The nonperiodic dynamical Mordell–Lang conjecture

Let $X$ be a quasiprojective variety defined over ${\mathbb C}$, let $V\subset X$ be a subvariety, let $\Phi$ be an endomorphism of $X$, and let $P\in X({\mathbb C})$. The orbit of $P$ under $\Phi$ is

$$\mathcal{O}_{\Phi}(P)=\{\Phi^n(P):n\geq 0\}.$$

A subvariety is periodic under $\Phi$ if there is a positive integer $N\geq 1$ such that $\Phi^N(V)\subseteq V$. Nonperiodic dynamical Mordell–Lang conjecture. If $V({\mathbb C})\cap\mathcal{O}_{\Phi}(P)$ is an infinite set, then $V$ contains a positive-dimensional subvariety that is periodic under $\Phi$. This is presented as a special case of the dynamical Mordell–Lang conjecture and links infinite orbit–subvariety intersections to positive-dimensional periodic geometry. Its resolution is not specified in the supplied text.