  Let X be a smooth curve defined over the algebraic numbers, let a,b be
algebraic numbers, and let f_l(x) be an algebraic family of rational maps
indexed by all l in X. We study whether there exist infinitely many l in X such
that both a and b are preperiodic for f_l. In particular we show that if P,Q
are polynomials over the algebraic numbers such that deg(P) >= 2+deg(Q), and
there exists l such that a is periodic for P(x)/Q(x) + l, but b is not
preperiodic for P(x)/Q(x) + l, then there exist at most finitely many l such
that both a and b are preperiodic for P(x)/Q(x)+l. We also prove a similar
result for certain two-dimensional families of endomorphisms of P^2.
