Intersection of orbits of loxodromic automorphisms of affine surfaces

We show the following result: If X₀ is an affine surface over a field K and $f, g$ are two loxodromic automorphisms with an orbit meeting infinitely many times, then f and g must share a common iterate. The proof uses the preliminary work of the author in [Abb23] on the dynamics of endomorphisms of affine surfaces and arguments from arithmetic dynamics. We then show a dynamical Mordell-Lang type result for surfaces in X₀ × X₀.