Periodic points of rational functions over finite fields

Abstract For a prime power and a rational function with coefficients in , let be the proportion of that is periodic with respect to . Furthermore, if is a positive integer, let be the set of prime powers coprime to and let be the expected value of as ranges over rational functions with coefficients in of degree . We prove that if is a positive integer no less than 2, then tends to 0 as increases in . This theorem generalizes our previous work, which held only for quadratic polynomials, and only in fixed characteristic. 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.