The Center For Advanced Mathematical Sciences invites you to a seminar entitled "Random matrix products when the top Lyapunov exponent is simple" by Richard Aoun, AUB.
Random matrix products theory has known deep contributions in the last few years due to its large spectrum of applications: expander graphs, geometric group theory, diophantine approximation... Roughly speaking, its goal is to study the behavior of a random walk on the general linear group GL(V ), where V is a finite dimensional vector space defined on any local field.
Fine results are known provided the law μ of the increments satisfies some algebraic assumption on the support of μ (irreducibility) and some dynamical assumption (proximality). In this talk, we present new results concerning this theory obtained with Pr. Yves Guivarc’h. In this work, we keep a dynamical assumption but consider no algebraic assumption on the support of μ. We show that this settings leads to new dynamics, essentially on a skew-product space. We show that there exists a unique stationary probability measure ν on the projective space of V corresponding to the top Lyapunov exponent. We describe the support of ν in terms of μ and relate it to the limit set of the semi-group of GL(V ) generated by the support of μ. Furthermore, we show that ν has Holder regularity and give new applications to random walks on the affine group.