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A nonhomogeneous orbit closure of a diagonal subgroup. (English) Zbl 1192.22006

In an unpublished paper, M. Rees exhibited a lattice \(\Gamma\) of \(\text{SL}(3, \mathbb R)\) and a point \(x\in G/\Gamma\) such that for the full diagonal group \(A\) the orbit closure \(\overline{Ax}\) is not homogeneous. This shows that factor actions of 1-parameter non-Ad-unipotent groups are obstructions to the topological rigidity of diagonal subgroups. Margulis’s conjecture essentially states that these are the only ones [G. A. Margulis and N. Qian, Ergodic Theory Dyn. Syst. 21, No.1, 121–164 (2001; Zbl 0976.22009)].
Here the author exhibits some counterexamples when \(G=\text{SL}(n,\mathbb R)\) for \(n\geq 6\) and \(A\) is some strict subgroup of the diagonal group of matrices with nonnegative entries.
The idea behind the construction can also be used to yield examples of “nonhomogeneous” orbits for diagonal toral endomorphisms. It turns out that a proper closed invariant set is not necessarily a subset of a finite union of rational affine tori.

MSC:

22E40 Discrete subgroups of Lie groups
37B99 Topological dynamics

Citations:

Zbl 0976.22009
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References:

[1] D. Berend, ”Minimal sets on tori,” Ergodic Theory Dynam. Systems, vol. 4, iss. 4, pp. 499-507, 1984. · Zbl 0563.58020
[2] A. Borel, Introduction aux Groupes Arithmétiques, Paris: Hermann, 1969. · Zbl 0186.33202
[3] 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
[4] D. Ferte, Dynamique topologique d’une action de groupe sur un espace homogène : exemples d’actions unipotente et diagonale.
[5] 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
[6] 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. 1 A, Amsterdam: North-Holland, 2002, pp. 813-930. · Zbl 1050.22026
[7] E. Lindenstrauss and B. Weiss, ”On sets invariant under the action of the diagonal group,” Ergodic Theory Dynam. Systems, vol. 21, iss. 5, pp. 1481-1500, 2001. · Zbl 1073.37006 · doi:10.1017/S0143385701001717
[8] G. Margulis, ”Problems and conjectures in rigidity theory,” in Mathematics: Frontiers and Perspectives, Providence, RI: Amer. Math. Soc., 2000, pp. 161-174. · Zbl 0952.22005
[9] G. A. Margulis and N. Qian, ”Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices,” Ergodic Theory Dynam. Systems, vol. 21, iss. 1, pp. 121-164, 2001. · Zbl 0976.22009 · doi:10.1017/S0143385701001109
[10] D. Meiri and Y. Peres, ”Bi-invariant sets and measures have integer Hausdorff dimension,” Ergodic Theory Dynam. Systems, vol. 19, iss. 2, pp. 523-534, 1999. · Zbl 0954.37004 · doi:10.1017/S014338579912100X
[11] G. Prasad and M. S. Raghunathan, ”Cartan subgroups and lattices in semi-simple groups,” Ann. of Math., vol. 96, pp. 296-317, 1972. · Zbl 0245.22013 · doi:10.2307/1970790
[12] G. Prasad and A. S. Rapinchuk, ”Irreducible tori in semisimple groups,” Internat. Math. Res. Notices, iss. 23, pp. 1229-1242, 2001. · Zbl 1057.22025 · doi:10.1155/S1073792801000587
[13] 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
[14] N. A. Shah, ”Uniformly distributed orbits of certain flows on homogeneous spaces,” Math. Ann., vol. 289, iss. 2, pp. 315-334, 1991. · Zbl 0702.22014 · doi:10.1007/BF01446574
[15] G. Tomanov, ”Actions of maximal tori on homogeneous spaces,” in Rigidity in Dynamics and Geometry (Cambridge, 2000), New York: Springer-Verlag, 2002, pp. 407-424. · Zbl 1012.22021
[16] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Basel: Birkhäuser, 1984. · Zbl 0571.58015
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