Positivity of the top Lyapunov exponent

Question

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.

Answer 1

The most comprehensive reference for this sort of thing seems to be Furstenberg, "Noncommuting random products", Trans. Amer. Math. Soc. 108 (1963), 377-428. I say this because I've seen it referenced in other places that consider similar questions; a quick glance through Furstenberg's paper suggests that some non-trivial study may be needed to answer your questions from his work.

A more direct treatment of your question can be found in Marcelo Viana's recent book "Lectures on Lyapunov Exponents" (Cambridge University Press). I'm not sure if the entire book is available yet but at the moment the first chapter is available on Viana's website. That chapter describes the case when $d=2$ and the matrices are chosen i.i.d., and quotes Furstenberg's paper as implying that the top Lyapunov exponent is positive as long as the monoid generated by $S$ satisfies a certain "pinching and twisting" condition; the proof, and the general case, seem to come later on in the book, in Chapters 6 and 7, which are not available online.

Answer 2

It should follow from a general theorem of Guivarc'h on simplicity of the Lyapunov spectrum for products of matrices with Markov dependence http://www.ams.org/mathscinet-getitem?mr=772409

Answer 3

I think that the canonical references (in the $SL(d, \mathbb{Z})$ case) is Goldsheid-Margulis and Goldsheid-Guivarc'h. For the positivity results to hold you need a moment condition (which is automatic when the support is finite, as it is in your case) and that the semi-group generated by the process is Zariski-dense (which is presumably what you mean by "large").

This is the canonical paper.

Goldsheid, I. Ya, and Yves Guivarc'h. "Zariski closure and the dimension of the Gaussian low of the product of random matrices. I." Probability theory and related fields 105.1 (1996): 109-142.

For groups other than $SL,$ there is a later paper by Guivarc'h alone. All these results are used in my preprint

Rivin, Igor. "Statistics of Random 3-Manifolds fibering over the circle." arXiv preprint arXiv:1401.5736 (2014).

which you might find interesting.