# Random matrix products when the top Lyapunov exponent is simple

## TL;DR

This paper studies the behavior of random matrix products on general linear groups over local fields, establishing the existence, uniqueness, and regularity of stationary measures without requiring irreducibility, and generalizing previous results.

## Contribution

It introduces new results on stationary measures for random matrix products with simple top Lyapunov exponent, without the irreducibility assumption, and describes their support and regularity.

## Key findings

- Existence and uniqueness of stationary measure $
u$ on $ extrm{P}(V)$.
- H"older regularity of the stationary measure.
- Description of the limit set related to the semi-group $T_{
u}$.

## 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.