Skip to main content
Log in

Invariant measures and orbit closures for unipotent actions on homogeneous spaces

  • Published:
Geometric & Functional Analysis GAFA Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • [BBo]F. Bien, A. Borel, Sous-groupes épimorphiques de groupes algebriques lineaires I, C.R. Acad. Sci. Paris 315 (1992).

  • [BoP]A. Borel, G. Prasad, Values of isotropic quadratic forms atS-integral points, Composito Math. 83 (1992), 347–372.

    Google Scholar 

  • [Bow]R. Bowen, Weak mixing and unique ergodicity on homogeneous spaces, Israel J. Math. 23 (1976), 267–273.

    Google Scholar 

  • [D1]S.G. Dani, Invariant measures and minimal sets of horospherical flows Invent. Math. 64 (1981), 357–385.

    Google Scholar 

  • [D2]S.G. Dani, On orbits of unipotent flows on homogeneous spaces, II Ergodic Theory and Dynamical Systems 6 (1986), 167–182.

    Google Scholar 

  • [DM1]S.G. Dani, G.A. Margulis, Orbit closures of generic unipotent flows on homogeneous spaces ofS L (3, ℝ), Math. Ann. 286 (1990), 101–128.

    Google Scholar 

  • [DM2]S.G. Dani, G.A. Margulis, Values of quadratic forms at primitive integral points, Invent. Math. 98 (1989), 405–425.

    Google Scholar 

  • [DM3]S.G. Dani, G.A. Margulis, On limit distributions of orbits of unipotent flows and integral solutions of quadratic inequalities, C.R. Acad. Sci. Paris, Ser. I 314 (1992), 698–704.

    Google Scholar 

  • [DSm]S.G. Dani, J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J. 5 (1984), 185–194.

    Google Scholar 

  • [EPe]R. Ellis, W. Perrizo, Unique ergodicity of flows on homogeneous spaces, Israel J. Math. 29 (1978), 276–284.

    Google Scholar 

  • [F1]H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. J. Math. 83 (1961), 573–601.

    Google Scholar 

  • [F2]H. Furstenberg, The unique ergodicity of the horocyle flow, In “Recent Advances in Topological Dynamics”, Lecture Notes in Math. 318, Springer, Berlin (1972), 95–115.

    Google Scholar 

  • [H]G.A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J. 2 (1936), 530–542.

    Google Scholar 

  • [M1]G.A. Margulis, Discrete subgroups and ergodic theory, in “Number Theory, Trace Formula and Discrete Groups”, Symposium in Honor of A. Selberg, Oslo 1987, Academic Press (1989), 377–398.

  • [M2]G.A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, Proc. of ICM Kyoto (1990), 193–215.

  • [MT]G.A. Margulis, G.M. Tomanov, Measure rigidity for algebraic groups over local fields, C.R. Acad. Sci. Paris, t. 315, Serie I (1992), 1221–1226.

    Google Scholar 

  • [Mo]S. Mozes, Epimorphic subgroups and invariant measures, Preprint.

  • [MoS]S. Mozes, N. Shah, On the space of ergodic measures of unipotent flows, preprint.

  • [P]W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771.

    Google Scholar 

  • [R]M.S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag (1972).

  • [Ra1]M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990), 449–482.

    Google Scholar 

  • [Ra2]M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), 229–309.

    Google Scholar 

  • [Ra3]M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. 134 (1991), 545–607.

    Google Scholar 

  • [Ra4]M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), 235–280.

    Google Scholar 

  • [Ra5]M. Ratner, Rigidity of horocycle flows, Ann. of Math. 115 (1982), 597–614.

    Google Scholar 

  • [Ra6]M. Ratner, Raghunathan's Conjectures forSL(2,ℝ), Israel. J. Math. 80 (1992), 1–31.

    Google Scholar 

  • [Ra7]M. Ratner, Raghunathan's conjectures forp-adic Lie groups, International Mathematics Research Notices, No. 5 (1993), 141–146.

    Google Scholar 

  • [S]N. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann. 289 (1991), 315–334.

    Google Scholar 

  • [V]W. Veech, Unique ergodicity of horospherical flows, Amer. J. Math. 99 (1977), 827–859.

    Google Scholar 

  • [W1]D. Witte, Rigidity of some translations on homogeneous spaces, Invent. Math. 81 (1985), 1–27.

    Google Scholar 

  • [W2]D. Witte, Measurable quotients of unipotent translations on homogeneous spaces, To appear in Transactions of AMS.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ratner, M. Invariant measures and orbit closures for unipotent actions on homogeneous spaces. Geometric and Functional Analysis 4, 236–257 (1994). https://doi.org/10.1007/BF01895839

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01895839

Keywords

Navigation