Evidence for the Dynamical Brauer-Manin Criterion

TL;DR

This paper investigates the dynamical Brauer-Manin criterion, providing evidence that it explains the absence of intersection points between orbits and subvarieties in dynamical systems over number fields, supported by probabilistic and unconditional results.

Contribution

It offers new evidence that the dynamical Brauer-Manin condition accounts for orbit-subvariety disjointness, including probabilistic insights and unconditional results for etale maps.

Findings

Probabilistic evidence supporting the criterion

Unconditional results for etale maps

Enhanced understanding of orbit and subvariety interactions

Abstract

Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a "Bruaer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.