Random matrix products when the top Lyapunov exponent is simple

Abstract

In the present paper, we treat random matrix products on the general linear group GL(V ), where V is a vector space over any local field, when the top Lyapunov exponent is simple, without the irreducibility assumption. In particular, we show the existence and uniqueness of the stationary measure ν on P(V ) that is related to the top Lyapunov exponent and we describe the projective subspace generated by its support. We observe that the dynamics takes place in an open subset of P(V ) which has the structure of a skew product space. Then, we relate this support to the limit set of the semigroup Tµ of GL(V ) generated by the random walk. Moreover, we show that ν has Hölder regularity and give some limit theorems concerning the behavior of the random walk and the probability of hitting a hyperplane. These results generalize known ones when Tµ acts strongly irreducibly and proximally (i-p for short) on V. In particular, when applied to the affine group in the so-called contracting case or more generally when the Zariski closure of Tµ is not necessarily reductive, the Hölder regularity of the stationary measure together with the description of the limit set are new. We mention that we do not use results from the i-p setting; rather we see it as a particular case. © European Mathematical Society 2020