Advertisement

Parabolic Perturbations of Unipotent Flows on Compact Quotients of SL\({(3,\mathbb{R})}\)

  • Davide RavottiEmail author
Article
  • 8 Downloads

Abstract

We consider a family of smooth perturbations of unipotent flows on compact quotients of SL\({(3,\mathbb{R})}\) which are not time-changes. More precisely, given a unipotent vector field, we perturb it by adding a non-constant component in a commuting direction. We prove that, if the resulting flow preserves a measure equivalent to Haar, then it is parabolic and mixing. The proof is based on a geometric shearing mechanism together with a non-homogeneous version of Mautner Phenomenon for homogeneous flows. Moreover, we characterize smoothly trivial perturbations and we relate the existence of non-trivial perturbations to the failure of cocycle rigidity of parabolic actions in SL\({(3,\mathbb{R})}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

I am grateful to Giovanni Forni for many enlightening suggestions. I would also like to thank Danijela Damjanovic, Livio Flaminio, Vinay Kumaraswamy, and my supervisor Corinna Ulcigrai for several helpful conversations. I thank the referees for their valuable comments and suggestions on a previous version of the paper. The research leading to these results has received funding from the European Research Council under the European Union Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 335989.

References

  1. 1.
    Avila, A., Forni, G., Ulcigrai, C.: Mixing for time-changes of heisenberg nilflows. J. Diff. Geom. (89):369–410 (2011)Google Scholar
  2. 2.
    Brezin J., Moore C.C.: Flows on homogeneous spaces: a new look. Am. J. Math. 103(3), 571–613 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Damjanovic D., Katok A.: Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic \({\mathbb{R}^k}\) actions. Discrete Contin. Dyn. Syst. 13, 985–1005 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dani S.G.: Invariant measures and minimal sets of horospherical flows. Invent. Math. 64, 357–385 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dani S.G., Margulis G.A.: Orbit closures of generic unipotent flows on homogeneous spaces of SL\({(3,\mathbb{R})}\). Math. Ann. 286, 101–128 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Flaminio L., Forni G.: Invariant distributions and time averages for horocycle flows. Duke Math. J. 119(3), 465–526 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Flaminio L., Forni G., Rodriguez Hertz F.: Invariant distributions for homogeneous flows and affine transformations. J. Mod. Dyn. 10, 33–79 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Forni G., Ulcigrai C.: Time-changes of horocycle flows. J. Mod. Dyn. 6(2), 251–273 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Furstenberg, H.: The unique ergodicity of the horocycle flow. In: Topological Dynamics Lecture Notes in Math., vol.318, pp.95–115 (1972)Google Scholar
  10. 10.
    Gallot S., Hulin D., Lafontaine J.: Riemannian Geometry. Springer, Berlin, Heidelberg (2004)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kanigowski, A., Vinhage, K., Wei, D.: Kakutani equivalence of unipotent flows. arXiv:1805.01501 (2018)
  12. 12.
    Katok A., Spatzier R.J.: First cohomology of anosov actions of higher rank abelian groups and applications to rigidity. Inst. Hautes Etudes Sci. Publ. Math. 79, 131–156 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marcus B.: Ergodic properties of horocycle flows for surfaces of negative curvature. Ann. Math. 105(1), 81–105 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Marcus B.: The horocycle flow is mixing of all degrees. Invent. Math. 46(3), 201–209 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Margulis G.A., Tomanov G.M.: Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math. 116, 347–392 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mieczkowski D.: The first cohomology of parabolic actions for some higher-rank abelian group and representation theory. J. Mod. Dyn. 1(1), 61–91 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mozes S.: Mixing of all orders of lie groups actions. Invent. Math. 107, 235–241 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pugh C., Shub M.: Ergodic elements of ergodic actions. Compos. Math. 23(1), 115–122 (1971)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ramirez F.A.: Cocycles over higher-rank abelian actions on quotients of semi-simple lie groups. J. Mod. Dyn. 3(3), 335–357 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ratner M.: Rigidity of time changes for horocycle flows. Acta Math. 156, 1–32 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ratner M.: On measure rigidity of unipotent subgroups of semisimple groups. Acta Math. 165, 229–309 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ratner M.: Strict measure rigidity for unipotent subgroups of solvable groups. Invent. Math. 101, 449–482 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ratner M.: On raghunathan’s measure conjecture. Ann. Math. 134(3), 545–607 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ratner M.: Raghunathan’s conjectures for cartesian products of real and p-adic groups. Duke Math. J. 77, 275–382 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ravotti, D.: Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows. Ergod. Theory Dyn. Syst.  https://doi.org/10.1017/etds.2018.19
  26. 26.
    Simonelli L.D.: Absolutely continuous spectrum for parabolic flows/maps. Discrete Contin. Dyn. Syst. 38(1), 263–292 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tiedrade Aldecoa R.: Spectral analysis of time-changes of horocycle flows. J. Mod. Dyn. 6(2), 275–285 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wang Z.J.: Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups. Geom. Funct. Anal. 25, 1956–2020 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsUniversity of Bristol, University WalkBristolUK

Personalised recommendations