Distribution of preperiodic points in one-parameter families of rational maps

Abstract

Let $f_t$ be a one-parameter family of rational maps defined over a number field $K$. We show that for all $t$ outside of a set of natural density zero, every $K$-rational preperiodic point of $f_t$ is the specialization of some $K(T)$-rational preperiodic point of $f$. Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of $K$-rational preperiodic points of $f$, giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family $f_t(z) = z^2 + t$ over the field of rational numbers.