Wreath products and proportions of periodic points

Abstract

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

23 pages