The dynamical Mordell–Lang problem

The talk is based on works of J.P.Bell, D.Ghioca, and T.J.Tucker. Let $x \in X$ be a point in a Noetherian space, let $f:X \rightarrow X$ be a continuous function, and let $Y \subset X$ be a closed set. We show that the set $S := {n \in \mathbb{N}: f^n(x) \in Y}$ is a union of finitely many arithmetic progressions and a set with Banach density zero. We also discuss some corollaries of this result.