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Topological mixing of Weyl chamber flows

Part of: Global differential geometry Variational problems in infinite-dimensional spaces Connections with other structures, applications Topological dynamics

Published online by Cambridge University Press:  17 February 2020

NGUYEN-THI DANG  and
OLIVIER GLORIEUX
NGUYEN-THI DANG
Affiliation:
IRMAR (CNRS-UMR 6625), University of Rennes 1, 35000Rennes, France email nguyen-thi.dang@univ-rennes1.fr
OLIVIER GLORIEUX
Affiliation:
University of Luxembourg Faculty of Science, Technology and Communication, Mathematics Research Unit 6, rue Coudenhove-Kalergi, L-1359Luxembourg-Kirchberg Bureau G104, Luxembourg email olivier.glrx@gmail.com
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Abstract

In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.

Keywords

Weyl flowinfinite volumeZariski dense discrete grouphigher ranktopological mixing

MSC classification

Primary: 54H20: Topological dynamics 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 53C30: Homogeneous manifolds
Secondary: 53C35: Symmetric spaces 58E40: Group actions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1342 - 1368
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Benoist, Y.. Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7(1) (1997), 1–47.Google Scholar
Benoist, Y.. Propriétés asymptotiques des groupes linéaires II. Adv. Stud. Pure Math. 26 (2000), 33–48.CrossRefGoogle Scholar
Breuillard, E. and Gelander, T.. On dense free subgroups of Lie groups. J. Algebra 261(2) (2003), 448–467.Google Scholar
Benoist, Y. and Quint, J.-F.. Random walks on reductive groups. Random Walks on Reductive Groups. Springer, Cham, 2016, pp. 153–167.CrossRefGoogle Scholar
Conze, J.-P. and Guivarc’h, Y.. Densité d’orbites d’actions de groupes linéaires et propriétés d’équidistribution de marches aléatoires. Rigidity in Dynamics and Geometry. Springer, Berlin, 2002, pp. 39–76.CrossRefGoogle Scholar
Dal’bo, F.. Topologie du feuilletage fortement stable. Ann. Inst. Fourier (Grenoble) 50(3) (2000), 981–993.CrossRefGoogle Scholar
Eberlein, P.. Geodesic flows on negatively curved manifolds I. Ann. of Math. (2) (1972), 492–510.CrossRefGoogle Scholar
Guivarc’h, Y., Ji, L. and Taylor, J. C.. Compactifications of Symmetric Spaces (Progress in Mathematics, 156) . Birkhäuser, Basel, 2012.Google Scholar
Helgason, S.. Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics, 34) . American Mathematical Society, Providence, RI, 2001, corrected reprint of the 1978 original.CrossRefGoogle Scholar
Kim, I.. Length spectrum in rank one symmetric space is not arithmetic. Proc. Amer. Math. Soc. 134(12) (2006), 3691–3696.CrossRefGoogle Scholar
Prasad, G. and Rapinchuk, A. S.. Zariski-dense subgroups and transcendental number theory. Math. Res. Lett. 12(2) (2005), 239–248.CrossRefGoogle Scholar
Quint, J.-F.. Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv. 77(3) (2002), 563–608.CrossRefGoogle Scholar
Sambarino, A.. The orbital counting problem for hyperconvex representations [sur le décompte orbital pour les representations hyperconvexes]. Ann. Inst. Fourier 65 (2015), 1755–1797.CrossRefGoogle Scholar
Sert, Ç.. Joint spectrum and large deviation principles for random matrix products. PhD Thesis, Université Paris-Saclay, 2016.Google Scholar
Thirion, X.. Sous-groupes discrets de SL(d, R) et equidistribution dans les espaces symetriques. PhD Thesis, Tours, 2007.Google Scholar
Thirion, X.. Propriétés de mélange du flot des chambres de Weyl des groupes de ping-pong. Bull. Soc. Math. France 137(3) (2009), 387–421.CrossRefGoogle Scholar
Tits, J.. Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque. J. Reine Angew. Math. 247 (1971), 196–220.Google Scholar