  For any $q$ a prime power, $j$ a positive integer, and $\phi$ a rational
function with coefficients in $\mathbb{F}_{q^j}$, let $P_{q,j}(\phi)$ be the
proportion of $\mathbb{P}^1(\mathbb{F}_{q^j})$ that is periodic with respect to
$\phi$. Now suppose that $d$ is an integer greater than 1. We show that if
$\gcd{(q,d!)}=1$, then as $j$ increases the expected value of $P_{q,j}(\phi)$,
as $\phi$ ranges over rational functions of degree $d$, tends to 0. This
theorem generalizes our previous work, which held only for polynomials, and
only when $d=2$. To deduce this result, we prove a uniformity theorem on
specializations of dynamical systems of rational functions with coefficients in
certain finitely-generated algebras over residually finite Dedekind domains.
This specialization theorem generalizes our previous work, which held only for
algebras of dimension one.
