I have a general question about the Oseledets Multiplicative Ergodic Theorem. In the context of the MET I'd like to know if there is some reasonably general sufficient condition which implies that the top Lyapunov exponent \lambda_1 is >0.
My specific situation involves a finite-state aperiodic irreducible Markov chain with values in SL(d,Z). Thus a random trajectory of this Markov chain is a sequence of matrices \omega=A_1,A_2,...A_n,... where all A_i come from a finite subset S of SL(d,Z). I also know that the stationary distribution on S for this chain is the uniform distribution on S, and that S generates a "large" subgroup of SL(d,Z).
As I understand it, in this situation, the MET implies that for a.e. trajectory \omega=A_1,A_2,...A_n,... of this Markov chain, for S_n=A_{n-1}...A_1 the operator norm of S_n grows as e^{\lambda_1 n} where \lambda_1 is the top Lyapunov exponent.
I'd like to be able to claim that in fact \lambda_1>0 in my case, so that ||S_n|| grows exponentially fast.
I hope that there is some general result implying positivity of \lambda_1 that could be applied here.
Thanks a lot,
Ilya Kapovich.