Periodic points of rational functions of large degree over finite fields

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.Comment: 10 pages, typos corrected. arXiv admin note: text overlap with arXiv:2103.1653