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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
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