Isolated Periodic Points in Several Nonarchimedean Variables

2015/11/03 by Alon Levy, Levy, Alon
Mathematics · #14T05 #37P20 #37P55 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #math.DS #math.NT #msc:14T05 #msc:37P20 #msc:37P55

paper · pdf · doi:10.48550/arxiv.1511.00793

26 pages

arxiv created 2015/11/03 · arxiv updated 2015/11/04

Abstract

Let φ: ℙⁿ_F → ℙⁿ_F where F is a complete valued field. If x is a fixed point, such that the action of φ on T_x has eigenvalues λ₁, …, λ_n, with λ₁, …, λ_r not contained in the multiplicative group generated by λ_r+1, …, λ_n, then φ 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 |λ_i| ≤ 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 ℚ-point with Zariski-dense forward orbit in two cases, extending results of Amerik, Bogomolov, and Rovinsky.

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