On a dynamical brauer-manin obstruction

Let φ: X → X be a morphism of a variety defined over a number field K, let V ⊂ X be a K-subvariety, and let O_φ(P) = {φⁿ(P): n ≥ 0} be the orbit of a point P ∈ X(K). We describe a local-global principle for the intersection V ∩ Oφ(P). This principle may be viewed as a dynamical analog of the Brauer- Manin obstruction. We show that the rational points of V (K) are Brauer-Manin unobstructed for power maps on ℙ² in two cases: (1) V is a translate of a torus. (2) V is a line and P has a preperiodic coordinate. A key tool in the proofs is the classical Bang- Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties.