  We prove a quantitative partial result in support of the Dynamical
Mordell-Lang Conjecture (also known as the DML conjecture) in positive
characteristic. More precisely, we show the following: given a field $K$ of
characteristic $p$, given a semiabelian variety $X$ defined over a finite
subfield of $K$ and endowed with a regular self-map $\Phi:X \longrightarrow X$
defined over $K$, given a point $\alpha\in X(K)$ and a subvariety $V\subseteq
X$, then the set of all non-negative integers $n$ such that $\Phi^n(\alpha)\in
V(K)$ is a union of finitely many arithmetic progressions along with a subset
$S$ with the property that there exists a positive real number $A$ (depending
only on $N$, $\Phi$, $\alpha$, $V$) such that for each positive integer $M$, we
have $$\#\left\{n\in S\colon~ n\le M\right\}\le A\cdot \left(1+\log
M\right)^{\dim V}.$$
