# The image size of iterated rational maps over finite fields

## TL;DR

This paper derives asymptotic formulas for the size of image sets of iterated rational maps over finite fields, linking Galois group properties to image set growth and applying results to bounds on periodic points in reductions of maps over number fields.

## Contribution

It introduces a method to estimate image set sizes of iterated rational maps over finite fields using Galois groups and Chebotarev Density, with applications to periodic points over number fields.

## Key findings

- Asymptotic formulas for image set sizes as a function of iteration
- Connection between Galois groups and image set growth
- Explicit bounds on periodic points in residue fields

## Abstract

Let $\varphi:\mathbb{P}^1(\mathbb F_q)\to\mathbb{P}^1(\mathbb F_q)$ be a rational map of degree $d>1$ on a fixed finite field. We give asymptotic formulas for the size of image sets $\varphi^n(\mathbb{P}^1(\mathbb F_q))$ as a function of $n$. This is done using properties of Galois groups of iterated maps, whose connection to the size of image sets is established via the Chebotarev Density Theorem. We apply our results in the following setting. For a rational map defined over a number field, consider the reduction of the map modulo each prime of the number field. We use our results to give explicit bounds on the proportion of periodic points in the residue fields.