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 defined on 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 $\nu$ on P$V$ that is relative 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 set of P$V$ which has the structure of a skew product space. Then, we relate this support to the limit set of the semi-group $T_{\mu }$ of GL$(V)$ generated by the random walk. Moreover, we show that $\nu$ 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 $T_{\mu }$ acts strongly irreducibly and proximally (i-p to abbreviate) on $V$. In particular, when applied to the affine group in the so-called contracting case or more generally when the Zariski closure of $T_{\mu }$ 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.