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Let $\mathbb{F}_q$ be the finite field with $q$ elements and$char(\mathbb{F}_q)$ odd. In this article we will describe completely thedynamics of the map $f(X)=c(X^{q+1}+aX^2)$, for $a=\{\pm1\}$ and$c\in\mathbb{F}_q^*$, over the finite field $\mathbb{F}_{q^2}$, and give somepartial results for $a\in\mathbb{F}_q^*\setminus\{\pm1\}$.

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