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

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 ${\mathbb{F}}_{q^2}$.