The dynamical Mordell-Lang conjecture in positive characteristic

Abstract:

Let $K$ be an algebraically closed field of prime characteristic $p$, let $N\in \mathbb {N}$, let $\Phi :\mathbb {G}_m^N\longrightarrow \mathbb {G}_m^N$ be a self-map defined over $K$, let $V\subset \mathbb {G}_m^N$ be a curve defined over $K$, and let $\alpha \in \mathbb {G}_m^N(K)$. We show that the set $S=\{n\in \mathbb {N}\colon \Phi ^n(\alpha )\in V\}$ is a union of finitely many arithmetic progressions, along with a finite set and finitely many $p$-arithmetic sequences, which are sets of the form $\{a+bp^{kn}\colon n\in \mathbb {N}\}$ for some $a,b\in \mathbb {Q}$ and some $k\in \mathbb {N}$. We also prove that our result is sharp in the sense that $S$ may be infinite without containing an arithmetic progression. Our result addresses a positive characteristic version of the dynamical Mordell-Lang conjecture, and it is the first known instance when a structure theorem is proven for the set $S$ which includes $p$-arithmetic sequences.