A HASSE PRINCIPLE FOR PERIODIC POINTS

Abstract

Let F be a global field, let φ ∈ F(x) be a rational map of degree at least 2, and let α ∈ F. We say that α is periodic if φn(α) = α for some n ≥ 1. A Hasse principle is the idea, or hope, that a phenomenon which happens everywhere locally should happen globally as well. The principle is well known to be true in some situations and false in others. We show that a Hasse principle holds for periodic points, and further show that it is sufficient to know that α is periodic on residue fields for every prime in a set of natural density 1 to know that α is periodic in F.