$p$-Adic interpolation of orbits under rational maps

TL;DR

This paper demonstrates a p-adic interpolation of orbits under rational maps over fields of characteristic zero, establishing a connection between dynamical systems and p-adic analysis, and proving cases of the dynamical Mordell-Lang conjecture.

Contribution

It introduces a method to interpolate orbits of rational maps p-adically, providing new tools for understanding dynamical systems over non-archimedean fields.

Findings

Existence of p-adic power series interpolating orbits

Application to the dynamical Mordell-Lang conjecture for split maps

Infinite non-archimedean completions with convergent power series

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 $g$ \'etale.