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 ⊂ N 0 S ⊂ N_0 which is a union of finitely many arithmetic progressions along with finitely many p p -sets of the form { ∑ j = 1 m c j p k j n j : n j ∈ N 0 } \{ ∑ ⱼ₌₁ᵐ c_j pk_jn_j : n_j ∈ N_0 \} ( c j ∈ Q c_j ∈ Q , k j ∈ N 0 k_j ∈ N_0 ), there exist a split torus X = G m k X = G_m^k defined over K = F p ¯ ( t ) K=F_p̄(t) , an endomorphism Φ Φ of X X , α ∈ X ( K ) α ∈ X(K) and a closed subvariety V ⊆ X V ⊆ X such that { n ∈ N 0 : Φ n ( α ) ∈ V ( K ) } = S \{ n ∈ N_0 : Φ ^n(α ) ∈ V(K) \} = S .