Graph components and dynamics over finite fields

TL;DR

This paper investigates the structure and behavior of dynamical systems generated by polynomials and rational maps over finite fields, focusing on the number of connected components and periodic points.

Contribution

It provides bounds on the average number of connected components and periodic points for these dynamical systems over finite fields.

Findings

Bounded average number of connected components

Bounded average number of periodic points

Quantitative insights into finite field dynamical systems

Abstract

For polynomials and rational maps of fixed degree over a finite field, we bound both the average number of connected components of their functional graphs as well as the average number of periodic points of their associated dynamical systems.