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 $Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots$ eventually stabilizes, where $Y\subset X$ 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 $s_0$, depending only on $X$ and $K$, such that whenever $x,y\in X(K)$ have the property that $f^s(x)=f^s(y)$ for some $s\ge 0$, 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 ${\mathbb{P}}^1$ where we allow for composition by multiple different maps $f_1,\dots ,f_r$.

27 pages