Periodic Points of Power Maps in Finite Matrix Groups and Algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918384609787904 |
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| author | Panja, Saikat |
| author_facet | Panja, Saikat |
| contents | Consider the power map $x\mapsto x^L$ for a prime $L\neq 2$ such that $L|q-1$ where $q$ is a power of a prime. We determine the periodic points under this map for $\operatorname{M}_n(q)$, the algebra of $n\times n $ matrices over a finite field of order $q$, and also for the group $\operatorname{GL}_n(q)=\operatorname{M}_n(q)^\times$. We compute the limit $ \lim\limits_{\substack{q\longrightarrow \infty\\v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{M}_\ell(q))\right|}{|\operatorname{M}_\ell(q)|}$ and consequently $\lim\limits_{\substack{q\longrightarrow \infty
v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{GL}_\ell(q))\right|}{|\operatorname{GL}_\ell(q)|}$, where $v_L$ denotes the $L$-adic valuation. We also compute the quantity $\lim\limits_{\substack{q\longrightarrow \infty
v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{Sp}_{2\ell}(q))\right|}{|\operatorname{Sp}_{2\ell}(q)|}$ and $\lim\limits_{\substack{q\longrightarrow \infty
v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{U}_\ell(q))\right|}{|\operatorname{U}_\ell(q)|}$; turns out these two limiting values are same.
In all the cases, it turns out that the regular semisimple elements play the role in determining the limiting values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12295 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Periodic Points of Power Maps in Finite Matrix Groups and Algebras Panja, Saikat Number Theory Group Theory Rings and Algebras 20G40, 20P05, 16R10, 16S50, 11T06, 37P25, 37P05, 37P35 Consider the power map $x\mapsto x^L$ for a prime $L\neq 2$ such that $L|q-1$ where $q$ is a power of a prime. We determine the periodic points under this map for $\operatorname{M}_n(q)$, the algebra of $n\times n $ matrices over a finite field of order $q$, and also for the group $\operatorname{GL}_n(q)=\operatorname{M}_n(q)^\times$. We compute the limit $ \lim\limits_{\substack{q\longrightarrow \infty\\v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{M}_\ell(q))\right|}{|\operatorname{M}_\ell(q)|}$ and consequently $\lim\limits_{\substack{q\longrightarrow \infty v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{GL}_\ell(q))\right|}{|\operatorname{GL}_\ell(q)|}$, where $v_L$ denotes the $L$-adic valuation. We also compute the quantity $\lim\limits_{\substack{q\longrightarrow \infty v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{Sp}_{2\ell}(q))\right|}{|\operatorname{Sp}_{2\ell}(q)|}$ and $\lim\limits_{\substack{q\longrightarrow \infty v_L(q-1)=c}}\dfrac{\left|\operatorname{Per}(x^L,\operatorname{U}_\ell(q))\right|}{|\operatorname{U}_\ell(q)|}$; turns out these two limiting values are same. In all the cases, it turns out that the regular semisimple elements play the role in determining the limiting values. |
| title | Periodic Points of Power Maps in Finite Matrix Groups and Algebras |
| topic | Number Theory Group Theory Rings and Algebras 20G40, 20P05, 16R10, 16S50, 11T06, 37P25, 37P05, 37P35 |
| url | https://arxiv.org/abs/2603.12295 |