Inner Galois Equidistribution in S-Hecke orbits

We obtain results on the so-called Andre-Pink-Zannier conjecture which is a special case of a the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Our methods rely on an ergodic theorem of Richard-Zamojski and we are able to obtain stronger conclusions that those of the Andre-Pink-Zannier conjecture in the special case we consider. We work under the assumption of the S-Shafarevich conjecture and S-semisimplicity conjecture which hold for Shimura varieties of abelian type.