On the distribution of orbits in affine varieties

Abstract

Given an affine variety $X$, a morphism $\varphi :X\to X$, a point $\alpha \in X$, and a Zariski closed subset $V$ of $X$, we show that the forward $\varphi$-orbit of $\alpha$ meets $V$ in at most finitely many infinite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the Dynamical Mordell-Lang Conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces.

References in corpus (3)

• The dynamical Mordell-Lang problem for etale maps
• Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture
• The Dynamical Mordell-Lang problem

Cited by in corpus (2)

• Height Gap Conjectures, $D$-Finiteness, and Weak Dynamical Mordell-Lang
• Rational dynamical systems, $S$-units, and $D$-finite power series