For a surjective self-morphism on a projective variety defined over a number field, the preimages question asks if the set of rational points on the iterated preimages of an invariant closed subscheme eventually stabilizes. We recently answered affirmatively in the case of the direct product of the projective line. In this talk, we introduce the preimages question, which is based on the joint work with Yohsuke Matsuzawa.