### Abstract

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) = {φ^{n}(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 ℙ^{2} 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.

Original language | English |
---|---|

Pages (from-to) | 235-250 |

Number of pages | 16 |

Journal | Journal de Theorie des Nombres de Bordeaux |

Volume | 21 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Jan 1 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal de Theorie des Nombres de Bordeaux*,

*21*(1), 235-250. https://doi.org/10.5802/jtnb.668