Several nonarchimedean variables, isolated periodic points, and Zhang's conjecture

Joint IAS/Princeton University Number Theory Seminar
Topic:Several nonarchimedean variables, isolated periodic points, and Zhang's conjecture
Speaker:Alon Levy
Affiliation:KTH Royal Institute of Technology, Stockholm
Date:Wednesday, November 25
Time/Room:4:30pm - 5:30pm/Fine 224, Princeton University

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.