Wreath products and proportions of periodic points

TL;DR

This paper investigates the distribution of periodic points of rational maps over number fields, demonstrating that for many such maps, the proportion of periodic points modulo primes tends to be small, aligning with heuristic predictions.

Contribution

The paper proves that for many rational functions over number fields, the proportion of periodic points modulo primes is small, confirming heuristic expectations.

Findings

Proportion of periodic points is small for many rational maps.

Results align with heuristic predictions about periodic points.

Provides new evidence for behavior of rational maps over finite fields.

Abstract

Let $\varphi: {\mathbb P}^1 \longrightarrow {\mathbb P}^1$ be a rational map of degree greater than one defined over a number field $k$. 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 $\varphi$.