Preperiodic points for families of rational maps

Let X be a smooth curve defined over Q̄, let a, b ∈ P(Q̄) and let fλ(x) ∈ Q̄(x) be an algebraic family of rational maps indexed by all λ ∈ X(C). We study whether there exist infinitely many λ ∈ X(C) such that both a and b are preperiodic for fλ. In particular, we show that if P,Q ∈ Q̄[x] such that deg(P ) 2 + deg(Q), and if a, b ∈ Q̄ such that a is periodic for P (x)/Q(x), but b is not preperiodic for P (x)/Q(x), then there exist at most finitely many λ ∈ C such that both a and b are preperiodic for P (x)/Q(x) + λ. We also prove a similar result for certain two-dimensional families of endomorphisms of P. As a by-product of our method, we extend a recent result of Ingram [‘Variation of the canonical height for a family of polynomials’, J. reine. angew. Math. 685 (2013), 73–97] for the variation of the canonical height in a family of polynomials to a similar result for families of rational maps.