On measures invariant under tori on quotients of semisimple groups. (English) Zbl 1316.22009

Einsiedler, Manfred; Lindenstrauss, Elon

Ann. Math. (2) 181, No. 3, 993-1031 (2015).

This paper is a significant contribution to the study of measure rigidity of abelian group actions in the setting of homogeneous dynamics. It is well known that the action of the diagonal subgroup of \(\text{SL}_2(\mathbb{R})\) on \(\text{SL}_2(\mathbb{Z})\backslash \text{SL}_2(\mathbb{R})\) admits an enormous variety of invariant probability measures (for example, realizing all possible entropies), and the same holds for actions of \(\mathbb{R}\)-split tori on quotients of real algebraic groups by lattices whenever the torus has rank one. Starting with the work of J. W. S. Cassels and H. P. F. Swinnerton-Dyer [Philos. Trans. R. Soc. Lond., Ser. A 248, 73–96 (1955; Zbl 0065.27905)], motivated by questions in number theory, and the work of H. Furstenberg [Math. Syst. Theory 1, 1–49 (1967; Zbl 0146.28502)], motivated by a structural approach to dynamics, it was realised that the case of tori of rank greater than or equal to two is entirely different, and the space of invariant measures in that setting is very prescribed. This phenomena is an instance of abelian measure rigidity. Despite the phenomenon not being entirely understood (in particular the case in which each element of the action has zero entropy remains open), this phenomenon has already found significant applications, most notably in the work of E. Lindenstrauss on quantum unique ergodicity [Ann. Math. (2) 163, No. 1, 165–219 (2006; Zbl 1104.22015)] and that of M. Einsiedler et al. [Ann. Math. (2) 164, No. 2, 513–560 (2006; Zbl 1109.22004)] on Littlewood’s problem in the Diophantine analysis or pairs of real numbers. These developments have their origin in part in abelian settings: D. J. Rudolph [Ergodic Theory Dyn. Syst. 10, No. 2, 395–406 (1990; Zbl 0709.28013)] showed that a probability measure on the circle invariant and ergodic under the action of multiplication modulo one by two multiplicatively independent natural numbers giving positive entropy to the action of one of them must be the Lebesgue measure. In applications of this and later quantitative versions, it is of particular importance that the entropy condition required, of exceeding some non-zero lower bound, is stable under weak*-limits of invariant measures. A. Katok and R. J. Spatzier [Ergodic Theory Dyn. Syst. 16, No. 4, 751–778 (1996; Zbl 0859.58021); correction ibid. 18, No. 2, 503–507 (1998)] initiated the extensions of this kind of result to higher rank, in some cases requiring in addition to an entropy condition a mixing condition, which considerably complicates some of the potential applications as mixing properties behave poorly under weak*-limits of invariant measures. In this paper the authors give a rather broad and practical classification of measures on homogeneous spaces \(\Gamma\backslash G\) invariant under the action of higher-rank tori. This extends earlier work of the authors with A. Katok on the measure rigidity behind the approach to the Littlewood problem and of the second author on the measure rigidity behind arithmetic quantum unique ergodicity. As is usual in these problems, the classification is less rigid and definitive than that given by the Ratner theory for the action of subgroups generated by unipotents because of the possible presences of actions of rank-one tori, where in some sense there is complete chaos for well-understood reasons, and because of the possible presence of zero entropy subactions, where it is simply not known whether exotic invariant measures could exist. The main result says that a probability measure on an arithmetic homogeneous quotient of a semi-simple \(S\)-algebraic group that is invariant and ergodic for a maximal split torus in at least one simple local factor has algebraic support that splits into the product of a torus, a homogeneous space on which the measure is the Haar measure up to finite index, a product of homogeneous spaces on each of which the action is essentially rank one (and hence inherently uncontrollable), and a homogeneous space on which every element of the action has zero entropy with respect to the measure. This is an important and satisfying result which is certain to find many applications. The methods used to prove this theorem are diverse and sophisticated, with a critical role played by the notion of leafwise measures for group actions.

