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 Φ : X −→ X be a regular self-map defined over K, let V ⊂ X be a subvariety defined over K, and let α ∈ X(K). The Dynamical Mordell-Lang Conjecture in characteristic p predicts that the set S = {n ∈ N : Φ(α) ∈ V } is a union of finitely many arithmetic progressions, along with finitely many p-sets, which are sets of the form {∑m i=1 cip kini : ni ∈ N } for some m ∈ 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 Φ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of X.