Wreath products and proportions of periodic points

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 $varphi$.