Abstract
We study in this paper contraction properties of a matrix semi-groupT ⊂GL(d,R) acting on the flag space ofR d; then we obtain properties of the Liapunoff exponents of theT-valued products of random matrices. The principal result is that, in this study, we can replaceT by its algebraic closureH inGL(d,R). This implies a “decomposition” of the action ofT in a proximal part and an isometric part; then we can write, modulo cohomology, the corresponding cocycle in a block-diagonal form, the blocks being similarities. In fact, we can express the multiplicities of the exponents in terms of the diagonal part of a conjugate of the groupH. So we obtain an extension of a recent result of Goldsheid and Margulis about the simplicity of Liapunoff’s spectrum [5]; this work uses their ideas as well as those of previous work [6].
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Bibliographie
R. Azencott,Espaces de Poisson des groupes localement compacts, Lecture Notes No. 148, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
A. Borel,Introduction aux groupes arithmétiques, Hermann, Paris, 1969.
P. Bougerol and J. Lacroix,Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics8, Birkhaüser, 1985.
H. Furstenberg,Boundary theory and stochastic processes on homogeneous spaces, Proc. Symp. Pure Math.26 (1972), 193–229.
I. Y. Goldsheid and G. A. Margulis,Simplicity of the Liapunoff spectrum for products of random matrices, Soviet Math.35 (2) (1987), 309–313.
Y. Guivarc’h and A. Raugi,Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrscheinlichkeitstheor. Verw. Geb69 (1985), 187–242.
Y. Guivarc’h and A. Raugi,Products of random matrices: convergence theorems, Contemporary Mathematics, Am. Math. Soc.50 (1986), 31–53.
V. I. Oseledec,A multiplicative ergodic theorem, Trans. Moscow Math. Soc.19 (1968), 197–231.
M. S. Ragunathan,A proof of Oseledec multiplicative theorem, Isr. J. Math.32 (1979), 356–362.
A. Raugi,Fonctions harmoniques et théorèmes limites pour les marches aléatoires sur les groupes, Bull. Soc. Math. France, mémoire54, 1977.
R. Zimmer,Ergodic Theory and Semi-simple Groups, Birkhauser, Boston-Basel-Stuttgart, 1984.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Guivarc’h, Y., Raugi, A. Propriétés de contraction d’un semi-groupe de matrices inversibles. Coefficients de Liapunoff d’un produit de matrices aléatoires indépendantes. Israel J. Math. 65, 165–196 (1989). https://doi.org/10.1007/BF02764859
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02764859