Periodic Points of Power Maps in Finite Matrix Groups and Algebras

Abstract

Consider the power map $x\mapsto x^L$ for a prime $L\ne 2$ such that $L|q-1$ where $q$ is a power of a prime. We determine the periodic points under this map for $M_n(q)$, the algebra of $n\times n$ matrices over a finite field of order $q$, and also for the group ${\mathrm{GL}}_n(q)=M_n(q{)}^{\times }$. We compute the limit $\underset{\begin{array}{c}q⟶\infty \\ v_L(q-1)=c\end{array}}{\lim }\frac{\left|\mathrm{Per}(x^L,M_{\ell }(q))\right|}{|M_{\ell }(q)|}$ and consequently $\underset{\begin{array}{c}q⟶\infty v_L(q-1)=c\end{array}}{\lim }\frac{\left|\mathrm{Per}(x^L,{\mathrm{GL}}_{\ell }(q))\right|}{|{\mathrm{GL}}_{\ell }(q)|}$, where $v_L$ denotes the $L$-adic valuation. We also compute the quantity $\underset{\begin{array}{c}q⟶\infty v_L(q-1)=c\end{array}}{\lim }\frac{\left|\mathrm{Per}(x^L,{\mathrm{Sp}}_{2\ell }(q))\right|}{|{\mathrm{Sp}}_{2\ell }(q)|}$ and $\underset{\begin{array}{c}q⟶\infty v_L(q-1)=c\end{array}}{\lim }\frac{\left|\mathrm{Per}(x^L,U_{\ell }(q))\right|}{|U_{\ell }(q)|}$; 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.

Preliminary version: 16 pages. Comments are always welcome!