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

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.