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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 an étale self-map of a quasiprojective variety X. [ABSTRACT FROM AUTHOR]