Time change for unipotent flows and rigidity

Abstract

We prove a dichotomy regarding the behavior of one-parameter unipotent flows on quotients of semisimple lie groups under time change. We show that if $u_t^{(1)}$ acting on $G_1/{\Gamma }_1$ is such a flow it satisfies exactly one of the following: (1) The flow is loosely Kronecker, and hence isomorphic after an appropriate time change to any other loosely Kronecker system. (2) The flow exhibits the following rigid behavior: if the one-parameter unipotent flow $u_t^{(1)}$ on $G_1/{\Gamma }_1$ is isomorphic after time change to another such flow $u_t^{(2)}$ on $G_2/{\Gamma }_2$, then $G_1/{\Gamma }_1$ is isomorphic to $G_2/{\Gamma }_2$ with the isomorphism taking $u_t^{(1)}$ to $u_t^{(2)}$ and moreover the time change is cohomologous to a trivial one. The full details will appear in [LW23].