More on the Concept of Anti-integrability for H\'enon Maps

TL;DR

This paper extends the theory of anti-integrability for Hénon maps to a broader limit where the Jacobian varies with the parameter, linking the dynamics to hyperbolic quadratic maps.

Contribution

It generalizes the anti-integrability framework for Hénon maps to cases where the Jacobian depends on the parameter, under hyperbolicity conditions.

Findings

The theory applies to limits where b/√a approaches a positive constant.

The dynamics relate to hyperbolic quadratic maps.

AI orbits can be continued to genuine orbits for large a.

Abstract

For the family of H\'{e}non 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 H\'{e}non 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.