×

Stationary measures and invariant subsets of homogeneous spaces. III. (English) Zbl 1279.22013

This important paper is the third in a series of papers dealing with the ergodic theory of homogeneous spaces under real and \(p\)-adic Lie groups.
For Parts I and II see [C. R., Math., Acad. Sci. Paris 347, No. 1–2, 9–13 (2009; Zbl 1155.22014); ibid. 349, No. 5–6, 341–345 (2011; Zbl 1211.22006)]. The authors have obtained random analogues of equidistribution properties of unipotent flows on homogeneous spaces of Lie groups obtained by Dani, Margulis and Ratner.
One of the main theorems of the paper asserts:
Let \(G\) be a real Lie group, \(\Lambda\) a lattice in \(G\) and \(X = G/\Lambda\). Let \(\Gamma\) be a compactly generated closed subsemigroup of \(G\). Further, assume that the Zariski closure of the semigroup \(Ad(\Gamma)\) is semisimple and without compact factors. Then, for every \(x \in G/\Lambda\), there exists a closed subgroup \(H\) of \(G\) containing \(\Gamma\) such that the closure of the orbit \(\Gamma x\) is the orbit \(Hx\). Moreover, \(Hx\) carries an \(H\)-invariant probability measure.
Under the assumption that \(G\) is simple and \(\Gamma\) is already Zariski dense in \(G\), the authors had proved this theorem in the first of the three papers referred to above. The above theorem is already new for the striking cases such as: (i) \(G =\mathrm{SL}_2(\mathbb{R}) \times\mathrm{SL}_2(\mathbb{R})\) and \(\Gamma\) is Zariski dense in \(G\) and; (ii) \(G =\mathrm{SO}(3,1)\) and \(\Gamma\) is Zariski dense in \(\mathrm{SO}(2,1)\).
The authors work more generally with the so-called weakly regular \(S\)-adic Lie groups. Such a topological group is isomorphic to a closed subgroup of products of finitely many real and \(p\)-adic Lie groups where weak regularity in a \(p\)-adic Lie group \(G_p\) is the property that two one-parameter subgroups \(\alpha, \beta : \mathbb{Q}_p \rightarrow G_p\) are equal if their derivatives at 1 are equal. The authors prove a general version of the above theorem in the context of weakly regular \(S\)-adic Lie groups.
We don’t state this general result but point out to two striking cases when it applies: (i) \(G =\mathrm{SL}_2(\mathbb{Q}_p)\) and \(\Gamma\) is Zariski dense and unbounded in \(G\) and; (ii) \(G =\mathrm{SL}_2(\mathbb{Q}_p) \times\mathrm{SL}_2(\mathbb{R})\) and the projection of \(\Gamma\) on each factor is Zariski dense and unbounded.
Several subsidiary results are proved of independent interest. For instance, an interesting byproduct of the main theorem is: Let \(\{g_n \}\) be a sequence of independent identically distributed random elements of \(\mathrm{SL}_2(\mathbb{Z})\) whose law has finite support and generates a non-solvable group. Then, starting from any irrational point \(x\) in the torus \(\mathbb{T}^2\), almost surely, the trajectory \(g_n \cdots g_1x\) equidistributes toward the Haar probability on \(\mathbb{T}^2\).
The powerful techniques introduced by the authors have already led to new, concrete, striking results in the subject and are expected to continue to do so.

MSC:

