Arithmetic properties of families of plane polynomial automorphisms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929497926795264 |
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| author | Zhang, Yugang |
| author_facet | Zhang, Yugang |
| contents | Given an algebraic family $f\colonΛ\times \mathbb{A}^2 \to Λ\times \mathbb{A}^2$ of plane polynomial automorphisms of Hénon type parameterized by a quasi-projective curve, defined over a number field $\mathbb{K},$ we investigate certain arithmetic properties of periodic points contained in a family of subvarieties $X \subset Λ\times \mathbb{A}^2 \twoheadrightarrow Λ$.
First, consider $X$ as a curve. We prove that the set of parameters $t\inΛ(\overline{\mathbb{Q}})$, such that $X_t$ is periodic, has bounded height. This generalizes a result of Patrick Ingram. Moreover, if $X$ is non-periodic, then under some mild conditions -- such as when the family is dissipative -- we show that there are, in fact, only finitely many periodic parameters. This extends a result of Charles Favre and Romain Dujardin.
Second, let $X$ be a family of curves. Assuming $X$ is non-degenerate, we establish a uniform bound on the number of periodic points in each curve $X_t$, $t\in Λ(\overline{\mathbb{Q}})$ and show that the set of these periodic points have bounded height in $Λ\times \mathbb{A}^2$ as well. We then examine in more detail the non-degeneracy property in the case of dissipative families of quadratic Hénon maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15952 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic properties of families of plane polynomial automorphisms Zhang, Yugang Dynamical Systems Complex Variables Number Theory Given an algebraic family $f\colonΛ\times \mathbb{A}^2 \to Λ\times \mathbb{A}^2$ of plane polynomial automorphisms of Hénon type parameterized by a quasi-projective curve, defined over a number field $\mathbb{K},$ we investigate certain arithmetic properties of periodic points contained in a family of subvarieties $X \subset Λ\times \mathbb{A}^2 \twoheadrightarrow Λ$. First, consider $X$ as a curve. We prove that the set of parameters $t\inΛ(\overline{\mathbb{Q}})$, such that $X_t$ is periodic, has bounded height. This generalizes a result of Patrick Ingram. Moreover, if $X$ is non-periodic, then under some mild conditions -- such as when the family is dissipative -- we show that there are, in fact, only finitely many periodic parameters. This extends a result of Charles Favre and Romain Dujardin. Second, let $X$ be a family of curves. Assuming $X$ is non-degenerate, we establish a uniform bound on the number of periodic points in each curve $X_t$, $t\in Λ(\overline{\mathbb{Q}})$ and show that the set of these periodic points have bounded height in $Λ\times \mathbb{A}^2$ as well. We then examine in more detail the non-degeneracy property in the case of dissipative families of quadratic Hénon maps. |
| title | Arithmetic properties of families of plane polynomial automorphisms |
| topic | Dynamical Systems Complex Variables Number Theory |
| url | https://arxiv.org/abs/2407.15952 |