The Dynamical Mordell-lang Conjecture in Positive Characteristic

Let K be an algebraically closed field of prime characteristic p, let N ∈ N, let Φ : Gm −→ Gm be a self-map defined over K, let V ⊂ Gm be a curve defined over K, and let α ∈ Gm(K). We show that the set S = {n ∈ N : Φn(α) ∈ 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+ bpkn : n ∈ N} for some a, b ∈ Q and some k ∈ 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.