Preperiodic points of polynomial dynamical systems over finite fields

For a prime $p$, positive integers $r,n$, and a polynomial $f$ with coefficients in ${\mathbb{F}}_{p^r}$, let $W_{p,r,n}(f)=f^n\left({\mathbb{F}}_{p^r}\right)\setminus f^{n+1}\left({\mathbb{F}}_{p^r}\right)$. As $n$ varies, the $W_{p,r,n}(f)$ partition the set of strictly preperiodic points of the dynamical system induced by the action of $f$ on ${\mathbb{F}}_{p^r}$. In this paper we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of ${\mathbb{F}}_{p^r}$ lying in a given $W_{p,r,n}(f)$. Moreover, when we generalize our definition of $W_{p,r,n}(f)$, we obtain both upper and lower bounds for the resulting averages.