Duke Mathematical Journal

Distribution of periodic torus orbits on homogeneous spaces

Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, and Akshay Venkatesh

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove results toward the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of PGLn(R). After attaching to each periodic orbit an integral invariant (the discriminant), our results have the following flavor: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most O(Δϵ) orbits of discriminant at most Δ. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We give examples of sequences of periodic orbits of this action that fail to become equidistributed even in higher rank. We also give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees

Article information

Source
Duke Math. J., Volume 148, Number 1 (2009), 119-174.

Dates
First available in Project Euclid: 22 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1240432194

Digital Object Identifier
doi:10.1215/00127094-2009-023

Mathematical Reviews number (MathSciNet)
MR2515103

Zentralblatt MATH identifier
1172.37003

Subjects
Primary: 37A17: Homogeneous flows [See also 22Fxx] 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]
Secondary: 11E99: None of the above, but in this section

Citation

Einsiedler, Manfred; Lindenstrauss, Elon; Michel, Philippe; Venkatesh, Akshay. Distribution of periodic torus orbits on homogeneous spaces. Duke Math. J. 148 (2009), no. 1, 119--174. doi:10.1215/00127094-2009-023. https://projecteuclid.org/euclid.dmj/1240432194


Export citation

References

  • Y. Benoist and H. Oh, Equidistribution of rational matrices in their conjugacy classes, Geom. Funct. Anal. 17, 2007, 1--32.
  • A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485--535.
  • J. Bourgain, A. A. Glibichuk, and S. V. Konyagin, Estimate for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2) 73 (2006), 380--398.
  • J. Bourgain and M.-C. Chang, Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors, C. R. Math. Acad. Sci. Paris 339 (2004), 463--466.
  • J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London. Ser. A. 248 (1955), 73--96.
  • S. G. Dani and G. A. Margulis, ``Limit distributions of orbits of unipotent flows and values of quadratic forms'' in I. M. Gel$^\prime$fand Seminar, Adv. Soviet Math. 16, Part 1, Amer. Math. Soc., Providence, 1993, 91--137.
  • W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), 73--90.
  • —, Extreme values of Artin $L$ -functions and class numbers, Compositio Math. 136 (2003), 103--115.
  • M. Einsiedler and A. Katok, Invariant measures on $G/\Gamma$ for split simple Lie-groups $G$, Comm. Pure Appl. Math. 56 (2003), 1184--1221.
  • —, Rigidity of measures --.-the high entropy case and non-commuting foliations, Israel J. Math. 148 (2005), 169--238.
  • M. Einsiedler, A. Katok, and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. (2) 164 (2006), 513--560.
  • M. Einsiedler and E. Lindenstrauss, ``Diagonaizable flows on locally homogeneous spaces and number theory'' in International Congress of Mathematicians, Vol. II (Madrid, 2006), Eur. Math. Soc., Zürich, 2006, 1731--1759.
  • M. Einsiedler, E. Lindenstrauss, Ph. Michel, and A. Venkatesh, Distribution of compact torus orbits, II, in preparation.
  • —, Distribution of periodic torus orbits and Duke's theorem for cubic fields, preprint,\arxiv0708.1113v1[math.NT]
  • A. Eskin and H. Oh, Representations of integers by an invariant polynomial and unipotent flows, Duke Math. J. 135 (2006), 481--506.
  • A. Gorodnik, Open problems in dynamics and related fields, J. Med. Dyn. 1 (2007), 1--35.
  • G. Harcos and Ph. Michel, The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points, II, Invent. Math. 163 (2006), 581--655.
  • T. W. Hungerford, Algebra, Grad. Texts in Math. 73, Springer, New York, 1980.
  • B. Kalinin and R. J. Spatzier, Rigidity of the measurable structure for algebraic actions of higher-rank A belian groups, Ergodic Theory Dynam. Systems 25 (2005), 175--200.
  • A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16 (1996), 751--778.; Correction, Ergodic Theory Dynam. Systems 18 (1998), 503--507.
  • A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, Cambridge, 1995.
  • D. Y. Kleinbock and G. A. Margulis, ``Bounded orbits of nonquasiunipotent flows on homogeneous spaces'' in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2 171, Amer. Math. Soc., Providence, 1996, 141--172.
  • E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), 165--219.
  • E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems 21 (2001), 1481--1500.
  • Yu. V. Linnik, Ergodic Properties of Algebraic Fields, Ergeb. Math. Grenzgeb. 45, Springer, New York, 1968.
  • G. A. Margulis, ``Oppenheim conjecture'' in Fields Medallists' Lectures, World Sci. Ser. 20th Century Math. 5, World Sci. Publ. River Edge, New Jersey, 1997.
  • —, ``Problems and conjectures in rigidity theory'' in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, 2000, 161--174.
  • G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), 347--392.
  • —, Measure rigidity for almost linear groups and its applications, J. Anal. Math. 69 (1996), 25--54.
  • S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems 15 (1995), 149--159.
  • H. Oh, Finiteness of compact maximal flats of bounded volume, Ergodic Theory Dynam. Systems 24 (2004), 217--225.
  • V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, Boston, 1994.
  • A. Popa, Central values of Rankin $L$-series over real quadratic fields, Compos. Math. 142 (2006), 811--866.
  • G. Prasad and M. S. Raghunathan, Cartan subgroups and lattices in semi-simple groups, Ann. of Math. (2) 96 (1972), 296--317.
  • M. S. Raghunathan, Discrete subgroups of Lie groups, Ergeb. Math. Grenzgeb 68, Springer, New York, 1972.
  • M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2) 134 (1991), 545--607.
  • —, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), 235--280.
  • M. Rees, Some $R^2$-anosov flows, preprint, 1982.
  • R. Remak, Über Grössenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Z ahlkörpers, Compositio Math. 10 (1952), 245--285.
  • D. J. Rudolph, $\times 2$ and $ \times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems 10 (1990), 395--406.
  • P. Sarnak, ``Reciprocal geodesics'' in Analytic Number Theory (Göttingen, Germany, 2005), Clay. Math. Proc. 7, Amer. Math. Soc., Providence, 2007, 217--237.
  • N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann. 289 (1991), 315--334.
  • C. L. Siegel, The average measure of quadratic forms with given determinant and signature, Ann. of Math. (2) 45 (1944), 667--685.
  • G. Tomanov, ``Actions of maximal tori on homogeneous spaces'' in Rigidity in Dynamics and Geometry (Cambridge, 2000), Springer, Berlin, 2002, 407--424.
  • G. Tomanov and B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces, Duke Math. J. 119 (2003), 367--392.