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

Abstract

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