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

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.

💡 Research Summary

The paper “Classification of Horospherical Invariant Measures in Higher Rank: The Full Story” presents a complete classification of Radon measures on the homogeneous space Γ\G that are invariant under horospherical subgroups, for a very broad class of discrete subgroups Γ of a connected semisimple real algebraic group G. The authors treat both Borel Anosov subgroups (the higher‑rank analogue of convex‑cocompact groups) and relatively Borel Anosov subgroups (the higher‑rank analogue of geometrically finite groups), without any restriction on the rank of G. Their results resolve two open problems posed by Landesberg–Lee–Lindenstrauss–Oh and by Oh, which asked whether every N‑ or NM‑invariant ergodic Radon measure must be a Burger–Roblin measure (or a trivial combination thereof) in the infinite‑volume setting.

Context and Background.
For a connected semisimple real algebraic group G, let P = MAN be a minimal parabolic subgroup with Langlands decomposition, N its unipotent radical, and M the centralizer of the maximal split torus A. The right multiplication of NM on Γ\G is called the (maximal) horospherical action. When Γ is a uniform lattice, the NM‑action is uniquely ergodic with Haar measure as the unique invariant probability. For non‑uniform lattices, Dani classified all NM‑invariant ergodic Radon measures, and Ratner’s theorem extended this to all unipotent flows. In the infinite‑volume case, Burger proved unique ergodicity for convex‑cocompact subgroups of PSL(2,R) under a Hausdorff‑dimension condition, and Roblin later gave a full classification for geometrically finite groups in rank one. Subsequent work by Edwards–Lee–Oh, Lee–Oh, and others introduced higher‑rank Burger–Roblin measures for Borel Anosov subgroups, establishing ergodicity for certain cases. However, a complete measure classification for higher‑rank Anosov subgroups remained open.

Main Contributions.

1. Theorem 1.5 (NM‑invariant case). For any Zariski‑dense Borel Anosov subgroup Γ < G, every NM‑invariant ergodic Radon measure on the P‑minimal set E_Γ is a constant multiple of a Burger–Roblin measure μ_{BR}^ν associated to a Patterson–Sullivan measure ν on the limit set Λ(Γ).
2. Theorem 1.6 (N‑invariant case). If, in addition, the identity component P⁰ acts minimally on E_Γ (which holds for products of rank‑one groups), the same classification holds for N‑invariant ergodic measures.
3. Theorem 1.12 (Relative Anosov case). For a Zariski‑dense relatively Borel Anosov subgroup, any NM‑invariant ergodic Radon measure is either a Burger–Roblin measure or is supported on a closed NM‑orbit inside E_Γ. The analogous statement holds for N‑invariant measures when P⁰‑minimality is satisfied.
4. Corollaries 1.9–1.11 translate the classification into concrete parametrizations: the space of NM‑ergodic measures is homeomorphic to the interior of the limit cone L_Γ ⊂ a⁺, which has dimension equal to rank G. In rank ≤ 3, every NM‑ergodic measure is supported on a directionally recurrent set R_{Γ,v}, answering the problem of Landesberg–Lee–Lindenstrauss–Oh. For arbitrary rank, the results answer Oh’s question affirmatively.

Methodology.
The authors avoid traditional tools such as continuous flows, mixing arguments, or Ratner’s measure‑classification machinery. Instead, they develop a purely geometric approach based on:

• Gromov‑hyperbolic geometry and guided limit sets to describe the dynamics of Γ on the Furstenberg boundary F = G/P.
• Iwasawa decomposition and the Cartan projection κ: G → a⁺ to parametrize horospherical leaves and to define the growth indicator (limit cone) L_Γ.
• Higher‑rank Patterson–Sullivan theory (ψ‑dimensional measures) to construct invariant conformal densities on F, following Quint’s framework.
• Ergodic decomposition for the N‑action, combined with the known NM‑ergodicity of Burger–Roblin measures (Lee–Oh, Kim), to isolate the possible ergodic components.
• Homeomorphism between int L_Γ and the space of ψ‑Patterson–Sullivan measures (Lee–Oh), which yields a concrete parametrization of all invariant measures.

Resolution of Open Problems.
Landesberg–Lee–Lindenstrauss–Oh’s Problem 1.2 asked whether any N‑invariant ergodic measure on E_Γ (for rank ≤ 3) must be a Burger–Roblin measure supported on a directionally recurrent set. Corollary 1.11 shows that indeed every NM‑ (and, when P⁰‑minimal, N‑) ergodic measure is supported on R_{Γ,v} for some v ∈ int L_Γ, thus fully solving the problem. Oh’s Problem 1.3 asked the same question without any rank restriction; Theorem 1.5 and its corollaries give an affirmative answer for all ranks.

Implications and Future Directions.
The paper establishes a Ratner‑type classification in the infinite‑volume, higher‑rank setting, opening avenues for:

• Studying statistical properties (mixing rates, equidistribution) of horospherical flows on infinite‑volume spaces.
• Extending the geometric method to other unipotent actions beyond NM (e.g., higher‑step unipotent subgroups).
• Investigating non‑Radon or non‑σ‑finite invariant measures, and connections with thermodynamic formalism.
• Applying the parametrization by the limit cone to problems in counting, orbit closures, and representation theory of Anosov subgroups.

In summary, the authors provide a comprehensive, geometric classification of horospherical invariant measures for a wide class of discrete subgroups in arbitrary semisimple groups, thereby completing a program that has been open for several years and bridging the gap between rank‑one dynamics and the rich structure of higher‑rank Anosov representations.