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.
Citation
Jason P. Bell. Dragos Ghioca. Thomas J. Tucker. "The dynamical Mordell-Lang problem for Noetherian spaces." Funct. Approx. Comment. Math. 53 (2) 313 - 328, December 2015. https://doi.org/10.7169/facm/2015.53.2.7
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