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 $\textrm{GL}(V)$, where $V$ is a vector space defined on any local field, when the top Lyapunov exponent is simple, without irreducibility assumption. In particular, we show the existence and uniqueness of the stationary measure $\nu$ on $\textrm{P}(V)$ 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 a open set of $\textrm{P}(V)$ 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 $\textrm{GL}(V)$ generated by the random walk. Moreover, we show that $\nu$ has H\"older 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 H\"older regularity of the stationary measure together with the description of the limit set are new. We mention that we don't use results from the i-p setting; rather we see it as a particular case.