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 $n \rightarrow \infty$ along certain arithmetic progressions, and compute simple explicit formulas for $\delta(L_\ell,q)$ when $\ell$ is a prime and $E$ belongs to a special family of supersingular elliptic curves.