Title: Bounded height in families of dynamical systems,
Abstract: Let a and b be algebraic numbers 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
parametrized by t. We prove that the set of all algebraic numbers t for which
there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded
Weil height. This is a special case of a more general result supporting a new
bounded height conjecture in dynamics. Our results fit into the general setting
of the principle of unlikely intersections in arithmetic dynamics.