Periodic Points of Power Maps in Finite Matrix Groups and Algebras

Abstract

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.