$p$-Adic interpolation of orbits under rational maps

Let $L$ be a field of characteristic zero, let $h:$\mathbb{P}$^1\to $\mathbb{P}$^1$bearationalmapdefinedover$L$,andlet$c\in $\mathbb{P}$^1(L)$\mathrm{.}Weshowthatthereexistsafinitelygeneratedsubfield$K$of$L$overwhichboth$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 $i\in {0,\ldots ,a-1}$thereexistsapowerseries$g_i(t)\in K_{\mathfrak{p}}[[t]]$thatconvergesontheclosedunitdiscof$K_{\mathfrak{p}}$suchthat$h^{an+i}©=g_i(n)$forallsufficientlylarge$n$. 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.