  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
\'etale. 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 $s \geq 0$, we
necessarily have $f^{s_{0}}(x) = f^{s_{0}}(y)$. We prove this holds for \'etale
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 $f_1,\dots,f_r$.
