Joint IAS/Princeton University Number Theory Seminar

We study dynamical systems in several variables over a complete valued field. If $x$ is a fixed point, we show that in many cases there exist fixed analytic subvarieties through $x$. These cases include all cases in which $x$ is attracting in some directions and repelling in others, which lets us separate attracting, repelling, and indifferent directions, generalizing results from complex hyperbolic dynamics. We use this for two purposes: first, we show that over the $p$-adics, if $x$ has no repelling directions, then it is isolated, that is there exists a $p$-adic neighborhood of $x$ containing no other periodic points; and second, we prove some cases of a conjecture of Shouwu Zhang that every reasonable dynamical system defined over a number field has a point defined over $\bar{\mathbb Q}$ with a Zariski-dense forward orbit.