Mixing, counting, and equidistribution for higher-rank homogeneous spaces of infinite volume
Ergodic Theory and Dynamical Systems Seminar
28th April 2022, 2:00 pm – 3:00 pm
Fry Building, 2.04
Mixing of diagonal flows is a key ingredient in many important results and applications of homogeneous dynamics. In joint work with Minju Lee and Hee Oh, we generalize an argument of Roblin that converts mixing of diagonal flows for Bowen-Margulis-Sullivan (BMS) measures to mixing for the Haar measure on quotients Γ\G, where G is a higher-rank semisimple Lie group and Γ is an infinite-covolume discrete subgroup of G. As a consequence of this, we are able to follow a strategy due to Eskin-McMullen and obtain equidistribution and counting results for groups Γ for which the diagonal flows on Γ\G are BMS-mixing. I will try to present some groups Γ where BMS-mixing is known, and highlight some similarities and differences with the classically studied settings, namely a) the finite-volume case, i.e. when Γ is a lattice, and b) when G has (real) rank one and Γ is a geometrically finite subgroup.