Preperiodic points and unlikely intersections

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of 1-dimensional complex dynamical systems. We show that for any fixed a,b∈C and any integer d≥2, the set of c∈C for which both a and b are preperiodic for zd+c is infinite if and only if ad=bd. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions ϕ,ψ∈C(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and, in particular, the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that ϕ and ψ are defined over Q¯. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with nonarchimedean Berkovich spaces playing an essential role.