On the distribution of orbits in affine varieties

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