Advertisement

Inventiones mathematicae

, Volume 116, Issue 1, pp 347–392 | Cite as

Invariant measures for actions of unipotent groups over local fields on homogeneous spaces

  • G. A. Margulis
  • G. M. Tomanov
Article

Keywords

Invariant Measure Local Field Homogeneous Space Unipotent Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-Z] Bernstein, I.N., Zelevinski, A.V.: Representation of the group GL(n, F) whereF is a non-archimedean local field. Russ. Math. Surv.31.3, 1–68 (1976)Google Scholar
  2. [Bo1] Borel, A.: Linear Algebraic Groups, second enlarged edition. Berlin Heidelberg New York: Springer 1991Google Scholar
  3. [Bo2] Borel, A.: Density properties of certain subgroups of semisimple groups. Ann. Math.72, 179–188 (1960)Google Scholar
  4. [Bo-Pra] Borel, A., Prasad, G.: Values of isotropic quadratic gorms atS-integral points. Compos. Math.83, 347–372 (1992)Google Scholar
  5. [Bo-Spr] Borel, A., Springer, T.A.: Rationality properties of linear algebraic groups. Tôhoku Math. J.20, 443–497 (1968)Google Scholar
  6. [D1] Dani, S.G.: Invariant measures of horospherical flows on noncompact homogeneous spaces. Invent. Math.47, 101–138 (1978)Google Scholar
  7. [D2] Dani, S.G.: On invariant measures, minimal sets, and a lemma of Margulis. Invent. Math.51, 239–260 (1979)Google Scholar
  8. [D3] Dani, S.G.: Invariant measures and minimal sets of horospherical flows. Invent. Math.64, 357–385 (1981)Google Scholar
  9. [D4] Dani, S.G.: On orbits of unipotent flows on homogeneous spaces. Ergod. Th. Dyn. Syst. 4, 25–34 (1984)Google Scholar
  10. [D5] Dani, S.G.: On orbits of unipotent flows on homogeneous spaces — II. Ergod. Th. Dyn. Syst. 6, 167–182 (1986)Google Scholar
  11. [D-Mar1] Dani, S.G., Margulis, G.A.: Values of quadratic forms at primitive integral points. Invent. Math.98, 405–424 (1989)Google Scholar
  12. [D-Mar2] Dani, S.G., Margulis, G.A.: Orbit closures of generic unipotent flows on homogeneous spaces of SL(3,R). Math. Ann.286, 101–128 (1990)Google Scholar
  13. [D-Mar3] Dani, S.G., Margulis, G.A.: Values of quadratic forms at integral points; an elementary approach. Enseign. Math.36, 143–174 (1990)Google Scholar
  14. [D-Mar4] Dani, S.G., Margulis, G.A.: Asymptotic behavior of trajectories of unipotent flows on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci.101, 1–17 (1991)Google Scholar
  15. [D-Mar5] Dani, S.G., Margulis, G.A.: On the limit distributions of orbits of unipotent flows and integral solutions of quadratic inequalities. C.R. Acad. Sci., Paris, Ser. I314, 698–704 (1992)Google Scholar
  16. [D-Mar6] Dani, S.G., Margulis, G.A.: Limit distributions of orbits of unipotent flows and values of quadratic forms (to appear)Google Scholar
  17. [D-Smi] Dani, S.G., Mmillie, J.: Uniform distribution of horocycle flows for Fuchsian groups. Duke Math. J.51, 185–194 (1984)Google Scholar
  18. [Led-Str] Ledrapier, F., Strelcyn, J.-M.: A proof of the estimation from below in Pesin's entropy formula. Ergodic Theory Dyn. Syst.2, 203–219 (1982)Google Scholar
  19. [Led-Y] Ledrapier, F., Young, L.-S.: The metric entropy of diffeomorphisms. I. Ann. Math.122, 503–539 (1985)Google Scholar
  20. [Ma] Mañé, R.: A proof of Pesin's formula. Ergodic Theory Dyn. Syst.1, 95–102 (1981)Google Scholar
  21. [Mar1] margulis, G.A.: On the action of unipotent groups in the space of lattices. In: Gelfand, I.M. (ed.) Proc. of the summer school on group representations. Bolyai Janos Math. Soc., Budapest, 1971, pp. 365–370. Budapest: Akadémiai Kiado 1975Google Scholar
