  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 f be a regular
self-map on X defined over K, let V be a subvariety of X defined over K, and
let x be a K-point of X. The Dynamical Mordell-Lang Conjecture in
characteristic p predicts that the set S consisting of all positive integers n
for which f^n(x) lies on V is a union of finitely many arithmetic progressions,
along with finitely many p-sets, which are sets consisting of all integers of
the form c_1 p^{k_1 n_1} + ... + c_m p^{k_m n_m} (as we vary n_1,..,n_m among
all positive integers), for some given positive integer m, some rational
numbers c_i and some non-negative integers k_i. 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 the dimension of V is at most 2, or when no iterate of f is a group
endomorphism which induces the action of a power of the Frobenius on a positive
dimensional algebraic subgroup of X.
