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

TL;DR

This paper classifies horospherical invariant measures for Anosov subgroups in higher rank semisimple groups, extending previous results from rank one and finite-volume spaces to a broader, infinite-volume setting using geometric methods.

Contribution

It provides a comprehensive classification of horospherical invariant measures for a wide class of discrete subgroups in higher rank, generalizing prior work and resolving open problems.

Findings

Classifies measures for Anosov subgroups in higher rank groups

Extends measure classification from rank one to higher ranks

Uses geometric methods independent of flow or ergodic theory

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.