On Dynamical Cancellation

Abstract

Let X be a projective variety and let f be a dominant endomorphism of X, both of which are defined over a number field K. We consider a question of the second author, Meng, Shibata, and Zhang, which asks whether the tower of K-points Y pKq Ď pf ´1pY qqpKq Ď pf ´2pY qqpKq Ď ¨¨¨eventually stabilizes, where Y Ă X is a subvariety invariant under f . We show this question has an affirmative answer when the map f is étale. We also look at a related problem of showing that there is some integer s0, depending only on X and K, such that whenever x, y P XpKq have the property that f s pxq " f s pyq for some s ě 0, we necessarily have f s 0 pxq " f s 0 pyq. We prove this holds for étale morphisms of projective varieties, as well as self-morphisms of smooth projective curves. We also prove a more general cancellation theorem for polynomial maps on P 1 where we allow for composition by multiple different maps f1, . . . , fr.