Rigidity of the escaping set of certain Hénon maps
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908771550232576 |
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| author | Bera, Sayani |
| author_facet | Bera, Sayani |
| contents | Let $H$ be a Hénon map of the form $H(x,y)=(y,p(y)-ax)$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the actions of automorphisms of $\mathbb{C}^2$ if the degree of $H=d\le |a|$. Specifically, every automorphism of $\mathbb{C}^2$ that preserves $U^+$, essentially takes the form $C \circ H^s$ where $s \in \mathbb{Z}$, and $C(x,y)=(ηx, η^d y)$ with $η$ some $(d^2-1)$-root of unity.
Consequently, we show that the automorphisms of the short $\mathbb{C}^2$'s, obtained as the sub-level sets of the (positive) Green's function corresponding to the Hénon map $H$ for strictly positive values, are essentially linear maps of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism groups of these short $\mathbb{C}^2$'s are the same, finite, and form a subgroup of $\mathbb{Z}_{d^2-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity of the escaping set of certain Hénon maps Bera, Sayani Complex Variables Dynamical Systems 37F80, 32M18, 32Q02, 32T05 Let $H$ be a Hénon map of the form $H(x,y)=(y,p(y)-ax)$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the actions of automorphisms of $\mathbb{C}^2$ if the degree of $H=d\le |a|$. Specifically, every automorphism of $\mathbb{C}^2$ that preserves $U^+$, essentially takes the form $C \circ H^s$ where $s \in \mathbb{Z}$, and $C(x,y)=(ηx, η^d y)$ with $η$ some $(d^2-1)$-root of unity. Consequently, we show that the automorphisms of the short $\mathbb{C}^2$'s, obtained as the sub-level sets of the (positive) Green's function corresponding to the Hénon map $H$ for strictly positive values, are essentially linear maps of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism groups of these short $\mathbb{C}^2$'s are the same, finite, and form a subgroup of $\mathbb{Z}_{d^2-1}$. |
| title | Rigidity of the escaping set of certain Hénon maps |
| topic | Complex Variables Dynamical Systems 37F80, 32M18, 32Q02, 32T05 |
| url | https://arxiv.org/abs/2502.19358 |