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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)\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 \geqslant 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$|⁠. [ABSTRACT FROM AUTHOR]

We prove a uniform version of the Dynamical Mordell–Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our 1st result, we assume |$X$| is a quasi-projective variety defined over a field |$K$| of characteristic |$0$|⁠ , endowed with the action of an étale endomorphism |$\Phi $|⁠ , and |$f\colon X\longrightarrow Y$| is a morphism with |$Y$| a quasi-projective variety defined over |$K$|⁠. Then for any |$x\in X(K)$|⁠ , if for each |$y\in Y(K)$|⁠ , the set |$S_{x,y}:=\{n\in{\mathbb{N}}\colon f(\Phi ^n(x))=y\}$| is finite, then there exists a positive integer |$N_x$| such that |$\sharp S_{x,y}\le N_x$| for each |$y\in Y(K)$|⁠. For our 2nd result, we let |$K$| be a number field, |$f:X\dashrightarrow{\mathbb{P}}^1$| is a rational map, and |$\Phi $| is an arbitrary endomorphism of |$X$|⁠. If |${\mathcal{O}}_{\Phi }(x)$| denotes the forward orbit of |$x$| under the action of |$\Phi $|⁠ , then either |$f({\mathcal{O}}_{\Phi }(x))$| is finite, or |$\limsup _{n\to \infty } h(f(\Phi ^n(x)))/\log (n)>0$|⁠ , where |$h(\cdot)$| represents the usual logarithmic Weil height for algebraic points. [ABSTRACT FROM AUTHOR]