic projection viewed from two different angles.
• Figure 2: (Left) Hyperbolic plateaus under the Möbius transformation $\mathcal{M}$. (Right) Magnification for $0.8\le \alpha<1$ and $-0.1\le \beta \le 0.1$.
• Figure 3: Hyperbolic plateaus in the $(\rho,\theta)$-coordinates.
• Figure 4: (Left) Hyperbolic plateaus on the $(\epsilon, r)$-plane. (Right) Magnification for $0.746\le \epsilon\le 0.762$ and $-0.010\le r\le 0.010$. When $r=0$, the blue part is contained in the attracting period-$3$ window of the quadratic map $Q_a$.

Theorems & Definitions (16)

• Remark 1
• Remark 2
• Remark 3
• Definition 4
• Definition 5
• Theorem 6
• Corollary 7: Anti-integrability for Hénon maps
• Corollary 8: $b/\sqrt{a}=$ a fixed nonzero constant
• Corollary 9
• Definition 10