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Equidistribution of divergent diagonal orbits in positive characteristic

Nguyen-Thi Dang, Frédéric Paulin, Rafael Sayous

TL;DR

The paper extends the theory of divergent diagonal orbits to function fields of positive characteristic, providing a complete algebraic classification of divergence and a function-field analogue of Margulis’s results. It then proves equidistribution for natural families of divergent diagonal orbits using a high-entropy method, carefully controlling mass in the thin part and constructing high-entropy partitions via dynamical neighborhoods. The main achievement is showing that suitably normalized averages of divergent diagonal orbits converge to the homogeneous measure on the lattice space ${ t X}$ (and, by corollary, on ${ t X}_1$), with an explicit constant via zeta-like data. The results connect the arithmetic of lattices over function fields to homogeneous dynamics and Bruhat–Tits tree geometry (notably in the $n=2$ case), and extend prior real-field and number-field results to the positive-characteristic setting with a precise discriminant-type invariant. These findings advance understanding of equidistribution in noncompact homogeneous spaces and pave the way for S-adic or adelic generalizations.

Abstract

Given a local field $\widehat K$ with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group $\operatorname{GL}_n(\widehat K)$ on homogeneous spaces of discrete lattices in ${\widehat K}^{\,n}$. We first give a function field version of results by Margulis and Tomanov-Weiss, characterizing the divergent diagonal orbits. When $n=2$, we relate the divergent diagonal orbits with the divergent orbits of the geodesic flow in the modular quotient of the Bruhat-Tits tree of $\operatorname{PGL}_2(\widehat K)$. Using the (high) entropy method by Einsiedler-Lindentraus et al, we then give a function field version of a result of David-Shapira on the equidistribution of a natural family of these divergent diagonal orbits, with height given by a new notion of discriminant of the orbits.

Equidistribution of divergent diagonal orbits in positive characteristic

TL;DR

The paper extends the theory of divergent diagonal orbits to function fields of positive characteristic, providing a complete algebraic classification of divergence and a function-field analogue of Margulis’s results. It then proves equidistribution for natural families of divergent diagonal orbits using a high-entropy method, carefully controlling mass in the thin part and constructing high-entropy partitions via dynamical neighborhoods. The main achievement is showing that suitably normalized averages of divergent diagonal orbits converge to the homogeneous measure on the lattice space (and, by corollary, on ), with an explicit constant via zeta-like data. The results connect the arithmetic of lattices over function fields to homogeneous dynamics and Bruhat–Tits tree geometry (notably in the case), and extend prior real-field and number-field results to the positive-characteristic setting with a precise discriminant-type invariant. These findings advance understanding of equidistribution in noncompact homogeneous spaces and pave the way for S-adic or adelic generalizations.

Abstract

Given a local field with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group on homogeneous spaces of discrete lattices in . We first give a function field version of results by Margulis and Tomanov-Weiss, characterizing the divergent diagonal orbits. When , we relate the divergent diagonal orbits with the divergent orbits of the geodesic flow in the modular quotient of the Bruhat-Tits tree of . Using the (high) entropy method by Einsiedler-Lindentraus et al, we then give a function field version of a result of David-Shapira on the equidistribution of a natural family of these divergent diagonal orbits, with height given by a new notion of discriminant of the orbits.

Paper Structure

This paper contains 20 sections, 31 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Background material
  3. Function fields over finite fields
  4. Lattices
  5. Homogeneous spaces of lattices
  6. Systoles of lattices
  7. Diagonal subgroups
  8. Homogeneous measures on diagonal orbits
  9. A classification of the divergent diagonal orbits
  10. A description of the divergent diagonal orbits
  11. Compact core and quasicenters of divergent diagonal orbits
  12. The mass behavior of the compact cores of divergent diagonal orbits
  13. Zigzag length and continued fractions
  14. Type and discriminant of the divergent diagonal orbits
  15. The high entropy method for equidistribution problem
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $x\in{\cal X}_1$. The diagonal orbit $A_1x$ is divergent if and only if there exists $g\in A_1\operatorname{GL}_n^1(K)$ such that $x=g\,R_v^{\,n}$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Proposition 4.1
  • Lemma 4.2
  • ...and 22 more