Duke Mathematical Journal

Joinings of higher-rank diagonalizable actions on locally homogeneous spaces

Manfred Einsiedler and Elon Lindenstrauss

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 classify joinings between a fairly general class of higher-rank diagonalizable actions on locally homogeneous spaces. In particular, we classify joinings of the action of a maximal R-split torus on G/Γ with G a simple Lie group of R-rank at least 2 and Γ<G a lattice. We deduce from this a classification of measurable factors of such actions as well as certain equidistribution properties

Article information

Source
Duke Math. J., Volume 138, Number 2 (2007), 203-232.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1181051030

Digital Object Identifier
doi:10.1215/S0012-7094-07-13822-5

Mathematical Reviews number (MathSciNet)
MR2318283

Zentralblatt MATH identifier
1118.37008

Subjects
Primary: 37A17: Homogeneous flows [See also 22Fxx]
Secondary: 22E46: Semisimple Lie groups and their representations 28D05: Measure-preserving transformations

Citation

Einsiedler, Manfred; Lindenstrauss, Elon. Joinings of higher-rank diagonalizable actions on locally homogeneous spaces. Duke Math. J. 138 (2007), no. 2, 203--232. doi:10.1215/S0012-7094-07-13822-5. https://projecteuclid.org/euclid.dmj/1181051030


Export citation

References

  • W. Arveson, An Invitation to $C\sp*$-Algebras, Grad. Texts in Math. 39, Springer, New York, 1976.
  • S. G. Dani, Bernoullian translations and minimal horospheres on homogeneous spaces, J. Indian Math. Soc. (N.S.) 40 (1976), 245--284.
  • A. Del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7 (1987), 531--557.
  • 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, Rigidity properties of $\mathbb Z\,\sp d$-actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 99--110.
  • M. Einsiedler and T. Ward, Entropy geometry and disjointness for zero-dimensional algebraic actions, J. Reine Angew. Math. 584 (2005), 195--214..
  • H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1--49.
  • E. Glasner, Ergodic Theory via Joinings, Math. Surveys Monogr. 101, Amer. Math. Soc., Providence, 2003.
  • B. Hasselblatt and A. Katok, ``Principal structures'' in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1--203.
  • B. Kalinin and A. Katok, Measurable rigidity and disjointness for $\mathbb Z\sp k$ actions by toral automorphisms, Ergodic Theory Dynam. Systems 22 (2002), 507--523.
  • B. Kalinin and R. Spatzier, Rigidity of the measurable structure for algebraic actions of higher-rank Abelian 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.; Corrections, Ergodic Theory Dynam. Systems 18 (1998), 503--507. $\!$;
  • D. Kleinbock, N. Shah, and A. Starkov, ``Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory'' in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 813--930.
  • A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math. 140, Birkhäuser, Boston, 2002.
  • F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, I: Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2) 122 (1985), 509--539.; II: Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (1985), 540--574. $\!$;
  • E. Lindenstrauss, Rigidity of multiparameter actions, Israel J. Math. 149 (2005), 199--226.
  • —, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), 165--219.
  • G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb. 17, Springer, Berlin, 1991.
  • —, ``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.
  • C. C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966), 154--178.
  • W. Parry, Topics in Ergodic Theory, Cambridge Tracts in Math. 75, Cambridge Univ. Press, Cambridge, 1981.
  • M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2) 118 (1983), 277--313.
  • —, On Raghunathan's measure conjecture, Ann. of Math. (2) 134 (1991), 545--607.
  • D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems 10 (1990), 395--406.
  • J. G. Sinaĭ, A weak isomorphism of transformations with invariant measure (in Russian), Dokl. Akad. Nauk SSSR 147 (1962), 797--800.
  • A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Transl. Math. Monogr. 190, Amer. Math. Soc., Providence, 2000.
  • V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191--220.
  • —, Lie Groups, Lie Algebras, and Their Representations, Prentice Hall Ser. Modern Anal., Prentice Hall, Englewood Cliffs, N.J., 1974.
  • D. Witte [D. W. Morris], Measurable quotients of unipotent translations on homogeneous spaces, Trans. Amer. Math. Soc. 345 (1994), 577--594.; Correction and extension, Trans. Amer. Math. Soc. 349 (1997), 4685--4688. $\!$;
  • R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monogr. Math. 81, Birkhäuser, Basel, 1984.