Residual Transitivity implies Minimality for Markoff Surfaces over $p$-adic Integers, by Means of $p$-adic Flows
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915173637292032 |
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| author | Jang, Seung Uk |
| author_facet | Jang, Seung Uk |
| contents | Let $X_D^\ast$ be the non-singuar locus of the Markoff surface $X_D\colon x^2+y^2+z^2=xyz+D$ and consider the set of its $p$-adic integer points $X_D^\ast(\mathbb{Z}_p)$. It is known to Bourgain, Gamburd, and Sarnak that the modulo $p$ transitivity by algebraic automorphisms of $X_0^\ast$ implies minimality of $X_0^\ast(\mathbb{Z}_p)$ by algebraic automorphisms. In this paper, we provide an alternative proof of this fact, by some techniques to study $p$-adic analytic flows. This establish a slight generalization to those parameters $D$ congruent to $0$ modulo $p^2$ or $(D-4)$ being a nonzero quadratic residue. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Residual Transitivity implies Minimality for Markoff Surfaces over $p$-adic Integers, by Means of $p$-adic Flows Jang, Seung Uk Dynamical Systems Number Theory Let $X_D^\ast$ be the non-singuar locus of the Markoff surface $X_D\colon x^2+y^2+z^2=xyz+D$ and consider the set of its $p$-adic integer points $X_D^\ast(\mathbb{Z}_p)$. It is known to Bourgain, Gamburd, and Sarnak that the modulo $p$ transitivity by algebraic automorphisms of $X_0^\ast$ implies minimality of $X_0^\ast(\mathbb{Z}_p)$ by algebraic automorphisms. In this paper, we provide an alternative proof of this fact, by some techniques to study $p$-adic analytic flows. This establish a slight generalization to those parameters $D$ congruent to $0$ modulo $p^2$ or $(D-4)$ being a nonzero quadratic residue. |
| title | Residual Transitivity implies Minimality for Markoff Surfaces over $p$-adic Integers, by Means of $p$-adic Flows |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2502.18976 |