# zbMATH — the first resource for mathematics

Wreath products and proportions of periodic points. (English) Zbl 1404.37124
Summary: 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 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 $$\varphi$$.

##### MSC:
 37P25 Dynamical systems over finite ground fields 11G35 Varieties over global fields 37P15 Dynamical systems over global ground fields 37P35 Arithmetic properties of periodic points
Full Text: