Intersection of orbits of loxodromic automorphisms of affine surfaces

Abstract

We show the following result: If $X_0$ 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 $X_0\times X_0$.