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

Abstract

Let $u_X^t$ be a unipotent flow on $X=SO(n,1)/\Gamma$, $u_Y^t$ 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 and extends a recent work of Dong, Kanigowski and Wei.