Reviewer: Thomas B. Ward (Durham)

MSC:

22E40	Discrete subgroups of Lie groups
22D40	Ergodic theory on groups
22F30	Homogeneous spaces
37A15	General groups of measure-preserving transformations and dynamical systems
37A17	Homogeneous flows
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)

Keywords:

Homogeneous dynamics; abelian rigidity; measure rigidity; higher-rank action

Citations:

Zbl 0065.27905; Zbl 0146.28502; Zbl 1104.22015; Zbl 1109.22004; Zbl 0709.28013; Zbl 0859.58021

References:

[1]	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., vol. 248, pp. 73-96, 1955. · Zbl 0065.27905 · doi:10.1098/rsta.1955.0010
[2]	S. G. Dani and G. A. Margulis, ”Orbit closures of generic unipotent flows on homogeneous spaces of \({ SL}(3,{\mathbf R})\),” Math. Ann., vol. 286, iss. 1-3, pp. 101-128, 1990. · Zbl 0679.22007 · doi:10.1007/BF01453567
[3]	M. Einsiedler and A. Katok, ”Invariant measures on \(G/\Gamma\) for split simple Lie groups \(G\),” Comm. Pure Appl. Math., vol. 56, iss. 8, pp. 1184-1221, 2003. · Zbl 1022.22023 · doi:10.1002/cpa.10092
[4]	M. Einsiedler and A. Katok, ”Rigidity of measures-the high entropy case and non-commuting foliations,” Israel J. Math., vol. 148, pp. 169-238, 2005. · Zbl 1097.37017 · doi:10.1007/BF02775436
[5]	M. Einsiedler, A. Katok, and E. Lindenstrauss, ”Invariant measures and the set of exceptions to Littlewood’s conjecture,” Ann. of Math., vol. 164, iss. 2, pp. 513-560, 2006. · Zbl 1109.22004 · doi:10.4007/annals.2006.164.513
[6]	M. Einsiedler and E. Lindenstrauss, ”Joinings of higher-rank diagonalizable actions on locally homogeneous spaces,” Duke Math. J., vol. 138, iss. 2, pp. 203-232, 2007. · Zbl 1118.37008 · doi:10.1215/S0012-7094-07-13822-5
[7]	M. Einsiedler and E. Lindenstrauss, ”Diagonalizable flows on locally homogeneous spaces and number theory,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1731-1759. · Zbl 1121.37028
[8]	M. Einsiedler and E. Lindenstrauss, ”On measures invariant under diagonalizable actions: the rank-one case and the general low-entropy method,” J. Mod. Dyn., vol. 2, iss. 1, pp. 83-128, 2008. · Zbl 1146.37006
[9]	M. Einsiedler and E. Lindenstrauss, ”Diagonal actions on locally homogeneous spaces,” in Homogeneous Flows, Moduli Spaces and Arithmetic, Providence, RI: Amer. Math. Soc., 2010, vol. 10, pp. 155-241. · Zbl 1225.37005
[10]	M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, ”Distribution of periodic torus orbits on homogeneous spaces,” Duke Math. J., vol. 148, iss. 1, pp. 119-174, 2009. · Zbl 1172.37003 · doi:10.1215/00127094-2009-023
[11]	M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, ”Distribution of periodic torus orbits and Duke’s theorem for cubic fields,” Ann. of Math., vol. 173, iss. 2, pp. 815-885, 2011. · Zbl 1248.37009 · doi:10.4007/annals.2011.173.2.5
[12]	H. Federer, Geometric Measure Theory, New York: Springer-Verlag, 1969, vol. 153. · Zbl 0176.00801
[13]	H. Furstenberg, ”Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,” Math. Systems Theory, vol. 1, pp. 1-49, 1967. · Zbl 0146.28502 · doi:10.1007/BF01692494
[14]	N. Jacobson, Lie Algebras, New York: Interscience Publishers (a division of John Wiley & Sons), 1962, vol. 10. · Zbl 0121.27504
[15]	A. Johnson, ”Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers,” Israel J. Math., vol. 77, iss. 1-2, pp. 211-240, 1992. · Zbl 0790.28012 · doi:10.1007/BF02808018
[16]	A. Johnson and D. J. Rudolph, ”Convergence under \(\times_q\) of \(\times_p\) invariant measures on the circle,” Adv. Math., vol. 115, iss. 1, pp. 117-140, 1995. · Zbl 0934.28010 · doi:10.1006/aima.1995.1052
[17]	A. Katok and R. J. Spatzier, ”Invariant measures for higher-rank hyperbolic abelian actions,” Ergodic Theory Dynam. Systems, vol. 16, iss. 4, pp. 751-778, 1996. · Zbl 0859.58021 · doi:10.1017/S0143385700009081