  22. [Mar2] margulis, G.A.: Formes quadratiques indefinies et flots unipotents sur les espaces homogénes. C.R. Acad. Sci., Paris, Ser. I304, 249–253 (1987)Google Scholar
  23. [Mar3] Margulis, G.A.: Discrete subgroups and ergodic theory. In: Aubert, K.E. et al. (eds.) Proc. of the conference “Number theory, trace formula and discrete groups” in honor of A. Selberg. Oslo 1987, pp. 377–388. London New York: Academic Press 1988Google Scholar
  24. [Mar4] Margulis, G.A.: Orbits of group actions and values of quadrtic forms at integral points. In: Festschrift in honour of I.I. Piatetski-Shapiro. (Isr. Math. Conf. Proc., vol. 3, pp. 127–151) Jerusalem: The Weizmann Science Press of Israel 1990Google Scholar
  25. [Mar5] margulis, G.A.: Discrete Subgroups of Semisimple Groups. Berin Heidelberg New York: Springer 1990Google Scholar
  26. [Mar6] Margulis, G.A.: Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications in number theory. In: Sutake, I. (ed.) Proceedings of the International Congress of Mathematicians. Kyoto 1990, pp. 193–215, Tokyo: The Mathematical Society of Japan and Berlin Heidelberg New York: Springer 1991Google Scholar
  27. [Mar-To] Margulis, G.A., Tomanov, G.M.: Measure rigidity for algebraic groups over local fields. C.R. Acad. Sci., Paris315, 1221–1226 (1992)Google Scholar
  28. [Mo] Moore, C.C.: The Mautner phenomenon for general unitary representations. Pac. J. Math.86, 155–169 (1980)Google Scholar
  29. [Pral] Prasad, G.: Elementary proof of a theorem of Bruhat-Tits-Rousseau and a theorem of Tits. Bull. Sci. Math. Fr.110, 197–202 (1982)Google Scholar
  30. [Pra2] Prasad, G.: Ratner's Theorem in S-arithmetic setting. In: Workshop on Lie Groups, Ergodic Theory and Geometry. (Publ., Math. Sci. Res. Inst., p. 53) Berlin Heidelberg New York: Springer 1992Google Scholar
  31. [R1] Ratner, M.: Horocycle flows: joining and rigidity of products. Ann. Math.118, 277–313 (1983)Google Scholar
  32. [R2] Ratner, M.: Strict measure rigidity for unipotent subgroups of solvable groups. Invent. Math.101, 449–482 (1990)Google Scholar
  33. [R3] Ratner, M.: On measure rigidity of unipotent subgroups of semi-simple groups. Acta. Math.165, 229–309 (1990)Google Scholar
  34. [R4] Ratner, M.: On Raghunathan's measure conjecture. Ann. Math.134, 545–607 (1992)Google Scholar
  35. [R5] Ratner, M.: Raghunathan topological conjecture and distributions of unipotent flows. Duke Math. J.63, 235–280 (1991)Google Scholar
  36. [R6] Ratner, M.: Invariant measures and orbit closures for unipotent actions on homogeneous spaces. (Publ., Math. Sci. Res. Inst.) Berlin Heidelberg New York: Springer (to appear)Google Scholar
  37. [R7] Ratner, M.: Raghunathan's conjectures forp-adic Lie Groups. Int. Math. Research Notices. Number 5, 141–146 (1993) (in Duke Math. J. 70:2)Google Scholar
  38. [Roh] Rohlin, V.A.: Lectures on the theory of entropy of transformations with invariant measures. Russ. Math. Surv.22:5, 1–54 (1967)Google Scholar
  39. [Sh] Shah, N.: Uniformly distributed orbits of certain flows on homogeneous spaces. Math. Ann.289, 315–334 (1991)Google Scholar
  40. [Tem] Tempelman, A.: Ergodic Theorems for Group Actions. Dordrecht Boston London: Kluwer 1992Google Scholar
  41. [W] Witte, D.: Rigidity of some translations on homogeneous spaces. Invent Math.81, 1–27 (1985)Google Scholar
  42. [Zi] Zimmer, R.: Ergodic Theory and Semisimple Groups. Boston Basel Stuttgart: Birkhäuser 1984Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • G. A. Margulis
    • 1
  • G. M. Tomanov
    • 1
  1. 1.Department of MathematicsYale UniversityYale StationUSA

Personalised recommendations