Periodic Points of Power Maps in Finite Matrix Groups and Algebras

TL;DR

This paper characterizes the periodic points of power maps in finite matrix groups and algebras, computes their asymptotic proportions, and highlights the role of regular semisimple elements in these limits.

Contribution

It provides explicit descriptions of periodic points under power maps for matrix groups and algebras over finite fields, including asymptotic limits as the field size grows.

Findings

Periodic points are characterized for matrix groups and algebras.

Asymptotic limits of proportions of periodic points are computed.

Regular semisimple elements determine the limiting behavior.

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.