Mixing of frame flow for rank one locally symmetric spaces and measure classification

Let G be a connected simple linear Lie group of rank one, and let I" < G be a discrete Zariski dense subgroup admitting a finite Bowen-Margulis-Sullivan measure m (BMS). We show that the right translation action of the one-dimensional diagonalizable subgroup is mixing on (I"\G, m (BMS)). Together with the work of Roblin, this proves ergodicity of the Burger-Roblin measure under the horospherical group N, establishes a classification theorem for N invariant Radon measures on I"\G, and provides precise asymptotics for the Haar measure matrix coefficients.