Local rigidity of restrictions of Weyl chamber flows

Danijela Damjanović; Anatole Katok

doi:10.1016/j.crma.2007.03.009

Dynamical Systems

Local rigidity of restrictions of Weyl chamber flows
[Rigidite locale des restrictions de flots des chambres de Weyl]

Présenté par : Étienne Ghys

Danijela Damjanović ¹ ; Anatole Katok ²

¹ Institut des hautes études scientifiques, 35, route de Chartres, 91440 Bures-sur-Yvette, France
² Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 503-508.

Résumé
Abstract

Dans cette Note on étudie des exemples d'actions partiellement hyperboliques : restrictions de flots des chambres de Weyl sur la variété $SL (n, R) / Γ$ ( $n ⩾ 4$ ). Nous démontrons que, génériquement, les restrictions de rang ⩾2 sont localement rigides. Notre approche combine la géométrie des feuilletages invariants et les propriétés algébriques du groupe $SL (n, R)$ . Cette approche est aussi applicable dans la démonstration de la rigidité locale des restrictions des autres flots des chambres de Weyl.

In this Note we consider examples of partially hyperbolic actions: restrictions of Weyl chamber flows on $SL (n, R) / Γ$ ( $n ⩾ 4$ ). We show that generic restrictions of rank at least two are locally rigid. Our approach combines the geometry of the invariant foliations for the action and the algebraic properties of the group $SL (n, R)$ . The method is applicable to restrictions of Weyl chamber flows on other homogeneous spaces.

Reçu le : 2006-04-12
Accepté le : 2007-02-09
Publié le : 2007-04-15

DOI : 10.1016/j.crma.2007.03.009

Affiliations des auteurs :

Danijela Damjanović ¹ ; Anatole Katok ²

¹ Institut des hautes études scientifiques, 35, route de Chartres, 91440 Bures-sur-Yvette, France
² Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

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     author = {Danijela Damjanovi\'c and Anatole Katok},
     title = {Local rigidity of restrictions of {Weyl} chamber flows},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {503--508},
     publisher = {Elsevier},
     volume = {344},
     number = {8},
     year = {2007},
     doi = {10.1016/j.crma.2007.03.009},
     language = {en},
}

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%A Anatole Katok
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Danijela Damjanović; Anatole Katok. Local rigidity of restrictions of Weyl chamber flows. Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 503-508. doi : 10.1016/j.crma.2007.03.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.009/

Bibliographie
Cité par

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[2] D. Damjanović, Central extensions of some simple Lie groups and rigidity of some partially hyperbolic algebraic actions, submitted for publication

[3] D. Damjanović; A. Katok Local rigidity of actions of higher rank Abelian groups and KAM method, ERA–AMS, Volume 10 (2004), pp. 142-154

[4] D. Damjanović; A. Katok Local rigidity of partially hyperbolic actions I. KAM method and actions on the torus www.math.psu.edu/katok_a/papers.html (submitted for publication)

[5] D. Damjanović; A. Katok Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic $R^{k}$ actions, Discr. Contin. Dyn. Syst., Volume 13 (2005), pp. 985-1005

[6] D. Damjanović, A. Katok, Local rigidity of partially hyperbolic actions, II. Restrictions of Weyl chamber flows on $SL (n, R) / Γ$ and algebraic K-theory, submitted for publication, www.math.psu.edu/katok_a/papers.html

[7] M. Guysinsky; A. Katok Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations, Math. Res. Lett., Volume 5 (1998), pp. 149-163

[8] M. Hirsch; C. Pugh; M. Shub Invariant Manifolds, Lecture Notes in Mathematics, vol. 583, Springer-Verlag, Berlin, 1977

[9] A. Katok; A. Kononenko Cocycle stability for partially hyperbolic systems, Math. Res. Lett., Volume 3 (1996), pp. 191-210

[10] A. Katok; R. Spatzier Subelliptic estimates of polynomial differential operators and applications to rigidity of Abelian actions, Math. Res. Lett., Volume 1 (1994), pp. 193-202

[11] A. Katok; R. Spatzier Differential rigidity of Anosov actions of higher rank Abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math., Volume 216 (1997), pp. 287-314

[12] J. Milnor Introduction to Algebraic K-Theory, Princeton University Press, 1971

[13] R. Steinberg, Générateurs, relations et revêtements de groupes algébriques, in: Colloq. théorie des groupes algébriques, Bruxelles, 1962, pp. 113–127

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