Wreath products and proportions of periodic points

2014/10/13 by Juul, Jamie, Kurlberg, Par, Madhu, Kalyani +1 · 5 citations
#11G50 #14G25 #37P05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1410.3378

Abstract

Let φ: \mathbb P¹ \longrightarrow \mathbb P¹ be a rational map of degree greater than one defined over a number field k. For each prime \mathfrak p of good reduction for φ, we let φ_{\mathfrak p} denote the reduction of φ modulo \mathfrak p. A random map heuristic suggests that for large \mathfrak p, the proportion of periodic points of φ_{\mathfrak p} in \mathbb P¹(\mathfrak o_k/\mathfrak p) should be small. We show that this is indeed the case for many rational functions φ.

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