  We study the problem of unlikely intersections for automorphisms of Markov
surfaces of positive entropy. We show for certain parameters that two
automorphisms with positive entropy share a Zariski dense set of periodic
points if and only if they share a common iterate. Our proof uses arithmetic
equidistribution for adelic line bundles over quasiprojective varieties, the
theory of laminar currents and quasi-Fuchsian representation theory.
