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Hello everyone,

I have a quick question for people working on quasi-periodic Schrodinger operators, Lyapunov exponents for Schrodinger cocycles or in other fields that might make them aware of this topic. There is an inductive tool used to prove positivity or continuity of the Lyapunov exponent called the Avalanche Principle. It says something about the growth of large products of $SL_2(\mathbb{R})$ matrices.

Does anyone know if this principle has been extended in some form to higher dimensional matrices? If so, can you point me to a paper where it appears? I would like/need to extend it myself, but if the work has been done already, I'd rather not repeat it.

Thanks.

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What is this principle? In layman's terms, if possible... :) Thanks! –  Felix Goldberg Nov 6 '12 at 13:36
    
Roughly speaking, this principle provides a sufficient criterion for the norm of a product of matrices in $SL_2(\mathbb{R})$ to be close to the product of norms. Originally, this principle was developed by M. Goldstein and W. Schlag (see ams.org/mathscinet-getitem?mr=1847592) to study Lyapunov exponents of Schrodinger cocycles/spectrum of discrete Schrodinger operators (the almost Mathieu operator en.wikipedia.org/wiki/Almost_Mathieu_operator being a prominent example). In the next comment, I'll try to give a formal statement of the original avalanche principle. –  Matheus Nov 6 '12 at 15:01
    
Let $A_1,\dots, A_n\in SL_2(\mathbb{R})$ with $\|A_i\|\geq\mu>n$ and $$|\log(\|A_{j+1}\|\|A_j\|)-\log\|A_{j+1} A_j\||<\log\mu/2$$ Then, $$|\log\|A_n\dots A_1\|+\sum\limits_{j=2}^n\log\|A_j\|-\sum\limits_{j=1}^{n-1}\log\|A_{j+1}A_j\||<‌​Cn/\mu$$ where $C$ is an absolute (numerical) constant. –  Matheus Nov 6 '12 at 15:06
    
@Matheus: Thanks a lot! –  Felix Goldberg Nov 6 '12 at 18:41

1 Answer 1

up vote 2 down vote accepted

Dear Silvius,

I think the following recent paper of W. Schlag answers your question.

Best,

Matheus

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Hey Matheus, thanks a lot for your answer. Well, it turns out that someone else in the community had a similar version of the AP as well, and my collaborator and I also figured out an extension. But I am grateful for your pointing out to me W. Schlag's paper, as the problem he solves there would have been our next project :) –  Silvius Nov 16 '12 at 20:26

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