A converse of dynamical Mordell--Lang conjecture in positive characteristic

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 $\left\{{\sum }_{j=1}^m{c}_j{p}^{k_j{n}_j}:n_j\in {\mathbb{N}}_0\right\}$ ($c_j\in \mathbb{Q}$, $k_j\in {\mathbb{N}}_0$), 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 $\left\{n\in {\mathbb{N}}_0:{\Phi }^n(\alpha )\in V(K)\right\}=S$.

6 pages

References in corpus (1)

• Dynamical Mordell-Lang conjecture for totally inseparable liftings of Frobenius