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

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