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

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.

• The paper provides a comprehensive classification of horosphrical invariant Radon measures for discrete subgroups of any semisimple real algebraic group.
• Key findings show that every NM-invariant ergodic Radon measure on the minimal set is a constant multiple of a Burger–Roblin measure.
• This classification accommodates measure ergodic extensions of Burger–Roblin measures, and resolves the concrete classification invariances as a consequence.

Context and motivation

The paper classifies horospherical invariant Radon measures for discrete subgroups of an arbitrary connected semisimple real algebraic group $\mathsf{G}$ with minimal parabolic $\mathsf{P} = MAN$. For a lattice $\Gamma < \mathsf{G}$, the $NM$-action on $\Gamma\backslash\mathsf{G}$ is uniquely ergodic (Furstenberg for $\mathrm{PSL}(2,\mathbb{R})$, Veech in general), and Dani classified all $NM$-invariant ergodic Radon measures for non-uniform lattices; Ratner's theorem extends this to unipotent flows. For infinite-covolume Zariski dense subgroups, the natural phase space is the unique $\mathsf{P}$-minimal set $E_\Gamma \subset \Gamma\backslash\mathsf{G}$, whose uniqueness in higher rank is due to Benoist. In rank one, Burger proved unique ergodicity of the horospherical action on $E_\Gamma$ for convex cocompact $\mathsf{P} = MAN$0 with limit set of Hausdorff dimension exceeding $\mathsf{P} = MAN$1, and Roblin classified all $\mathsf{P} = MAN$2-invariant ergodic Radon measures on $\mathsf{P} = MAN$3 for geometrically finite $\mathsf{P} = MAN$4: each is either supported on a closed orbit or a multiple of the Burger–Roblin measure. Prior to this work, no analogous classification existed in higher rank even for a single explicit example: Landesberg–Lee–Lindenstrauss–Oh obtained rigidity only for products of at most three rank-one groups and only on directionally recurrent sets $\mathsf{P} = MAN$5, and posed two open problems — whether every ergodic measure is supported on some $\mathsf{P} = MAN$6 when $\mathsf{P} = MAN$7, and Oh's question whether every ergodic measure is a Burger–Roblin measure without any rank restriction.

Main results

The paper resolves both open problems completely. For a Zariski dense Borel Anosov subgroup $\mathsf{P} = MAN$8 (with $\mathsf{P} = MAN$9 arbitrary), every $\Gamma < \mathsf{G}$0-invariant ergodic Radon measure on $\Gamma < \mathsf{G}$1 is a constant multiple of a Burger–Roblin measure of $\Gamma < \mathsf{G}$2. When the $\Gamma < \mathsf{G}$3-action on $\Gamma < \mathsf{G}$4 is minimal (e.g., $\Gamma < \mathsf{G}$5 a product of rank-one groups), the same holds for $\Gamma < \mathsf{G}$6-invariant measures; note that minimality is genuinely needed, since otherwise Burger–Roblin measures fail to be $\Gamma < \mathsf{G}$7-ergodic. Via the Lee–Oh homeomorphism between $\Gamma < \mathsf{G}$8 (the interior of the limit cone) and Patterson–Sullivan measures on the limit set, the set of ergodic measures is homeomorphic to $\Gamma < \mathsf{G}$9 — a continuous family of mutually singular measures, reflecting the corresponding structure of higher-rank Patterson–Sullivan measures. Combined with work of Burger–Landesberg–Lee–Oh, this yields that for $NM$0 every ergodic measure is supported on some directional recurrent set $NM$1, resolving the first open problem.

For relatively Borel Anosov subgroups (higher-rank analogues of geometrically finite groups, e.g., cusped Hitchin representations), the classification gains the expected second alternative: every $NM$2-invariant ergodic Radon measure on $NM$3 is either a multiple of a Burger–Roblin measure or supported on a closed $NM$4-orbit.

The most general framework replaces the Borel parabolic by any standard parabolic $NM$5 associated to $NM$6. The authors introduce $NM$7-horospherical foliations $NM$8, where the $NM$9-action combines the action on the $\Gamma\backslash\mathsf{G}$0-boundary with translation by the partial Iwasawa cocycle $\Gamma\backslash\mathsf{G}$1. For a Zariski dense $\Gamma\backslash\mathsf{G}$2-hypertransverse subgroup — a class containing all (relatively) $\Gamma\backslash\mathsf{G}$3-Anosov subgroups and their subgroups — the main theorem states that the $\Gamma\backslash\mathsf{G}$4-invariant ergodic Radon measures on the recurrence locus $\Gamma\backslash\mathsf{G}$5 are exactly the constant multiples of Burger–Roblin measures $\Gamma\backslash\mathsf{G}$6 attached to divergence-type Patterson–Sullivan measures $\Gamma\backslash\mathsf{G}$7. The proof requires only non-arithmeticity of the Jordan spectrum $\Gamma\backslash\mathsf{G}$8, which holds automatically under Zariski density by Benoist's theorem.

