Height arguments toward the dynamical Mordell-Lang problem in arbitrary characteristic

We use height arguments to prove two results about the dynamical Mordell-Lang problem. (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any iteration is not an integer. (ii) Let $f\times g:X\times C\rightarrow X\times C$ be an endomorphism, where $f$ and $g$ are surjective endomorphisms of a projective variety $X$ and a projective curve $C$, respectively. If the degree of $g$ is greater than the first dynamical degree of $f$, then the return sets of the system $(X\times C,f\times g)$ have the same form as the return sets of the system $(X,f)$. Using the second result, we deal with the case of split self-maps of products of curves, for which the degrees of the factors are pairwise distinct. In the cases that the height argument cannot be applied, we find examples which show that the return set can be very complicated -- more complicated than experts once imagined -- even for endomorphisms of tori with zero entropy. One may compare them with the conjectures and results stated in [CGSZ21] and [XY25].