The dynamical Mordell-Lang problem for Noetherian spaces

Abstract

Let $X$ be a Noetherian space, let $\Phi:X\longrightarrow X$ be a continuous function, let $Y\subseteq X$ be a closed set, and let $x\in X$. We show that the set $S:=\{n\in\mathbb{N}\colon \Phi^n(x)\in Y\}$ is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety $X$, any rational map $\Phi:X\longrightarrow X$, any subvariety $Y\subseteq X$, and any point $x\in X$ whose orbit under $\Phi$ is in the domain of definition for $\Phi$, the set $S$ is a finite union of arithmetic progressions together with a set of Banach density zero. This answers a question posed by Laurent Denis [7]. We prove a similar result for the backward orbit of a point and provide some quantitative bounds.