On the functional graph of $f(X)=c(X^{q+1}+a{X}^2)$ over quadratic extensions of finite fields

Abstract

Let ${\mathbb{F}}_q$ be the finite field with $q$ elements and $char({\mathbb{F}}_q)$ odd. In this article we will describe completely the dynamics of the map $f(X)=c(X^{q+1}+a{X}^2)$, for $a=\{\pm 1\}$ and $c\in {\mathbb{F}}_q^{\ast }$, over the finite field ${\mathbb{F}}_{q^2}$, and give some partial results for $a\in {\mathbb{F}}_q^{\ast }\setminus \{\pm 1\}$.