On the DML(1) property for regular endomorphisms of affine spaces: the $\mathbb{G}_m$-case

She Yang; Aoyang Zheng

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arXiv · 2609.29107

On the DML(1) property for regular endomorphisms of affine spaces: the $\mathbb{G}_m$-case

She Yang·Aoyang Zheng

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Abstract

Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ and let $C\subseteq\mathbb{A}_{\mathbb{C}}^N$ be an irreducible curve. Suppose $C$ has an infinite intersection with the $f$-orbit of a point $x\in\mathbb{A}^N(\mathbb{C})$. Then the normalization of $C$ is isomorphic to either $\mathbb{A}^1$ or $\mathbb{G}_m$. We prove that $C$ is $f$-periodic in the latter case, as expected by the dynamical Mordell-Lang conjecture.

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BibTeX RIS

She Yang, Aoyang Zheng. 2026-09-24. On the DML(1) property for regular endomorphisms of affine spaces: the $\mathbb{G}_m$-case. https://arxiv.org/abs/2609.29107

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