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Directional measure rigidity for time-changed first row unipotent actions

  • Davide Ravotti
  • Daren Wei
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Abstract

We study flows on finite volume homogeneous manifolds arising as directional restrictions of time changes of higher rank abelian unipotent actions. We establish Ratner-type rigidity results in this setting. For first row actions on finite volume quotients of SLn(ℝ), we show that natural quantitative assumptions on the cocycle force the flow to be algebraic and its ergodic invariant measure to be homogeneous. These results extend to higher dimensional subactions and to a general abstract setting. We also show that, on compact quotients of SL3(ℝ), the smooth time change is smoothly conjugate to a linear algebraic action by a leaf preserving diffeomorphism if and only if the cocycle of the directional flow has bounded deviations along one direction. We also obtain precise asymptotics for compact root increments. The principal thresholds in our results are sharp.

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Cite

BibTeXRIS
BibTeX
@misc{ravotti2026directional,
  title = {{Directional measure rigidity for time-changed first row unipotent actions}},
  author = {Davide Ravotti and Daren Wei},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03903},
  eprint = {2610.03903},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03903},
}
RIS
TY  - UNPB
TI  - Directional measure rigidity for time-changed first row unipotent actions
AU  - Davide Ravotti
AU  - Daren Wei
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03903
UR  - https://arxiv.org/abs/2610.03903
AB  - We study flows on finite volume homogeneous manifolds arising as directional restrictions of time changes of higher rank abelian unipotent actions. We establish Ratner-type rigidity results in this setting. For first row actions on finite volume quotients of SL_n(ℝ), we show that natural quantitative assumptions on the cocycle force the flow to be algebraic and its ergodic invariant measure to be homogeneous. These results extend to higher dimensional subactions and to a general abstract setting. We also show that, on compact quotients of SL_3(ℝ), the smooth time change is smoothly conjugate to a linear algebraic action by a leaf preserving diffeomorphism if and only if the cocycle of the directional flow has bounded deviations along one direction. We also obtain precise asymptotics for compact root increments. The principal thresholds in our results are sharp.
ER  -