Wreath products and proportions of periodic points. (English) Zbl 1404.37124

Juul, Jamie; Kurlberg, Pär; Madhu, Kalyani; Tucker, Tom J.

Int. Math. Res. Not. 2016, No. 13, 3944-3969 (2016).

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