On the distribution of orbits in affine varieties

Given an affine variety $X$, a morphism $\phi:X\to X$, a point $\alpha\in X$, and a Zariski closed subset $V$ of $X$, we show that the forward $\phi$-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.