Isolated Periodic Points in Several Nonarchimedean Variables

TL;DR

This paper studies isolated periodic points in nonarchimedean dynamics, generalizing known results to higher dimensions and proving a conjecture about Zariski-dense orbits for certain algebraic points.

Contribution

It introduces a new framework for analyzing fixed points and formal subvarieties in nonarchimedean dynamics, extending results to higher dimensions and proving Zhang's conjecture in specific cases.

Findings

Generalized Rivera-Letelier's results to higher dimensions

Proved existence of neighborhoods without other periodic points

Confirmed Zhang's conjecture for certain algebraic points

Abstract

Let $\varphi: \mathbb{P}^{n}_{F} \to \mathbb{P}^{n}_{F}$ where $F$ is a complete valued field. If $x$ is a fixed point, such that the action of $\varphi$ on $T_{x}$ has eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$, with $\lambda_{1}, \ldots, \lambda_{r}$ not contained in the multiplicative group generated by $\lambda_{r+1}, \ldots, \lambda_{n}$, then $\varphi$ has a codimension-$r$ fixed formal subvariety. Under mild assumptions, this subvariety is analytic. We use this to prove two results. First, we generalize results of Rivera-Letelier on isolated periodic points to higher dimension: if $F$ is $p$-adic, and each $|\lambda_{i}| \leq 1$, then there is an analytic neighborhood of $x$ without any other periodic points. And second, we prove Zhang's conjecture that there exists a $\overline{\mathbb{Q}}$-point with Zariski-dense forward orbit in two cases, extending results of Amerik, Bogomolov, and Rovinsky.