Method

The argument is geometric rather than flow-theoretic: it uses no diagonal flows or ergodic theorems, and does not rely on Besicovitch-type covering arguments (which require the product-of-rank-one hypothesis used by Landesberg–Lee–Lindenstrauss–Oh). This is what permits handling arbitrary semisimple $\Gamma\backslash\mathsf{G}$9 and full supports rather than smaller recurrent subsets.

The core is to show that any $\mathrm{PSL}(2,\mathbb{R})$0-invariant ergodic Radon measure on $\mathrm{PSL}(2,\mathbb{R})$1 is quasi-invariant under translations $\mathrm{PSL}(2,\mathbb{R})$2 on the $\mathrm{PSL}(2,\mathbb{R})$3-factor; a standard lemma of Aaronson–Sarig type then forces the measure to have the Burger–Roblin density form, and the Hopf–Tsuji–Sullivan dichotomy identifies the boundary component as divergence-type. Quasi-invariance under $\mathrm{PSL}(2,\mathbb{R})$4 for loxodromic $\mathrm{PSL}(2,\mathbb{R})$5 is established via a finite-to-one "push" map built from the extension lemma (Yang's coarse closing lemma) applied to translates of axes of $\mathrm{PSL}(2,\mathbb{R})$6, together with shadow estimates.

Two technical obstacles distinguish this from the authors' earlier work on mapping class group actions on measured lamination spaces. First, Teichmüller geodesics enjoy a quantitative squeezing property (derived there from Minsky's contraction theorem); here only Gromov hyperbolicity is available, giving merely the coarse contracting property, which is insufficient for the previous cocycle approximation scheme. Second, cocycles are vector-valued rather than scalar. To overcome both, the paper develops a robust connection between alignments of geodesics in the model Gromov hyperbolic space $\mathrm{PSL}(2,\mathbb{R})$7 and projective geometry via Tits representations: shadows defined through uniform transversality to hyperplanes $\mathrm{PSL}(2,\mathbb{R})$8 control Iwasawa cocycles (Quint's estimates), and a new local Lipschitz estimate for $\mathrm{PSL}(2,\mathbb{R})$9 — with Lipschitz constant independent of the group element — yields a uniform comparison between $NM$0 and $NM$1 whenever both $NM$2 and $NM$3 lie in suitable double shadows (the key local regularity theorem).

A new notion central to the proof is the guided limit set $NM$4: points of the conical limit set accumulated along translates of an axis of a fixed loxodromic element with controlled alignment. The first main step shows that any invariant ergodic Radon measure charges guided limit sets; since these are independent of the choice of loxodromic element up to constants, the measure concentrates on all of them simultaneously. As a byproduct, the authors obtain a strengthening of the Hopf–Tsuji–Sullivan dichotomy: divergence-type Patterson–Sullivan measures are supported not merely on the conical limit set but on each guided limit set, giving them full measure.

Closed-orbit alternatives are handled separately: using proper discontinuity of the $NM$5-action on $NM$6 (Kim–Oh–Wang) and the cocompact stabilizer structure of parabolic limit points, orbits over the parabolic limit set $NM$7 are shown to be closed in $NM$8.

Limitations and open questions

Several qualifications are stated explicitly. The classification on $NM$9 assumes non-arithmeticity of $\mathsf{P}$0; without Zariski density this must be imposed as a hypothesis. The passage from $\mathsf{P}$1-ergodic to $\mathsf{P}$2-ergodic classification requires $\mathsf{P}$3-minimality of $\mathsf{P}$4, which fails for Hitchin representations among others; outside this case the $\mathsf{P}$5-classification is left open beyond the ergodic decomposition of Burger–Roblin measures. Whether every $\mathsf{P}$6-orbit in $\mathsf{P}$7 is closed — automatic in rank one — is asserted not to hold for general $\mathsf{P}$8 (illustrated by an explicit $\mathsf{P}$9 example), so the classification covers $E_\Gamma \subset \Gamma\backslash\mathsf{G}$0 rather than all of $E_\Gamma \subset \Gamma\backslash\mathsf{G}$1. Finally, it remains unknown whether every transverse subgroup is hypertransverse, or whether the Hilbert-metric realization of Canary–Zhang–Zimmer can always be chosen Gromov hyperbolic on the convex hull of the limit set; the results here apply precisely within the hypertransverse class.

Conclusion

This paper completes the program of classifying horospherical invariant measures in higher rank: for Zariski dense (relatively) Borel Anosov subgroups of arbitrary semisimple real algebraic groups, the ergodic invariant Radon measures on the minimal set are exactly the Burger–Roblin measures, plus closed-orbit measures in the relatively Anosov case. The geometric method — guided limit sets combined with Tits-representation control of vector-valued Iwasawa cocycles — bypasses flows, ergodic theorems, and covering arguments, and applies uniformly across rank and across all parabolics. It simultaneously strengthens the Hopf–Tsuji–Sullivan dichotomy by showing divergence-type Patterson–Sullivan measures charge guided limit sets.