Please note special day, time, and location.   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 Q-bar with a Zariski-dense forward orbit.