On the simplicity of the spectrum of Lyapunov's characteristic indices of a product of random matrices

It is proved that the spectrum of Lyapunov's characteristic indices of a product of random matrices [4] is simple when multipliers form a stationary Markov chain on the group $SL(m,R)$ and the transitional probability of a chain satisfies some regularity conditions. When multipliers are independent and their distribution is absolutely continuous with respect to the Haar's measure on $SL(m,R)$ the simplicity of the spectrum of the characteristic indiced is a well-known result proved by V. N. Tutubalin [2] and (in a less explicite form) by H. Furstenberg [1]. The method of proof in the present paper is based on the development of some ideas of H. Furstenberg in [1] and generalizes the method of representations used in [8] (see also [9]–[11])