Algebraic Stability for Skew Products
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910555885797376 |
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| author | Birkett, Richard A. P. |
| author_facet | Birkett, Richard A. P. |
| contents | In this article we study algebraic stability for rational skew products in two dimensions $ϕ: X \dashrightarrow X$, i.e. maps of the form $ϕ(x, y) = (ϕ_1(x), ϕ_2(x, y))$. We prove that when $X$ is a birationally ruled surface and $ϕ_1$ has no superattracting cycles, then we can always find a smooth surface $\hat X$ and an algebraic stabilisation $π: (\hat ϕ, \hat X) \to (ϕ, X)$ which is a birational morphism. We provide an example of a skew product $ϕ$ where $ϕ_1$ has a superattracting fixed point and $ϕ$ is not algebraically stable on any model.
Our techniques involve transforming the stabilisation issue into a combinatorial dynamical problem for a 'non-Archimedean skew product' $ϕ_*: \mathbb P^1_{\text{an}}(\mathbb K) \to \mathbb P^1_{\text{an}}(\mathbb K)$ on the Berkovich projective line over the Puiseux series, $\mathbb K$. The Fatou-Julia theory for $ϕ_*$ is instrumental to our approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic Stability for Skew Products Birkett, Richard A. P. Dynamical Systems Algebraic Geometry 37F80, 37P50 (Primary), 14J26, 32H50 (Secondary) In this article we study algebraic stability for rational skew products in two dimensions $ϕ: X \dashrightarrow X$, i.e. maps of the form $ϕ(x, y) = (ϕ_1(x), ϕ_2(x, y))$. We prove that when $X$ is a birationally ruled surface and $ϕ_1$ has no superattracting cycles, then we can always find a smooth surface $\hat X$ and an algebraic stabilisation $π: (\hat ϕ, \hat X) \to (ϕ, X)$ which is a birational morphism. We provide an example of a skew product $ϕ$ where $ϕ_1$ has a superattracting fixed point and $ϕ$ is not algebraically stable on any model. Our techniques involve transforming the stabilisation issue into a combinatorial dynamical problem for a 'non-Archimedean skew product' $ϕ_*: \mathbb P^1_{\text{an}}(\mathbb K) \to \mathbb P^1_{\text{an}}(\mathbb K)$ on the Berkovich projective line over the Puiseux series, $\mathbb K$. The Fatou-Julia theory for $ϕ_*$ is instrumental to our approach. |
| title | Algebraic Stability for Skew Products |
| topic | Dynamical Systems Algebraic Geometry 37F80, 37P50 (Primary), 14J26, 32H50 (Secondary) |
| url | https://arxiv.org/abs/2408.02658 |