JournalsjemsVol. 22, No. 7pp. 2135–2182

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
Random matrix products when the top Lyapunov exponent is simple cover
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Abstract

In the present paper, we treat random matrix products on the general linear group GL(V)(V), where VV 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\nu on PVV 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 PVV which has the structure of a skew product space. Then, we relate this support to the limit set of the semi-group TμT_{\mu} of GL(V)(V) generated by the random walk. Moreover, we show that ν\nu 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 TμT_{\mu} acts strongly irreducibly and proximally (i-p to abbreviate) on VV. In particular, when applied to the affine group in the so-called contracting case or more generally when the Zariski closure of TμT_{\mu} 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