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On the distribution of orbits in affine varieties

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