  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.
