THE DYNAMICAL MORDELL-LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

Let K be an algebraically closed field of prime characteristic p, let X be a semiabelian variety defined over a finite subfield of K, let Phi : X -> X be a regular self-map defined over K, let V subset of X be a subvariety defined over K, and let alpha is an element of X(K). The dynamical Mordell-Lang conjecture in characteristic p predicts that the set S = {n is an element of N: Phi(n)(alpha) is an element of V} is a union of finitely many arithmetic progressions, along with finitely many p-sets, which are sets of the form {Sigma(m)(i=1) c(i) p(ki ni) : n(i) is an element of N} for some m is an element of N, some rational numbers ci and some non-negative integers ki. We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case X is an algebraic torus, we can prove the conjecture in two cases: either when dim(V) <= 2, or when no iterate of 8 is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of X. We end by proving that Vojta's conjecture implies the dynamical Mordell-Lang conjecture for tori with no restriction.