  Let $L$ be a field of characteristic zero, let $h:\mathbb{P}^1\to
\mathbb{P}^1$ be a rational map defined over $L$, and let $c\in
\mathbb{P}^1(L)$. We show that there exists a finitely generated subfield $K$
of $L$ over which both $c$ and $h$ are defined along with an infinite set of
inequivalent non-archimedean completions $K_{\mathfrak{p}}$ for which there
exists a positive integer $a=a(\mathfrak{p})$ with the property that for $i\in
\{0,\ldots ,a-1\}$ there exists a power series $g_i(t)\in
K_{\mathfrak{p}}[[t]]$ that converges on the closed unit disc of
$K_{\mathfrak{p}}$ such that $h^{an+i}(c)=g_i(n)$ for all sufficiently large
$n$. As a consequence we show that the dynamical Mordell-Lang conjecture holds
for split self-maps $(h,g)$ of $\mathbb{P}^1 \times X$ with $g$ \'etale.
