# Random matrix products when the top Lyapunov exponent is simple

### Richard Aoun

American University of Beirut, Lebanon### Yves Guivarc'h

Université de Rennes I, France

## 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 $ν$ 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 $T_{μ}$ 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 $T_{μ}$ 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 $T_{μ}$ 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.

## Cite this article

Richard Aoun, Yves Guivarc'h, Random matrix products when the top Lyapunov exponent is simple. J. Eur. Math. Soc. 22 (2020), no. 7, pp. 2135–2182

DOI 10.4171/JEMS/962