The Dynamical Mordell-Lang problem
Number Theory
2014-01-28 v1 Algebraic Geometry
Dynamical Systems
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.
Cite
@article{arxiv.1401.6659,
title = {The Dynamical Mordell-Lang problem},
author = {Jason P. Bell and Dragos Ghioca and Thomas J. Tucker},
journal= {arXiv preprint arXiv:1401.6659},
year = {2014}
}