A Hasse Principle for Periodic Points

Let $F$ be a global field, let $\vp \in \Fx$ be a rational map of degree at least 2, and let $\a \in F$. We say that $\a$ is periodic if $\vpn (\a) = \a$ for some $n \geq 1$. 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 $\a$ is periodic on residue fields for every prime in a set of natural density density 1 to know that $\a$ is periodic in $F$.