  Fix an odd prime $p$. If $r$ is a positive integer and $f$ a polynomial with
coefficients in $\mathbb{F}_{p^r}$, let $P_{p,r}(f)$ be the proportion of
$\mathbb{P}^1(\mathbb{F}_{p^r})$ that is periodic with respect to $f$. We show
that as $r$ increases, the expected value of $P_{p,r}(f)$, as $f$ ranges over
quadratic polynomials, is less than $22/(\log{\log{p^r}})$. This result follows
from a uniformity theorem on specializations of dynamical systems of rational
functions over residually finite Dedekind domains. The specialization theorem
generalizes previous work by Juul et al. that holds for rings of integers of
number fields. Moreover, under stronger hypotheses, we effectivize this
uniformity theorem by using the machinery of heights over general global
fields; this version of the theorem generalizes previous work of Juul on
polynomial dynamical systems over rings of integers of number fields. From
these theorems we derive effective bounds on image sizes and periodic point
proportions of families of rational functions over finite fields.
