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Orbits of automorphism groups of affine surfaces over $p$-adic fields

Serge Cantat, Seung Uk Jang

Abstract

We study orbit closures and stationary measures for groups of automorphisms of $p$-adic affine surfaces.

Orbits of automorphism groups of affine surfaces over $p$-adic fields

Abstract

We study orbit closures and stationary measures for groups of automorphisms of -adic affine surfaces.

Paper Structure

This paper contains 44 sections, 23 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Affine varieties over local fields
  3. Stationary measures
  4. Orbit closures
  5. From automorphisms to flows
  6. Local fields of characteristic $0$
  7. Analytic diffeomorphisms of the polydisk
  8. Reduction and the Bell-Poonen theorem
  9. Automorphisms
  10. Reduction
  11. Good polydisks (see CantatXie2018)
  12. The distribution $L_\Gamma(x)$ and its dimension $s_\Gamma(x)$
  13. A general finiteness result
  14. Automorphisms of affine surfaces
  15. Elliptic, parabolic, loxodromic
  16. ...and 29 more sections

Key Result

Theorem 2.1

Let $f$ be an analytic endomorphism of the polydisk ${\mathcal{U}}={{\mathfrak{o}}_{\mathbf{K}}}^m$ with $f\equiv {\rm id} \;({\mathrm{mod}}\; p^c)$ for some real number $c>1/(p-1)$. Then, $f$ is an analytic diffeomorphism of ${\mathcal{U}}$ and there exists a unique analytic flow $\Phi\colon {{\ma

Figures (1)

  • Figure 1: This picture represents, for a given $p$, the maximal length of an orbit of $g$ modulo $p$ divided by $p\log(p)$. Blue points correspond to $g_1({{\mathbf{x}}},{{\mathbf{y}}})=({{\mathbf{y}}}+{{\mathbf{x}}}^2+5,-{{\mathbf{x}}})$, orange points to $g_2({{\mathbf{x}}},{{\mathbf{y}}})=(-{{\mathbf{y}}}, {{\mathbf{x}}}+{{\mathbf{y}}}^3+2)$, and green points to $g_3=g_2\circ h_0\circ g_1$ where $h_0({{\mathbf{x}}},{{\mathbf{y}}}) = (2{{\mathbf{x}}}+{{\mathbf{y}}},{{\mathbf{x}}}+{{\mathbf{y}}})$ is as above.

Theorems & Definitions (58)

  • Theorem 2.1: Bell-Poonen
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 48 more