More on the Concept of Anti-integrability for Hénon Maps

Abstract

For the family of Hénon maps $(x,y)\mapsto (\sqrt{a}(1-x^2)-by,x)$ 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 $x\mapsto \frac{1}{\hat{r}}(1-x^2)$ is hyperbolic.