Frontière de furstenberg, propriétés de contraction et théorèmes de convergence

Joint spectrum and large deviation principle for random matrix products

“…For the point 2. of this theorem, we note that for a probability measure as in Theorem 1.3, the fact that λ µ belongs to the interior of a + was obtained by Guivarc'h-Raugi [18] and Goldsheid-Margulis [15]. Our result gives a more precise location for λ µ in case µ is, moreover, boundedly supported.…”

Central limit theorem on hyperbolic groups

“…It was then extended by Le Page in [36] for more general semigroups when the law has a finite exponential moment, that is, when there exists α > 0 such that G N (g) α dµ(g) < ∞. Thanks to later works of Guivarc'h and Raugi in [28] and Gol'dsheȋd and Margulis in [21], the assumptions in the Le Page theorem were clarified: the sole remaining but still unwanted assumption was that µ had a finite exponential moment.…”

“…Another important ingredient in the proof of the log-regularity of ν * is the simplicity of the first Lyapunov exponent due to Guivarc'h in [25] and [28].…”

Matrix random products with singular harmonic measure

“…If supp µ generates a Zariski dense subgroup of SL(d, R), then the Lyapunov spectrum is simple (≡ the vector λ lies in the interior of the positive Weyl chamber), see [GR85,GM89], so that the associated Lyapunov flags are full (≡ contain subspaces of all the intermediate dimensions). The space B = B(d) of full flags in R d is also known under the name of the Furstenberg boundary of the associated symmetric space S, see [Fur63b] for its definition and [Kai89,GJT98] for its relation with the boundaries of various compactifications of Riemannian symmetric spaces.…”