Wreath products and proportions of periodic points

Abstract

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