Unlikely intersection questions in arithmetic dynamics

Given an abelian variety G defined over a field of characteristic 0, the intersection between a subvariety X of G with another subvariety Y of G for which dim(X) + dim(Y) < dim(G) is generally expected to be empty. Rooted in this basic observation, Pink-Zilber and Bombieri-Masser-Zannier formulated the following conjecture regarding unlikely intersections in arithmetic geometry (over a field of characteristic 0). So, let X be an irreducible subvariety of dimension d of the abelian variety G, and assume X is not contained in any proper algebraic subgroup of G. Then the intersection of X with the union of all algebraic subgroups of G of codimension d+1 is predicted not to be Zariski dense in X. Motivated by this conjecture, we present a couple of related questions in arithmetic dynamics.