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Classification of horospherical invariant measures in higher rank: The Full Story

Inhyeok Choi, Dongryul M. Kim

TL;DR

This work classifies horospherical invariant Radon measures for Zariski dense discrete subgroups in arbitrary semisimple real groups, extending higher-rank Burger–Roblin theory beyond rank-one settings. It introduces theta-horospherical foliations and Burger–Roblin measures built from divergence-type Patterson–Sullivan data, then proves that ergodic NM-invariant measures on guided recurrence sets are precisely these BR measures for non-arithmetic spectra, with a parallel closed-orbit dichotomy for relatively Anosov subgroups. The proof is geometric and flow-free, relying on guided limit sets, alignments of geodesics, and cocycle–projection approximations via Tits representations to establish quasi-invariance and rigidity. The results resolve key open problems and provide a robust Hopf–Tsuji–Sullivan dichotomy strengthening in the higher-rank setting, with broad implications for Anosov and relatively Anosov dynamics on homogeneous spaces. Overall, the paper delivers a full higher-rank horospherical measure classification, unifying rank-one intuitions with higher-rank geometric structures and boundary dynamics.

Abstract

In this paper, we classify horospherical invariant Radon measures for Anosov subgroups of arbitrary semisimple real algebraic groups. This generalizes the works of Burger and Roblin in rank one to higher ranks. At the same time, this extends the works of Furstenberg, Veech, and Dani, and a special case of Ratner's theorem for finite-volume homogeneous spaces to infinite-volume Anosov homogeneous spaces. Especially, this resolves the open problems proposed by Landesberg--Lee--Lindenstrauss--Oh and by Oh. Our measure classification is in fact for a more general class of discrete subgroups, including relatively Anosov subgroups with respect to any parabolic subgroups, not necessarily minimal. Our method is rather geometric, not relying on continuous flows or ergodic theorems.

Classification of horospherical invariant measures in higher rank: The Full Story

TL;DR

This work classifies horospherical invariant Radon measures for Zariski dense discrete subgroups in arbitrary semisimple real groups, extending higher-rank Burger–Roblin theory beyond rank-one settings. It introduces theta-horospherical foliations and Burger–Roblin measures built from divergence-type Patterson–Sullivan data, then proves that ergodic NM-invariant measures on guided recurrence sets are precisely these BR measures for non-arithmetic spectra, with a parallel closed-orbit dichotomy for relatively Anosov subgroups. The proof is geometric and flow-free, relying on guided limit sets, alignments of geodesics, and cocycle–projection approximations via Tits representations to establish quasi-invariance and rigidity. The results resolve key open problems and provide a robust Hopf–Tsuji–Sullivan dichotomy strengthening in the higher-rank setting, with broad implications for Anosov and relatively Anosov dynamics on homogeneous spaces. Overall, the paper delivers a full higher-rank horospherical measure classification, unifying rank-one intuitions with higher-rank geometric structures and boundary dynamics.

Abstract

In this paper, we classify horospherical invariant Radon measures for Anosov subgroups of arbitrary semisimple real algebraic groups. This generalizes the works of Burger and Roblin in rank one to higher ranks. At the same time, this extends the works of Furstenberg, Veech, and Dani, and a special case of Ratner's theorem for finite-volume homogeneous spaces to infinite-volume Anosov homogeneous spaces. Especially, this resolves the open problems proposed by Landesberg--Lee--Lindenstrauss--Oh and by Oh. Our measure classification is in fact for a more general class of discrete subgroups, including relatively Anosov subgroups with respect to any parabolic subgroups, not necessarily minimal. Our method is rather geometric, not relying on continuous flows or ergodic theorems.
Paper Structure (39 sections, 43 theorems, 223 equations, 2 figures)

This paper contains 39 sections, 43 theorems, 223 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Work of Landesberg--Lee--Lindenstrauss--Oh and Problems
  3. Main results for Borel Anosov subgroups
  4. Relatively Borel Anosov subgroups
  5. Measure classification in the full generality
  6. Horospherical foliations
  7. Burger--Roblin measures
  8. Hypertransverse subgroups
  9. Measure classification
  10. On the proof
  11. Organization
  12. Notation
  13. Geometry of Gromov hyperbolic spaces
  14. Contracting
  15. Alignment
  16. ...and 24 more sections

Key Result

Theorem 1.1

Suppose that $\mathop{\mathrm{\mathsf{G}}}\nolimits$ is a product of rank one groups. Let $\Gamma < \mathop{\mathrm{\mathsf{G}}}\nolimits$ be a Zariski dense Borel Anosov subgroup and $v \in \operatorname{int} \mathfrak{a}^+$. Let $\mathcal{L}_{\Gamma} \subset \mathfrak{a}^+$ denote the limit coneTh

Figures (2)

  • Figure 1: A squeezing geodesic $\gamma$ (left) and a contracting geodesic $\gamma$ (right)
  • Figure 2: Alignment of geodesics and points.

Theorems & Definitions (73)

  • Theorem 1.1: LLLO_Horospherical
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Example 1.8
  • Corollary 1.9
  • Corollary 1.10
  • Corollary 1.11
  • Theorem 1.12
  • ...and 63 more