An undecidable problem in dynamics

Given a variety $X$ with a self-map $\phi$ and subvarieties $Y$ and $Z$ of $X$, we consider the set of $n$ for which ${\phi }^n(Y)$ intersects $Z$ nontrivially. Extending a beautiful construction of Lee and Nam, we show this gives rise to an undecidable problem for a fixed ambient variety $X$ and morphism $\phi$. We further use this construction to give a counterexample to a conjecture of the author, Satriano, and Sierra.