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 subset of N 0 which is a union of finitely many arithmetic progressions along with finitely many p-sets {Sigma m } of the form j =1 c j p k j n j : n j E N 0 (cj E Q, k j E N 0 ), there exist a split torus X = G k m defined over K = F p ( t ), an endomorphism Phi of X , alpha E X ( K ) and a closed subvariety V subset of X such that {n E N 0 : Phi n(alpha) E V ( K ) } = S .