The dynamical Mordell–Lang conjecture

Let $f:X\to X$ be an endomorphism of a quasi-projective variety over a field $K$ of characteristic $0$, and let $V$ be a closed subvariety of $X$. For $x\in X(K)$, define the return set

$$\{n\in\mathbb{N}\mid f^n(x)\in V(K)\}.$$

Dynamical Mordell–Lang conjecture. For every $x\in X(K)$, the return set is a finite union of arithmetic progressions. This conjecture is a central problem in arithmetic dynamics. The paper proves it for products of endomorphisms of an affine curve and a projective curve over $\overline{\mathbb{Q}}$, while the general statement remains open.