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.

Background

The conjecture asserts a strong sparsity property of orbit intersections with proper subvarieties once the orbit is Zariski dense. It is a far-reaching analogue of classical Mordell–Lang phenomena in an arithmetic–dynamical setting.

In this paper the conjecture is used as a conditional ingredient relating arithmetic and dynamical degrees: prior results verify the Kawaguchi–Silverman conjecture under this dynamical Mordell–Lang assumption for certain cohomologically hyperbolic maps. The general statement of the dynamical Mordell–Lang conjecture remains open.