Intersections of polynomial orbits

I'll prove this theorem:
Let f and g be polynomials (in one variable) over the complex numbers, and assume deg(f) and deg(g) are both >1. If a,b are complex numbers for which the orbits {a, f(a), f(f(a)), f(f(f(a))), ...} and {b, g(b), g(g(b)), g(g(g(b))), ...} have infinite intersection, then f and g have a common iterate, i.e., some f(f(f(f(...(f(x))...)))) = g(g(g(...(g(x))...))).

Then I'll explain how this relates to the Mordell conjecture (a curve of genus > 1 over a number field K has only finitely many points in K).

Michael Zieve