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Consider irrational rotation $T:S^1\to S^1, T(x) = x + \alpha$ where $\alpha\notin \mathbb{Q}$ (you may assume additional number theoretic properties of $\alpha$, say $\alpha = \sqrt{2}$ is already interesting for me). Assume that we have a very concrete smooth map $g:S^1 \to GL(n, \mathbb{R})$ (it is given by some rather unpleasant but finite formula). I want to understand the behavior of $f(m, x) = g(T^{m-1}x)g(T^{m-2}x)\ldots g(Tx)g(x)$ for big $m$. As far as I know, this is done by the multiplicative ergodic theorem which reads as follows:

There are $k\le n$ real numbers $\lambda_1 > \ldots > \lambda_k$ and for a.e. $x\in S^1$ there are $k$ subspaces $\mathbb{R}^n = V_1(x) \supset V_2(x) \supset \ldots \supset V_k(x)$ such that for $v\in V_l(x) \backslash V_{l+1}(x)$ we have $\frac{\log ||f(m, x)v||}{m}\to \lambda_l$.

I want two things:

  1. Some rigorous algorithm to prove that none of $\lambda_l$ is equal to $1$ so that if I will do some finite computation with the function $g$ with enough precision I will prove it;

  2. Quantitive and, more important, uniform in $x$ limit from above. Actually, I will be satisfied with the bound of the following form: for $v\in V_l(x)$ we have $$||f(m, x)v|| \le C_\varepsilon (\lambda_l + \varepsilon)^m ||v||,$$ where $\varepsilon > 0$ is arbitrary small and $C_\varepsilon$ does not depends on $x$.

Note that for $n = 1$ both of these tasks are doable: in that case $\lambda_1 = \int \log |g(x)|dx$ so we just have to calculate this integral with enough precision to deduce 1. For 2. we can subdivide $S^1$ into many small arcs and approximate the integral by the Riemann sum and then say that irrational rotation visits each arc approximately equal number of times (to quantify the first part we need some knowledge about modulus of continuity of $g$ (that is smoothness), while for the second part we need to know that $\alpha$ does not have any good rational approximations though with some abstract $C_\varepsilon$ the bound from 2. is true for any $g\in C(S^1)$ and any $\alpha \notin\mathbb{Q}$).

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  • $\begingroup$ Have you tried checking out Chapter 6 of Duarte and Klein's book? It deals with Cocycles whose base dynamics are irrational rotations satisfying diophantine approximation, so you should be able to find a partial answer there (sorry, I don't know the book myself, a colleague recommended it after I showed him your question). $\endgroup$
    – Dan Rust
    Commented Jul 16, 2019 at 21:42
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    $\begingroup$ Interesting question. A couple of comments: (1) the MET. actually gives you something stronger than you state- $\mathbb R^n$ may be expressed as a direct sum of subspaces with expansion rates $\lambda_1$,...,$\lambda_k$. (2) I guess you want none of the $\lambda_i$ to be 0? (although of course you can multiply the generator by a constant, adding a constant to the $\lambda_i$’s.). $\endgroup$
    – Anthony Quas
    Commented Jul 17, 2019 at 9:58
  • $\begingroup$ Check this paper. You need domination for your questions. This kind of cocycle is called "quasi periodic cocycle". As @DanRust mentioned, Duarte and Klein's book explain it. $\endgroup$
    – Adam
    Commented Jul 17, 2019 at 13:44
  • $\begingroup$ @RR thanks, I'll take a look into this! $\endgroup$
    – Aleksei Kulikov
    Commented Jul 18, 2019 at 14:51
  • $\begingroup$ Property (2) is true for $\ell=1$, as a consequence of Theorem 1 of this old paper by Furman, or as a consequence of the more general Theorem A.3 of this paper by Morris. More generally, property (2) holds for any value of $\ell$ such that $V_\ell$ is a continuous subbundle (e.g. the dominated subbundle of a dominated splitting): just restrict the cocycle to this subbundle and use Morris' theorem. $\endgroup$
    – Jairo Bochi
    Commented Jul 18, 2019 at 19:10

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