Unlikely intersections problem for automorphisms of Markov surfaces

Abstract

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.