Time change rigidity for unipotent flows

TL;DR

This paper establishes a dichotomy for unipotent flows on quotients of semisimple Lie groups under time change, showing they are either loosely Kronecker or exhibit strong rigidity with unique isomorphism properties.

Contribution

It proves a new dichotomy theorem characterizing the behavior of time-changed unipotent flows, revealing a rigid classification and isomorphism conditions.

Findings

Flows are either loosely Kronecker or rigid.

Rigid flows are uniquely determined up to isomorphism and trivial time change.

The classification applies to all one-parameter unipotent flows on such quotients.

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.