Random matrix products when the top Lyapunov exponent is simple.

Academic Journal

Journal of the European Mathematical Society (EMS Publishing); 2020, Vol. 22 Issue 7, p2135-2182, 48p

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 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. Then, we relate this support to the limit set of the semi-group Tμ of GL(V) generated by the random walk. Moreover, we show that ν has H\"older regularity and give some limit theorems concerning the behavior of the random walks: exponential convergence in direction, large deviation estimates of the probability of hitting an hyperplane. These results generalize known ones when Tμ 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, 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. [ABSTRACT FROM AUTHOR]