On the functional graph of $f(X)=X(X^{q-1}-c)^{q+1},$ over quadratic extensions of finite fields

TL;DR

This paper analyzes the structure and dynamics of a specific rational function over quadratic extensions of finite fields, revealing detailed behavior of its functional graph.

Contribution

It provides a detailed description of the dynamics of the map $f(X)=X(X^{q-1}-c)^{q+1}$ over quadratic extensions of finite fields, a novel analysis in this context.

Findings

Characterization of the functional graph of $f$ over $F_{q^2}$

Identification of cycle structures and fixed points

Insights into the iterative behavior of the map

Abstract

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements. In this paper we will describe the dynamics of the map $f(X)=X(X^{q-1}-c)^{q+1},$ with $c\in\mathbb{F}_{q}^{\ast},$ over the finite field $\mathbb{F}_{q^2}$.