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- W2617884511 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 Holder 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 Holder 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." @default.
- W2617884511 created "2017-06-05" @default.
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- W2617884511 date "2017-01-01" @default.
- W2617884511 modified "2023-10-18" @default.
- W2617884511 title "Random matrix products when the top Lyapunov exponent is simple" @default.
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