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arXiv · 2502.08081

Time change rigidity for unipotent flows

Elon Lindenstrauss·Daren Wei
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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 $\mathbf{G}_{1}/Γ_1$ 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 $\mathbf{G}_1/Γ_1$ is measurably isomorphic after time change to another such flow $u^{(2)} _ t$ on $\mathbf{G}_2/Γ_ 2$, then $\mathbf{G}_1/Γ_1 $ is isomorphic to $\mathbf{G}_2/ Γ_2$ 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.

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Elon Lindenstrauss, Daren Wei. 2025-02-12. Time change rigidity for unipotent flows. https://arxiv.org/abs/2502.08081

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