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

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 \ ∑ⱼ₌₁ᵐ cⱼ pkⱼnⱼ : nⱼ ∈ N₀ \ (cⱼ ∈ Q, kⱼ ∈ N₀), there exist a split torus X = Gₘᵏ defined over K=Fₚ̄(t), an endomorphism Φ of X, α ∈ X(K) and a closed subvariety V ⊆ X such that \ n ∈ N₀ : Φⁿ(α) ∈ V(K) \ = S.