  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.
