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Let $\varphi: \mathbb{P}^{n}_{F} \to \mathbb{P}^{n}_{F}$ where $F$ is acomplete 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 groupgenerated by $\lambda_{r+1}, \ldots, \lambda_{n}$, then $\varphi$ has acodimension-$r$ fixed formal subvariety. Under mild assumptions, thissubvariety is analytic. We use this to prove two results. First, we generalizeresults 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 analyticneighborhood of $x$ without any other periodic points. And second, we proveZhang's conjecture that there exists a $\overline{\mathbb{Q}}$-point withZariski-dense forward orbit in two cases, extending results of Amerik,Bogomolov, and Rovinsky.

Isolated Periodic Points in Several Nonarchimedean Variables