Let
K
K
be an algebraically closed field of prime characteristic
p
p
, let
N
∈
N
N\in \mathbb {N}
, let
Φ
:
G
m
N
⟶
G
m
N
\Phi :\mathbb {G}_m^N\longrightarrow \mathbb {G}_m^N
be a self-map defined over
K
K
, let
V
⊂
G
m
N
V\subset \mathbb {G}_m^N
be a curve defined over
K
K
, and let
α
∈
G
m
N
(
K
)
\alpha \in \mathbb {G}_m^N(K)
. We show that the set
S
=
{
n
∈
N
:
Φ
n
(
α
)
∈
V
}
S=\{n\in \mathbb {N}\colon \Phi ^n(\alpha )\in V\}
is a union of finitely many arithmetic progressions, along with a finite set and finitely many
p
p
-arithmetic sequences, which are sets of the form
{
a
+
b
p
k
n
:
n
∈
N
}
\{a+bp^{kn}\colon n\in \mathbb {N}\}
for some
a
,
b
∈
Q
a,b\in \mathbb {Q}
and some
k
∈
N
k\in \mathbb {N}
. We also prove that our result is sharp in the sense that
S
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
S
which includes
p
p
-arithmetic sequences.