In this paper, we prove the converse of the dynamical Mordell–Lang conjecture in positive characteristic: For every subset S ⊂ N0 which is a union of finitely many arithmetic progressions along with finitely many p-sets of the form {Σmj=1 cjpkjnj: nj ∈ N0} (cj ∈ Q, kj ∈ N0), there exist a split torus X = Gkm defined over K = Fp(t), an endomorphism Φ of X, α ∈ X(K) and a closed subvariety V ⊆ X such that {n ∈ N0: Φn(α) ∈ V (K)} = S.
The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2024-00334558). The authors thank Dragos Ghioca for helpful comments.
