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
$\unicode[STIX]{x1D6F7}:X\longrightarrow X$
be a regular self-map defined over
$K$
, let
$V\subset X$
be a subvariety defined over
$K$
, and let
$\unicode[STIX]{x1D6FC}\in X(K)$
. The dynamical Mordell–Lang conjecture in characteristic
$p$
predicts that the set
$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$
is a union of finitely many arithmetic progressions, along with finitely many
$p$
-sets, which are sets of the form
$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$
for some
$m\in \mathbb{N}$
, 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
$\dim (V)\leqslant 2$
, or when no iterate of
$\unicode[STIX]{x1D6F7}$
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.