Fixed point proportions for Galois groups of non-geometric iterated extensions

TL;DR

This paper investigates the behavior of fixed point proportions in Galois groups of iterated extensions of rational maps over number fields, extending previous results and establishing necessary and sufficient conditions for the proportion of periodic points to tend to zero.

Contribution

It generalizes prior work by providing weaker sufficient conditions for the vanishing of fixed point proportions and identifies these conditions as necessary in specific cases like polynomial maps.

Findings

Proportion of periodic points tends to zero under certain conditions.

Weaker sufficient conditions are identified for the property to hold.

Necessary conditions are established for specific families of functions.

Abstract

Given a map $\varphi:\mathbb{P}^1\rightarrow \mathbb{P}^1$ of degree greater than 1 defined over a number field $k$, one can define a map $\varphi_\mathfrak{p}:\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})\rightarrow \mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})$ for each prime $\mathfrak{p}$ of good reduction, induced by reduction modulo $\mathfrak{p}$. It has been shown that for a typical $\varphi$ the proportion of periodic points of $\varphi_\mathfrak{p}$ should tend to $0$ as $|\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})|$ grows. In this paper, we extend previous results to include a weaker set of sufficient conditions under which this property holds. We are also able to show that these conditions are necessary for certain families of functions, for example, functions of the form $\varphi(x)=x^d+c$, where $0$ is not a preperiodic point of this map. We study the proportion of periodic points by looking at the fixed point proportion of the Galois groups of certain extensions associated to iterates of the map.