Dynamical Uniform Bounds for Fibers and a Gap Conjecture

 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.