[18]	A. Katok and R. J. Spatzier, ”Corrections to: “Invariant measures for higher-rank hyperbolic abelian actions”,” Ergodic Theory Dynam. Systems, vol. 18, iss. 2, pp. 503-507, 1998. · doi:10.1017/S0143385798110969
[19]	D. Y. Kleinbock, ”Nondense orbits of flows on homogeneous spaces,” Ergodic Theory Dynam. Systems, vol. 18, iss. 2, pp. 373-396, 1998. · Zbl 0915.58078 · doi:10.1017/S0143385798100408
[20]	A. W. Knapp, Lie Groups Beyond an Introduction, Boston: Birkhäuser, 1996, vol. 140. · Zbl 0862.22006 · doi:10.1007/978-1-4757-2453-0
[21]	E. Lindenstrauss, ”Invariant measures and arithmetic quantum unique ergodicity,” Ann. of Math., vol. 163, iss. 1, pp. 165-219, 2006. · Zbl 1104.22015 · doi:10.4007/annals.2006.163.165
[22]	R. Lyons, ”On measures simultaneously \(2\)- and \(3\)-invariant,” Israel J. Math., vol. 61, iss. 2, pp. 219-224, 1988. · Zbl 0654.28010 · doi:10.1007/BF02766212
[23]	G. A. Margulis, ”Oppenheim conjecture,” in Fields Medallists’ Lectures, River Edge, NJ: World Sci. Publ., 1997, vol. 5, pp. 272-327. · doi:10.1142/9789812385215_0035
[24]	G. A. Margulis and G. M. Tomanov, ”Invariant measures for actions of unipotent groups over local fields on homogeneous spaces,” Invent. Math., vol. 116, iss. 1-3, pp. 347-392, 1994. · Zbl 0816.22004 · doi:10.1007/BF01231565
[25]	G. A. Margulis and G. M. Tomanov, ”Measure rigidity for almost linear groups and its applications,” J. Anal. Math., vol. 69, pp. 25-54, 1996. · Zbl 0864.22005 · doi:10.1007/BF02787100
[26]	D. Prasad, ”An analogue of a conjecture of Mazur: a question in Diophantine approximation on tori,” in Contributions to Automorphic Forms, Geometry, and Number Theory, Baltimore, MD: Johns Hopkins Univ. Press, 2004, pp. 699-709. · Zbl 1154.11330
[27]	M. Ratner, ”Factors of horocycle flows,” Ergodic Theory Dynam. Systems, vol. 2, iss. 3-4, pp. 465-489 (1983), 1982. · Zbl 0536.58029 · doi:10.1017/S0143385700001723
[28]	M. Ratner, ”Rigidity of horocycle flows,” Ann. of Math., vol. 115, iss. 3, pp. 597-614, 1982. · Zbl 0506.58030 · doi:10.2307/2007014
[29]	M. Ratner, ”Horocycle flows, joinings and rigidity of products,” Ann. of Math., vol. 118, iss. 2, pp. 277-313, 1983. · Zbl 0556.28020 · doi:10.2307/2007030
[30]	M. Ratner, ”On measure rigidity of unipotent subgroups of semisimple groups,” Acta Math., vol. 165, iss. 3-4, pp. 229-309, 1990. · Zbl 0745.28010 · doi:10.1007/BF02391906
[31]	M. Ratner, ”On Raghunathan’s measure conjecture,” Ann. of Math., vol. 134, iss. 3, pp. 545-607, 1991. · Zbl 0763.28012 · doi:10.2307/2944357
[32]	M. Ratner, ”Raghunathan’s topological conjecture and distributions of unipotent flows,” Duke Math. J., vol. 63, iss. 1, pp. 235-280, 1991. · Zbl 0733.22007 · doi:10.1215/S0012-7094-91-06311-8
[33]	M. Ratner, ”Raghunathan’s conjectures for Cartesian products of real and \(p\)-adic Lie groups,” Duke Math. J., vol. 77, iss. 2, pp. 275-382, 1995. · Zbl 0914.22016 · doi:10.1215/S0012-7094-95-07710-2
[34]	M. Rees, Some \(R^2\)-Anosov flows, 1982.
[35]	D. J. Rudolph, ”\(\times 2\) and \(\times 3\) invariant measures and entropy,” Ergodic Theory Dynam. Systems, vol. 10, iss. 2, pp. 395-406, 1990. · Zbl 0709.28013 · doi:10.1017/S0143385700005629
[36]	T. A. Springer, Linear Algebraic Groups, Second ed., Boston: Birkhäuser, 1998, vol. 9. · Zbl 0927.20024 · doi:10.1007/978-0-8176-4840-4
[37]	G. Tomanov, ”Orbits on homogeneous spaces of arithmetic origin and approximations,” in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Tokyo: Math. Soc. Japan, 2000, vol. 26, pp. 265-297. · Zbl 0960.22006
[38]	G. Tomanov, ”Actions of maximal tori on homogeneous spaces,” in Rigidity in Dynamics and Geometry, New York: Springer-Verlag, 2002, pp. 407-424. · Zbl 1012.22021
[39]	M. Waldschmidt, ”Open Diophantine problems,” Mosc. Math. J., vol. 4, iss. 1, pp. 245-305, 312, 2004. · Zbl 1066.11030