  Let K be an algebraically closed field of prime characteristic p, let N be a
positive integer, let f be a self-map on the algebraic torus T=G_m^N defined
over K, let V be a curve in T defined over K, and let x be a K-point of T. We
show that the set S consisting of all positive integers n for which f^n(x) is
contained in V is a union of finitely many arithmetic progressions, along with
a finite set and with finitely many p-arithmetic sequences, which are sets of
the form {b + ap^{kn}: n is a positive integer} where a and b are given
rational numbers and k is a positive integer. We also prove that our result is
sharp in the sense that S may be infinite without containing an arithmetic
progression. Our result addresses a positive characteristic version of the
dynamical Mordell-Lang conjecture and it is the first known instance when a
structure theorem is proven for the set S which includes p-arithmetic
sequences.
