Propriétés de contraction d’un semi-groupe de matrices inversibles. Coefficients de Liapunoff d’un produit de matrices aléatoires indépendantes
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- Guivarc’h, Y. & Raugi, A. Israel J. Math. (1989) 65: 165. doi:10.1007/BF02764859
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We study in this paper contraction properties of a matrix semi-groupT ⊂GL(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 ; this work uses their ideas as well as those of previous work .