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

Junyi Xie, She Yang

correcthigh confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the full pDML statement for automorphisms of projective surfaces in characteristic p, including the precise shape of return sets and the “moreover” clause that p-sets arise only in the bounded-degree case (Theorem 1.1), and it reduces the problem to the parabolic case and solves it via a new height argument using Gizatullin’s fibration (Theorem 1.3) . By contrast, the candidate solution claims the general bounded-degree case still needed work as of 2025-04-03 and labels the problem likely open; that is contradicted by the paper’s results and references establishing the bounded-degree and hyperbolic cases and completing the parabolic case.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript resolves the pDML problem for automorphisms of projective surfaces in characteristic p using a clean trichotomy and a novel height-growth method to settle the parabolic case. The result is definitive and of high interest to arithmetic dynamics. The exposition is clear overall, though a few technical points could be elaborated for accessibility.