Time change for unipotent flows and rigidity

Lindenstrauss, Elon; Wei, Daren

Mathematics Research Reports, Tome 4 (2023), pp. 11-22

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 $u_t^{\left(1\right)}$ 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 $u_t^{\left(1\right)}$ on $G_1/{\Gamma }_1$ is isomorphic after time change to another such flow $u_t^{\left(2\right)}$ on $G_2/{\Gamma }_2$, then $G_1/{\Gamma }_1$ is isomorphic to $G_2/{\Gamma }_2$ with the isomorphism taking $u_t^{\left(1\right)}$ to $u_t^{\left(2\right)}$ and moreover the time change is cohomologous to a trivial one.

The full details will appear in a later publication.

Classification : 37A17, 37A20, 37A35, 37C10
Keywords: time change, rigidity, unipotent flows, Kakutani equivalence

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