Uniform Dynamical Mordell–Lang Conjecture

Let $X$ and $Y$ be quasi-projective varieties over a field $K$ of characteristic $0$. Let $f\colon X\longrightarrow Y$ be a morphism defined over $K$, let $\Phi\colon X\longrightarrow X$ be an endomorphism defined over $K$, and let $x\in X(K)$. Uniform Dynamical Mordell–Lang Conjecture. If

$$|\mathcal{O}_\Phi(x)\cap f^{-1}(y)|<\infty$$

for all $y\in Y(K)$, then there is a constant $N$ such that

$$|\mathcal{O}_\Phi(x)\cap f^{-1}(y)|<N$$

for all $y\in Y(K)$. This strengthens the Dynamical Mordell–Lang prediction by requiring a uniform bound across the fibers of $f$. It is known for étale endomorphisms by the theorem stated in the paper, and the conjecture remains open for arbitrary endomorphisms.