Rigidity of joinings for time changes of unipotent flows on quotients of Lorentz groups

Abstract

AbstractLet $u_{X}^{t}$ be a unipotent flow on $X=\mathrm {SO}(n,1)/\Gamma $ , $u_{Y}^{t}$ be a unipotent flow on $Y=G/\Gamma ^{\prime }$ . Let $\tilde {u}_{X}^{t}$ , $\tilde {u}_{Y}^{t}$ be time changes of $u_{X}^{t}$ , $u_{Y}^{t}$ , respectively. We show the disjointness (in the sense of Furstenberg) between $u_{X}^{t}$ and $\tilde {u}_{Y}^{t}$ (or $\tilde {u}_{X}^{t}$ and $u_{Y}^{t}$ ) in certain situations. Our method refines the works of Ratner’s shearing argument. The method also extends a recent work of Dong, Kanigowski, and Wei [Rigidity of joinings for some measure preserving systems. Ergod. Th. & Dynam. Sys.42 (2022), 665–690].