Conformal measure rigidity and ergodicity of horospherical foliations

Abstract

In this paper, we prove two main theorems: conformal measure rigidity and ergodicity of horospherical foliations, especially in higher rank. Both theorems are new even for relatively Anosov groups. First, we establish a higher rank extension of rigidity theorems of Sullivan, Tukia, Yue, and Kim-Oh for representations of rank one discrete subgroups of divergence type, in terms of the push-forwards of conformal measures via boundary maps. We consider a certain class of higher rank discrete subgroups, which we call hypertransverse subgroups. It includes all rank one discrete subgroups, Anosov subgroups, relatively Anosov subgroups, and their subgroups. Our proof of the rigidity theorem generalizes the idea of Kim-Oh to self-joinings of higher rank hypertransverse subgroups, overcoming the lack of $\mathrm{CAT}(-1)$ geometry on symmetric spaces. In contrast to the work of Sullivan, Tukia, and Yue, our argument is closely related to the study of horospherical foliations. We also show the ergodicity of horospherical foliations with respect to Burger-Roblin measures. This generalizes the classical work of Hedlund, Burger, and Roblin in rank one and of Lee-Oh for Borel Ansov subgroups in higher rank. Moreover, we describe the ergodic decomposition of Burger-Roblin measures and Bowen-Margulis-Sullivan measures when a given parabolic subgroup is minimal.