Periodic Points of Power Maps in Finite Matrix Groups and Algebras

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