The Dynamical Mordell-Lang problem

TL;DR

This paper proves that for a broad class of dynamical systems on Noetherian spaces, the set of times when an orbit hits a particular subset is structured as a finite union of arithmetic progressions plus a negligible set, extending to algebraic varieties.

Contribution

It establishes a general structure theorem for orbit intersection sets in dynamical systems on Noetherian spaces, generalizing previous results in algebraic dynamics.

Findings

The set of hitting times is a union of finitely many arithmetic progressions and a set of Banach density zero.

The result applies to rational self-maps on quasi-projective varieties.

A similar structure holds for backward orbits.

Abstract

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point on X. We show that the set S consisting of all nonnegative integers n such that f^n(x) is in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety X, any rational self-map map f on X, any subvariety Y of X, and any point x in X whose orbit under f is in the domain of definition for f, the set S is a finite union of arithmetic progressions together with a set of Banach density zero. We prove a similar result for the backward orbit of a point.