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

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.