Dynamical Mordell-Lang problem for automorphisms of surfaces in positive characteristic

TL;DR

The paper resolves the p-DML problem for automorphisms of projective surfaces over fields of positive characteristic by giving an explicit description of return sets: they are finite unions of arithmetic progressions together with finitely many p-sets of the form $\{\frac{c_0+c_1q^m}{q-1}: m\in\mathbb{N}\}$, where $q$ is a power of $p$ and $q-1\mid c_0+c_1$. The authors classify automorphisms into three dynamical types and treat the parabolic case using a height-argument augmented by Gizatullin's theorem, which yields a fibration preserved by the automorphism. The abelian-surface case is handled with an explicit height-based analysis on a fibration to an elliptic base, while the general case is completed via Albanese reductions and an auxiliary finiteness lemma for automorphism groups. They also establish a converse-type result: every admissible return-set form occurs, with constructions showing the necessity of p-sets in the dense-orbit situation and their absence in bounded-degree unbounded growth. Overall, the work extends dynamical Mordell-Lang results to positive characteristic for surface automorphisms and provides a precise, applicable description of all possible return-sets.

Abstract

We solve the dynamical Mordell-Lang problem in positive characteristic for automorphisms of projective surfaces.

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• Proposition 2.6
• Theorem 2.7