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

TL;DR

This work proves the dynamical Mordell–Lang conjecture for the product of endomorphisms on a split variety $(X\times Y)$ with $X$ affine and $Y$ projective over $\overline{\mathbb{Q}}$, by reducing to the key case $(\mathbb{A}^1\times\mathbb{P}^1, f\times g)$ with $\deg f=\deg g>1$. It leverages a valuation-growth lemma for non-exceptional rational maps and analyzes orbit intersections with curves via the projective closure, obtaining finiteness results unless the curve is invariant; this yields the DML(1) property and, hence, the full DML statement for the split product. The main contribution is establishing DML for a broad class of split endomorphisms on mixed affine–projective varieties, including a detailed treatment of the $\mathbb{A}^1\times\mathbb{P}^1$ case and its extensions. This advances arithmetic dynamics by broadening the applicability of DML to structured products of dynamical systems and provides a framework for further generalizations.

Abstract

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

Theorems & Definitions (6)

• Theorem 1.1
• Proposition 1.2
• Corollary 1.3
• Lemma 2.1
• Lemma 2.2
• proof