Dynamical Mordell–Lang conjecture (generic orbit form)
Prove that for any projective variety X over the algebraic closure of Q, any dominant rational self-map f: X → X, and any point x with a well-defined forward orbit, if the orbit O_f(x) is Zariski dense in X then the orbit is generic in the sense that for every proper subvariety Y of X the set {n ≥ 0 | f^n(x) ∈ Y} is finite.
References
The Dynamical Mordell-Lang conjecture proposed by Ghioca and Tucker asserts that for every $x\in X_{f}(\overline{})$, if $O_f(x)$ is Zariski dense, then the $f$-orbit of $x$ is generic c.f. (see also Conjecture 1.2).
Therefore, in order to prove the DML(1) property for regular endomorphisms, our result solves the $\mathbb{G}_m$-case and leaves the $\mathbb{A}1$-case open.
When we were typing this manuscript, A.I. claimed that it can strengthen Theorem \ref{mainthm} by changing the phrase strictly less than $d$" intonot greater than $d$." If so, then this gives a (maybe) new proof of the DML conjecture for regular endomorphisms of $\mathbb{A}2$. After comprehending the arguments in this article, the reader may realize that the remaining component is exactly ``the DML conjecture for regular endomorphisms of projective surfaces". We will try to comprehend and arrange its proof for this result later.