Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911475215368192 |
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| author | Bera, Sayani Verma, Kaushal |
| author_facet | Bera, Sayani Verma, Kaushal |
| contents | Let $H$ be a polynomial automorphism of $\mathbb{C}^2$ of positive entropy and degree $d \ge 2$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the action of holomorphic automorphisms of $\mathbb{C}^2$. Specifically, every holomorphic automorphism of $\mathbb{C}^2$ that preserves $U^+$ essentially takes the form $L \circ H^s$ where $s \in \mathbb{Z}$ and $L$ belongs to a finite cyclic group of affine maps that preserve the escaping set.
Second, note that the sub-level sets $\{G^+ < c\}$, $c > 0$, of the Greens function $G^+$ associated with the map $H$ are canonical examples of Short $\mathbb{C}^2$s. As a consequence of the above theorem, we show that the holomorphic automorphisms of these Short $\mathbb{C}^2$s are affine automorphisms of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism group of these Short $\mathbb{C}^2$s are the same for every $c>0$ and is a finite cyclic group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$ Bera, Sayani Verma, Kaushal Complex Variables Dynamical Systems 37F80, 32M18, 32Q02, 32T05 Let $H$ be a polynomial automorphism of $\mathbb{C}^2$ of positive entropy and degree $d \ge 2$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the action of holomorphic automorphisms of $\mathbb{C}^2$. Specifically, every holomorphic automorphism of $\mathbb{C}^2$ that preserves $U^+$ essentially takes the form $L \circ H^s$ where $s \in \mathbb{Z}$ and $L$ belongs to a finite cyclic group of affine maps that preserve the escaping set. Second, note that the sub-level sets $\{G^+ < c\}$, $c > 0$, of the Greens function $G^+$ associated with the map $H$ are canonical examples of Short $\mathbb{C}^2$s. As a consequence of the above theorem, we show that the holomorphic automorphisms of these Short $\mathbb{C}^2$s are affine automorphisms of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism group of these Short $\mathbb{C}^2$s are the same for every $c>0$ and is a finite cyclic group. |
| title | Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$ |
| topic | Complex Variables Dynamical Systems 37F80, 32M18, 32Q02, 32T05 |
| url | https://arxiv.org/abs/2601.07681 |