Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve

We prove the dynamical Mordell-Lang conjecture for product of endomorphisms of an affine curve and a projective curve over $\overline{\mathbb{Q}}$.

💡 Research Summary

The paper addresses the Dynamical Mordell‑Lang (DML) conjecture for split self‑maps on the product of an affine curve X and a projective curve Y defined over (\overline{\mathbb Q}). The DML conjecture predicts that for a morphism (f\colon X\to X) and a closed subvariety (V\subset X), the set of integers (n) for which the orbit (f^n(x)) meets (V) is a finite union of arithmetic progressions. While the conjecture is known for étale maps, for polynomial endomorphisms of (\mathbb A^2), and in several other special cases, the authors prove it for the much broader class of split maps ((f,g)) acting on (X\times Y).

The core of the argument is a reduction to the simplest non‑trivial case: (X=\mathbb A^1) and (Y=\mathbb P^1). Proposition 1.2 treats this situation. After eliminating trivial cases (degree‑one maps, maps that become polynomial after conjugation, etc.) the authors assume (\deg f=\deg g=d>1) and that no iterate of (g) is conjugate to a polynomial. Under this hypothesis (g) has no exceptional points in the sense of Silverman.

Two technical lemmas are pivotal. Lemma 2.1 (Silverman, 1993) states that for a rational map without exceptional points, any non‑preperiodic orbit cannot converge to a point with maximal speed in any non‑archimedean absolute value. Lemma 2.2 shows that for a non‑preperiodic point (x_0) of a polynomial‑like map (f) of degree (d>1), there exists a non‑archimedean place (v) and constants (c_1,c_2>0) such that for large (n) one has \