A converse of dynamical Mordell--Lang conjecture in positive characteristic (2403.05107v1)

Abstract: In this paper, we prove the converse of the dynamical Mordell--Lang conjecture in positive characteristic: For every subset $S \subseteq \mathbb{N}0$ which is a union of finitely many arithmetic progressions along with finitely many $p$-sets of the form $\left { \sum{j=1}^{m} c_j p^{k_jn_j} : n_j \in \mathbb{N}_0 \right }$ ($c_j \in \mathbb{Q}$, $k_j \in \mathbb{N}_0$), there exist a split torus $X = \mathbb{G}_m^k$ defined over $K=\overline{\mathbb{F}_p}(t)$, an endomorphism $\Phi$ of $X$, $\alpha \in X(K)$ and a closed subvariety $V \subseteq X$ such that $\left { n \in \mathbb{N}_0 : \Phi^n(\alpha) \in V(K) \right } = S$.