The dynamical Mordell-Lang conjecture in positive characteristic

Abstract

Let K K be an algebraically closed field of prime characteristic p p , let N ∈ N N\in \mathbb {N} , let Φ : G m N ⟶ G m N \Phi :\mathbb {G}_m^N\longrightarrow \mathbb {G}_m^N be a self-map defined over K K , let V ⊂ G m N V\subset \mathbb {G}_m^N be a curve defined over K K , and let α ∈ G m N ( K ) \alpha \in \mathbb {G}_m^N(K) . We show that the set S = { n ∈ N : Φ n ( α ) ∈ V } 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 p -arithmetic sequences, which are sets of the form { a + b p k n : n ∈ N } \{a+bp^{kn}\colon n\in \mathbb {N}\} for some a , b ∈ Q a,b\in \mathbb {Q} and some k ∈ N k\in \mathbb {N} . We also prove that our result is sharp in the sense that S 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 S which includes p p -arithmetic sequences.