Periods of rational maps modulo primes

TL;DR

This paper proves that for rational maps over number fields, the property of a point not being in another's forward orbit persists modulo a positive density of primes, with evidence against higher-dimensional analogs.

Contribution

It establishes a positive density result for primes where orbit relations hold modulo primes and explores limitations in higher dimensions.

Findings

A positive proportion of primes preserve orbit relations modulo primes.

The result extends to multiple maps and points.

Heuristic and numerical evidence suggest higher-dimensional analogs are unlikely.

Abstract

Let $K$ be a number field, let $\phi \in K(t)$ be a rational map of degree at least 2, and let $\alpha, \beta \in K$. We show that if $\alpha$ is not in the forward orbit of $\beta$, then there is a positive proportion of primes ${\mathfrak p}$ of $K$ such that $\alpha \mod {\mathfrak p}$ is not in the forward orbit of $\beta \mod {\mathfrak p}$. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $\alpha$ by a hypersurface, such as the ramification locus of a morphism $\phi : {\mathbb P}^{n} \to {\mathbb P}^{n}$.