Let
L
L
be a field of characteristic zero, let
h
:
P
1
→
P
1
h:\mathbb {P}^1\to \mathbb {P}^1
be a rational map defined over
L
L
, and let
c
∈
P
1
(
L
)
c\in \mathbb {P}^1(L)
. We show that there exists a finitely generated subfield
K
K
of
L
L
over which both
c
c
and
h
h
are defined along with an infinite set of inequivalent non-archimedean completions
K
p
K_{\mathfrak {p}}
for which there exists a positive integer
a
=
a
(
p
)
a=a(\mathfrak {p})
with the property that for
i
∈
{
0
,
…
,
a
−
1
}
i\in \{0,\ldots ,a-1\}
there exists a power series
g
i
(
t
)
∈
K
p
[
[
t
]
]
g_i(t)\in K_{\mathfrak {p}}[[t]]
that converges on the closed unit disc of
K
p
K_{\mathfrak {p}}
such that
h
a
n
+
i
(
c
)
=
g
i
(
n
)
h^{an+i}(c)=g_i(n)
for all sufficiently large
n
n
. As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps
(
h
,
g
)
(h,g)
of
P
1
×
X
\mathbb {P}^1 \times X
with
g
g
an étale self-map of a quasiprojective variety
X
X
.