A Local-Global Criterion for Dynamics on P^1

TL;DR

This paper establishes a local-global criterion for the intersection of a rational map's orbit and a finite set on the projective line, advancing the understanding of dynamical systems over number and function fields.

Contribution

It provides a specific local-global criterion for dynamical intersections on P^1, a special case of the dynamical Brauer-Manin conjecture.

Findings

Proves a criterion linking local and global orbit intersections

Applies to rational maps over number and function fields

Supports the dynamical Brauer-Manin conjecture

Abstract

Let K be a number field or a function field, let F:P^1 --> P^1 be a rational map of degree at least two defined over K, let P be a point in P^1(K) having infinite F-orbit, and let Z be a finite subset of Z. We prove a local-global criterion for the intersection of the F-orbit of P and the finite set Z. This is a special case of a dynamical Brauer-Manin criterion suggested by Hsia and Silverman.