Israel Journal of Mathematics

, Volume 65, Issue 2, pp 165–196 | Cite as

Propriétés de contraction d’un semi-groupe de matrices inversibles. Coefficients de Liapunoff d’un produit de matrices aléatoires indépendantes

  • Yves Guivarc’h
  • Albert Raugi


We study in this paper contraction properties of a matrix semi-groupTGL(d,R) acting on the flag space ofRd; 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].


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  1. 1.
    R. Azencott,Espaces de Poisson des groupes localement compacts, Lecture Notes No. 148, Springer-Verlag, Berlin-Heidelberg-New York, 1970.MATHGoogle Scholar
  2. 2.
    A. Borel,Introduction aux groupes arithmétiques, Hermann, Paris, 1969.MATHGoogle Scholar
  3. 3.
    P. Bougerol and J. Lacroix,Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics8, Birkhaüser, 1985.Google Scholar
  4. 4.
    H. Furstenberg,Boundary theory and stochastic processes on homogeneous spaces, Proc. Symp. Pure Math.26 (1972), 193–229.Google Scholar
  5. 5.
    I. Y. Goldsheid and G. A. Margulis,Simplicity of the Liapunoff spectrum for products of random matrices, Soviet Math.35 (2) (1987), 309–313.MathSciNetGoogle Scholar
  6. 6.
    Y. Guivarc’h and A. Raugi,Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrscheinlichkeitstheor. Verw. Geb69 (1985), 187–242.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Y. Guivarc’h and A. Raugi,Products of random matrices: convergence theorems, Contemporary Mathematics, Am. Math. Soc.50 (1986), 31–53.MathSciNetGoogle Scholar
  8. 8.
    V. I. Oseledec,A multiplicative ergodic theorem, Trans. Moscow Math. Soc.19 (1968), 197–231.MathSciNetGoogle Scholar
  9. 9.
    M. S. Ragunathan,A proof of Oseledec multiplicative theorem, Isr. J. Math.32 (1979), 356–362.CrossRefGoogle Scholar
  10. 10.
    A. Raugi,Fonctions harmoniques et théorèmes limites pour les marches aléatoires sur les groupes, Bull. Soc. Math. France, mémoire54, 1977.Google Scholar
  11. 11.
    R. Zimmer,Ergodic Theory and Semi-simple Groups, Birkhauser, Boston-Basel-Stuttgart, 1984.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1989

Authors and Affiliations

  • Yves Guivarc’h
    • 1
  • Albert Raugi
    • 1
  1. 1.IRMAR, Institut Mathématiques de RennesUniversité Rennes IRennes CedexFrance

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