Preperiodic Points in Families of Rational Maps

Recently, Gao and Habegger have established the `relative Manin-Mumford conjecture' concerning the distribution of torsion points in families of abelian varieties. Inspired by the analogy between torsion points in abelian varieties and preperiodic points in a dynamical system, Zhang has proposed a dynamical analog of the Manin-Mumford conjecture. For instance, when can two rational maps share infinitely many common preperiodic points? In this talk we discuss progress towards answering a dynamical relative Manin-Mumford question, concerning the distribution of preperiodic points of families of rational maps. Though related questions have been considered in the dynamical setting by various authors, many problems remain unsolved. The talk will feature results with Harry Schmidt and with Laura DeMarco.