Density of Periodic Points for Latt\`es maps over Finite Fields

Let $L_d$ be the Latt`es map associated to the multiplication-by-$d$ endomorphism of an elliptic curve $E$ defined over a finite field ${\mathbb{F}}_q$. We determine the density $\delta (L_d,q)$ of periodic points for $L_d$ in ${\mathbb{P}}^1({\mathbb{F}}_q)$. We show that the periodic point densities $\delta (L_d,q^n)$ converge as $n\to \mathrm{\infty }$ along certain arithmetic progressions, and compute simple explicit formulas for $\delta (L_{\mathrm{\ell }},q)$ when $\mathrm{\ell }$ is a prime and $E$ belongs to a special family of supersingular elliptic curves.