Bounded Height In Families Of Dynamical Systems

Let a, b is an element of (Q) over bar be such that exactly one of a and b is an algebraic integer, and let f(t)(z) := z(2)+t be a family of polynomials parameterized by t is an element of(Q) over bar. We prove that the set of all t is an element of(Q) over bar for which there exist m, n >= 0 such that f(t)(m) (a) = f(t)(n) (b) has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics.