Evidence for the Dynamical Brauer-Manin Criterion

Let φ : X → X be a morphism of a variety over a number field K. We consider local conditions and a “Brauer-Manin” condition, defined by Hsia and Silverman, for the orbit of a point P ∈ X(K) to be disjoint from a subvariety V ⊆ X, i.e., for Oφ(P )∩ V = ∅. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection Oφ(P ) ∩ V ; this evidence stems from a probabilistic argument as well as unconditional results in the case of étale maps.