Time change for unipotent flows and rigidity

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 a strong rigidity property.

Contribution

It proves a new dichotomy result characterizing the behavior of unipotent flows under time change, revealing a rigidity phenomenon for their isomorphisms.

Findings

Unipotent flows are either loosely Kronecker or rigid under time change.

Rigidity implies isomorphism of the underlying spaces and triviality of the time change.

The results are detailed in a forthcoming publication [LW23].

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 $G_{1}/\Gamma_1$ 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 $u^{(1)} _ t$ on $G_1/\Gamma_1$ is isomorphic after time change to another such flow $u^{(2)} _ t$ on $G_2/\Gamma _ 2$, then $G_1/\Gamma_1 $ is isomorphic to $G_2/ \Gamma_2$ with the isomorphism taking $u^{(1)} _ t$ to $u^{(2)} _ t$ and moreover the time change is cohomologous to a trivial one. The full details will appear in [LW23].