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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 \(\varphi\).

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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