## Duke Mathematical Journal

### Joinings of higher-rank diagonalizable actions on locally homogeneous spaces

#### 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 ${\mathbb R}$-split torus on $G/\Gamma$ with $G$ a simple Lie group of ${\mathbb R}$-rank at least $2$ and $\Gamma \lt G$ 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

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

#### 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

#### 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.