Isolated Periodic Points in Several Nonarchimedean Variables

Abstract

Let $\phi :{\mathbb{P}}_F^n\to {\mathbb{P}}_F^n$ where $F$ is a complete valued field. If $x$ is a fixed point, such that the action of $\phi$ on $T_x$ has eigenvalues ${\lambda }_1,\dots ,{\lambda }_n$, with ${\lambda }_1,\dots ,{\lambda }_r$ not contained in the multiplicative group generated by ${\lambda }_{r+1},\dots ,{\lambda }_n$, then $\phi$ 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 $|{\lambda }_i|\le 1$, 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.

26 pages

References in corpus (1)

• Generic parabolic points are isolated in positive characteristic