On Dynamical Cancellation. (25th March 2022) - Bell, Jason P; Matsuzawa, Yohsuke; Satriano, Matthew
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 2nd author, Meng, Shibata, and Zhang, who asks whether the tower of K -points Y(K)(f^-1(Y))(K)(f^-2(Y))(K)… eventually stabilizes, where YX 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 s_0, depending only on X and K, such that whenever x, y X(K) have the property that f^s(x) = f^s(y) for some s 0, we necessarily have f^s_0(x) = f^s_0(y) . 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 f_1, …, f_r .