Résumé : 

Given an ergodic pmp system $(X,m,T)$ and an integrable map $F:X\to SL(d,\mathbf{R})$, the Multiplicative Ergodic Theorem of Oseledets describes the asymptotic behavior of the products $F_n(x)=F(T^{n-1}x)\cdots F(Tx)F(x)$ by the Lyapunov spectrum $\Lambda=(\lambda_1\ge\dots\ge \lambda_d)$ with $\sum \lambda_i=0$, and a certain measurable family of flags on $\mathbf{R}^d$. In this talk I will describe a joint work with Uri Bader, in which we describe a class of systems for which we can prove simplicity of the spectrum: $\lambda_1>\lambda_2>\dots>\lambda_d$, and its continuity under certain perturbations. This class of systems covers many interesting examples. The proofs use ideas of "Boundary theory" for groups, that appear in the recent proofs of super-rigidity of representations of lattices and cocycles.