A Hasse Principle for Periodic Points

TL;DR

This paper proves a Hasse principle for periodic points of rational maps over global fields, showing local periodicity at almost all primes implies global periodicity.

Contribution

It establishes that a Hasse principle holds for periodic points and that local periodicity at a density 1 set of primes suffices for global periodicity.

Findings

Hasse principle holds for periodic points.

Local periodicity at density 1 primes implies global periodicity.

Provides conditions under which local-global principles apply to dynamical systems.

Abstract

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 $F$.