Preperiodic points of polynomial dynamical systems over finite fields

For a prime p, positive integers r, n, and a polynomial f with coefficients in F-pr, let W-p,W-r,W-n(f) = f(n) (F-pr) \ f(n+1) (F-pr). As n varies, the W-p,W-r,W-n(f) partition the set of strictly preperiodic points of the dynamical system induced by the action of f on F-pr. 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 F-pr lying in a given W-p,W-r,W-n(f). Moreover, when we generalize our definition of W-p,W-r,W-n(f), we obtain both upper and lower bounds for the resulting averages.