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

Let f t f_t be a one-parameter family of rational maps defined over a number field K K . We show that for all t t outside of a set of natural density zero, every K K -rational preperiodic point of f t f_t is the specialization of some K ( T ) K(T) -rational preperiodic point of f f . Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of K K -rational preperiodic points of f 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 f_t(z) = z^2 + t over the field of rational numbers.