Time change rigidity for unipotent flows

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 $u^{(1)}_t$ acting on $\mathbf{G}_{1}/\Gamma_1$ is such a flow it satisfies exactly one of the following: (1) The flow is loosely Kronecker, and hence measurably 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 $u^{(1)} _ t$ on $\mathbf{G}_1/\Gamma_1$ is measurably isomorphic after time change to another such flow $u^{(2)} _ t$ on $\mathbf{G}_2/\Gamma _ 2$, then $\mathbf{G}_1/\Gamma_1 $ is isomorphic to $\mathbf{G}_2/ \Gamma_2$ with the isomorphism taking $u^{(1)}_t$ to $u^{(2)}_t$ and moreover the time change is cohomologous to a trivial one up to a renormalization.