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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2505.15346 |
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| _version_ | 1866916749230735360 |
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| author | Arai, Zin Chen, Yi-Chiuan |
| author_facet | Arai, Zin Chen, Yi-Chiuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | More on the Concept of Anti-integrability for Hénon Maps Arai, Zin Chen, Yi-Chiuan Dynamical Systems 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. |
| title | More on the Concept of Anti-integrability for Hénon Maps |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2505.15346 |