摘要：I will present two recent results linking number theory with dynamical systems.The first result exhibit all pairs of complex polynomials f(x) and  g(x)  for which there are complex numbers c and d such that the orbits {c, f(c), f(f(c)), ...} and {d, g(d), g(g(d)), ...} have infinite intersection. The second result asserts that any polynomial with rational coefficients induces a function on the rational numbers which is at most 6-to-1 outside a finite set. I will also present conjectural generalizations of these two results which simultaneously generalize the Mordell-Lang conjecture and Mazur's theorem on rational torsion on elliptic curves, respectively.