Time change for unipotent flows and rigidity

Elon Lindenstrauss; Daren Wei

doi:10.5802/mrr.15

Elon Lindenstrauss ¹; Daren Wei ²

¹ Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel
² Department of Mathematics, National University of Singapore, 119076, Singapore

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

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 $G_{1} / Γ_{1}$ 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 $u_{t}^{(1)}$ on $G_{1} / Γ_{1}$ is isomorphic after time change to another such flow $u_{t}^{(2)}$ on $G_{2} / Γ_{2}$ , then $G_{1} / Γ_{1}$ is isomorphic to $G_{2} / Γ_{2}$ with the isomorphism taking $u_{t}^{(1)}$ to $u_{t}^{(2)}$ and moreover the time change is cohomologous to a trivial one.

The full details will appear in a later publication.

Received: 2023-01-06
Revised: 2023-04-20
Published online: 2023-09-29

DOI: 10.5802/mrr.15

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

Author's affiliations:

Elon Lindenstrauss ¹; Daren Wei ²

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Elon Lindenstrauss; Daren Wei. Time change for unipotent flows and rigidity. Mathematics Research Reports, Volume 4 (2023), pp. 11-22. doi : 10.5802/mrr.15. https://mrr.centre-mersenne.org/articles/10.5802/mrr.15/

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