Wreath Products and Proportions of Periodic Points

Let $\varphi : \mathbb {P}^1 \longrightarrow \mathbb {P}^1$ be a rational map of degree greater than 1 defined over a number field $k$ with ring of integers $\mathfrak {o}_k$. 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 ...