Uniform bound on common periodic points for families of regular plane polynomial automorphisms
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918331925135360 |
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| author | Abboud, Marc Zhang, Yugang |
| author_facet | Abboud, Marc Zhang, Yugang |
| contents | Given two one-dimensional families $f$ and $g$ of regular plane polynomial automorphisms parameterised by an algebraic curve $B$, all defined over some number field $K$, such that one of them is dissipative, we prove that at any parameter $b\in B(\mathbb{C})$, either $f_b$ and $g_b$ share a common iterate, or the number of their common periodic points $\mathrm{Per}(f_b) \cap \mathrm{Per}(g_b)$ is bounded by a uniform constant $D$ (independent of the parameter $b$). We thus extend a result of Mavraki and Schmidt for rational maps to our setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10310 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform bound on common periodic points for families of regular plane polynomial automorphisms Abboud, Marc Zhang, Yugang Dynamical Systems Algebraic Geometry Complex Variables Number Theory Given two one-dimensional families $f$ and $g$ of regular plane polynomial automorphisms parameterised by an algebraic curve $B$, all defined over some number field $K$, such that one of them is dissipative, we prove that at any parameter $b\in B(\mathbb{C})$, either $f_b$ and $g_b$ share a common iterate, or the number of their common periodic points $\mathrm{Per}(f_b) \cap \mathrm{Per}(g_b)$ is bounded by a uniform constant $D$ (independent of the parameter $b$). We thus extend a result of Mavraki and Schmidt for rational maps to our setting. |
| title | Uniform bound on common periodic points for families of regular plane polynomial automorphisms |
| topic | Dynamical Systems Algebraic Geometry Complex Variables Number Theory |
| url | https://arxiv.org/abs/2602.10310 |