  Let $\varphi: {\mathbb P}^1 \longrightarrow {\mathbb P}^1$ be a rational map
of degree greater than one defined over a number field $k$. For each prime
${\mathfrak p}$ of good reduction for $\varphi$, we let $\varphi_{\mathfrak p}$
denote the reduction of $\varphi$ modulo ${\mathfrak p}$. A random map
heuristic suggests that for large ${\mathfrak p}$, the proportion of periodic
points of $\varphi_{\mathfrak p}$ in ${\mathbb P}^1({\mathfrak o}_k/{\mathfrak
p})$ should be small. We show that this is indeed the case for many rational
functions $\varphi$.