22E40 Discrete subgroups of Lie groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Y. Benoist, ”Propriétés asymptotiques des groupes linéaires,” Geom. Funct. Anal., vol. 7, iss. 1, pp. 1-47, 1997. · Zbl 0947.22003 · doi:10.1007/PL00001613
[2] Y. Benoist and J. Quint, ”Mesures stationnaires et fermés invariants des espaces homogènes,” Ann. of Math., vol. 174, iss. 2, pp. 1111-1162, 2011. · Zbl 1241.22007 · doi:10.4007/annals.2011.174.2.8
[3] Y. Benoist and J. Quint, ”Mesures stationnaires et fermés invariants des espaces homogènes II,” C. R. Math. Acad. Sci. Paris, vol. 349, iss. 5-6, pp. 341-345, 2011. · Zbl 1211.22006 · doi:10.1016/j.crma.2011.01.015
[4] Y. Benoist and J. Quint, ”Stationary measures and invariant subsets of homogeneous spaces (II),” J. Amer. Math. Soc., vol. 26, iss. 3, pp. 659-734, 2013. · Zbl 1211.22006 · doi:10.1016/j.crma.2011.01.015
[5] Y. Benoist and J. Quint, ”Random walks on finite volume homogeneous spaces,” Invent. Math., vol. 187, iss. 1, pp. 37-59, 2012. · Zbl 1244.60009 · doi:10.1007/s00222-011-0328-5
[6] Y. Benoist and J. Quint, Lattices in \(S\)-adic Lie groups. · Zbl 1291.22013
[7] J. Bourgain, A. Furman, E. Lindenstrauss, and S. Mozes, ”Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus,” J. Amer. Math. Soc., vol. 24, iss. 1, pp. 231-280, 2011. · Zbl 1239.37005 · doi:10.1090/S0894-0347-2010-00674-1
[8] L. Breiman, ”The strong law of large numbers for a class of Markov chains,” Ann. Math. Statist., vol. 31, pp. 801-803, 1960. · Zbl 0104.11901 · doi:10.1214/aoms/1177705810
[9] L. Clozel, H. Oh, and E. Ullmo, ”Hecke operators and equidistribution of Hecke points,” Invent. Math., vol. 144, iss. 2, pp. 327-351, 2001. · Zbl 1144.11301 · doi:10.1007/s002220100126
[10] S. G. Dani and G. A. Margulis, ”Limit distributions of orbits of unipotent flows and values of quadratic forms,” in I. M. Gel\cprime fand Seminar, Providence, RI: Amer. Math. Soc., 1993, vol. 16, pp. 91-137. · Zbl 0814.22003
[11] A. Eskin and G. Margulis, ”Recurrence properties of random walks on finite volume homogeneous manifolds,” in Random Walks and Geometry, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 431-444. · Zbl 1064.60092
[12] A. Eskin, S. Mozes, and N. Shah, ”Unipotent flows and counting lattice points on homogeneous varieties,” Ann. of Math., vol. 143, iss. 2, pp. 253-299, 1996. · Zbl 0852.11054 · doi:10.2307/2118644
[13] A. Eskin and H. Oh, ”Ergodic theoretic proof of equidistribution of Hecke points,” Ergodic Theory Dynam. Systems, vol. 26, iss. 1, pp. 163-167, 2006. · Zbl 1092.11023 · doi:10.1017/S0143385705000428
[14] Y. Guivarc’h and A. N. Starkov, ”Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms,” Ergodic Theory Dynam. Systems, vol. 24, iss. 3, pp. 767-802, 2004. · Zbl 1050.37012 · doi:10.1017/S0143385703000440
[15] M. Kneser, ”Erzeugende und Relationen verallgemeinerter Einheitengruppen,” J. Reine Angew. Math., vol. 214/215, pp. 345-349, 1964. · Zbl 0141.02603 · doi:10.1515/crll.1964.214-215.345
[16] G. A. Margulis, Discrete subgroups of semisimple Lie groups, New York: Springer-Verlag, 1991. · Zbl 0732.22008
[17] 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
[18] S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, New York: Springer-Verlag, 1993. · Zbl 0925.60001
[19] D. Montgomery and L. Zippin, Topological Transformation Groups, New York: Interscience Publishers, 1955. · Zbl 0068.01904
[20] S. Mozes and N. Shah, ”On the space of ergodic invariant measures of unipotent flows,” Ergodic Theory Dynam. Systems, vol. 15, iss. 1, pp. 149-159, 1995. · Zbl 0818.58028 · doi:10.1017/S0143385700008282
[21] R. Muchnik, ”Semigroup actions on \(\mathbb T^n\),” Geom. Dedicata, vol. 110, pp. 1-47, 2005. · Zbl 1071.37008 · doi:10.1007/s10711-003-0816-x
[22] G. Prasad and M. S. Raghunathan, ”Topological central extensions of semisimple groups over local fields,” Ann. of Math., vol. 119, iss. 1, pp. 143-201, 1984. · Zbl 0552.20025 · doi:10.2307/2006967
[23] M. S. Raghunathan, Discrete Subgroups of Lie Groups, New York: Springer-Verlag, 1972, vol. 68. · Zbl 0254.22005
[24] 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
[25] 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
[26] N. A. Shah, ”Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements,” in Lie Groups and Ergodic Theory, Bombay: Tata Inst. Fund. Res., 1998, vol. 14, pp. 229-271. · Zbl 0951.22006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.