Rigidity of periodic points for loxodromic automorphisms of affine surfaces

Abstract

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. When the base field is not a number field or the function field of a curve we use the theory of Moriwaki heights to prove the result.