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 $u_t^{(1)}$ acting on ${\mathbf{G}}_1/\hat{I}{“}_1$ 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 $u_t^{(1)}$ on ${\mathbf{G}}_1/\hat{I}{“}_1$ is measurably isomorphic after time change to another such flow $u_t^{(2)}$ on ${\mathbf{G}}_2/\hat{I}{“}_2$, then ${\mathbf{G}}_1/\hat{I}{“}_1$ is isomorphic to ${\mathbf{G}}_2/\hat{I}{“}_2$ with the isomorphism taking $u_t^{(1)}$ to $u_t^{(2)}$ and moreover the time change is cohomologous to a trivial one up to a renormalization.

115 pages