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

Abstract

Let G be a connected simple linear Lie group of rank one, and let Γ < 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 (Γ\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 Γ\G, and provides precise asymptotics for the Haar measure matrix coefficients.

A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds

“…This result has been proved by Winter [12] on the frame bundle Γ\G as a (nontrivial) corollary of Roblin's work on…”

“…Taking the product locally with the Haar measure of N gives a N -invariant measure. Therefore, additional work (done by Winter in [12]) is needed to get a unique ergodicity result on the frame bundle Γ\G. Thus, in the setting of hyperbolic manifolds, our result (the same as Winter's result) is stronger than Roblin's theorem.…”

“…where dλ zN = dn and dµ [12] proved that its lift m F BM to Γ\G is also mixing, as soon as the group Γ is Zariski dense. As a corollary, he gets the following equidistribution result of averages pushed by the flow.…”

Statistical Regularity of Apollonian Gaskets

“…Although we didn't use the results from [21] directly, that paper together with the data produced from the IGL project gave us the main inspiration of this paper. The technique employed in this paper is mainly from [34], [26], [23]. Thanks are also due to Prof. Curt McMullen for his enlightening comments and corrections.…”

The case of equality in the dichotomy of Mohammadi–Oh

“…Proof. For 1 and 2 see [17]. For 3 compare the definitions of BMS and BR measure, taking into account the fact that N parametrizes the unstable horospheres in the frame bundle.…”