Rigidity of periodic points for loxodromic automorphisms of affine surfaces

We show that two automorphisms of an affine surface with dynamical degree strictly larger than 1 share a Zariski dense set of periodic points if and only if they have the same periodic points. We construct canonical heights for these automorphisms and use arithmetic equidistribution for adelic line bundles over quasiprojective varieties following the work of Yuan and Zhang. Finally, we extend the local arithmetic Hodge index theorem to a certain class of vertical adelic line bundles over quasiprojective varieties to conclude the proof.