Preperiodic points of polynomial dynamical systems over finite fields

Abstract

For a prime [Formula: see text], positive integers [Formula: see text], and a polynomial [Formula: see text] with coefficients in [Formula: see text], let [Formula: see text]. As [Formula: see text] varies, the [Formula: see text] partition the set of strictly preperiodic points of the dynamical system induced by the action of [Formula: see text] on [Formula: see text]. 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 [Formula: see text] lying in a given [Formula: see text]. Moreover, when we generalize our definition of [Formula: see text], we obtain both upper and lower bounds for the resulting averages.