A Hasse Principle for Periodic Points

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