THE DYNAMICAL MORDELL-LANG CONJECTURE for ENDOMORPHISMS of SEMIABELIAN VARIETIES DEFINED over FIELDS of POSITIVE CHARACTERISTIC

Let be an algebraically closed field of prime characteristic, let be a semiabelian variety defined over a finite subfield of, let be a regular self-map defined over, let be a subvariety defined over, and let. The dynamical Mordell-Lang conjecture in characteristic predicts that the set is a union of finitely many arithmetic progressions, along with finitely many -sets, which are sets of the form for some, some rational numbers and some non-negative integers. We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case is an algebraic torus, we can prove the conjecture in two cases: either when, or when no iterate of is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of. We end by proving that Vojta's conjecture implies the dynamical Mordell-Lang conjecture for tori with no restriction.