$p$-adic interpolation of orbits under rational maps

Abstract:

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 $X$.