有限矩阵群和代数中幂映射的周期点
Periodic Points of Power Maps in Finite Matrix Groups and Algebras
AI总结：
本文研究了有限矩阵群和代数中幂映射的周期点，计算了相关极限值，并发现正则半单元素在确定极限值中起关键作用。
摘要：
考虑对于素数L≠2，使得L|q-1（其中q是素数的幂）的幂映射x→x^L。我们确定此映射在有限域阶数为q的n×n矩阵群M_n(q)以及群GL_n(q)=M_n(q)×中的周期点。我们计算极限lim_{q→∞,v_L(q-1)=c} |Per(x^L,M_ℓ(q))| / |M_ℓ(q)|以及相应的lim_{q→∞,v_L(q-1)=c} |Per(x^L,GL_ℓ(q))| / |GL_ℓ(q)|，其中v_L表示L-adic估值。我们还计算了lim_{q→∞,v_L(q-1)=c} |Per(x^L,Sp_{2ℓ}(q))| / |Sp_{2ℓ}(q)|和lim_{q→∞,v_L(q-1)=c} |Per(x^L,U_ℓ(q))| / |U_ℓ(q)|；结果表明这两个极限值相同。在所有情况下，发现正则半单元素在确定极限值中起作用。
英文摘要：
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.
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