Iteration of Polynomials $A{X}^d+C$ Over Finite Fields

For a polynomial $f(X)=A{X}^d+C\in {\mathbb{F}}_p[X]$ with $A\mathrm{\ne }0$ and $d\geq 2$,weprovethatif$d;|;p-1$and$f^i(0)\neq f^j(0)$for$0\leq i<j\leq N$, then $\mathrm{\#}{f}^N({\mathbb{F}}_p)\sim \frac{2p}{(d-1)N},$ where $f^N$ is the $N$-th iteration of $f$.