# What is Birkhoff's ergodic theorem for $GL_n$?

I was wondering about the general setup of a "noncommutative" ergodic theory. Instead of measurable maps into $\mathbb R$, we should consider measurable maps into $GL_n(\mathbb R)$, and instead of adding real numbers, we have matrix multiplication etc.. I have the following specific questions:

1. To consider ergodic theoretical questions for maps from a probability space $X$ to $GL_n(\mathbb R)$, what is a suitable norm on $GL_n(\mathbb R)$ ?

2. More specifically, suppose in some situation one needs to define a Hölder continuous map $X \rightarrow GL_n(\mathbb R )$. Would one here be taking a sub-multiplicative norm on $GL_n(\mathbb R )$, or would one consider its logarithm? Are there issues with taking the logarithm of the norm or would everything work okay?

3. What is the analogous statement of Birkhoff's ergodic theorem?

4. What some good general references in which such matters are described, and proved?

Thanks,

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What you are looking for is perhaps the multiplicative ergodic theorem. At one point I tried to learn about this seriously, but I never got far enough, so I can't say anything too serious about it. (See en.wikipedia.org/wiki/Oseledets_theorem) –  Akhil Mathew Sep 24 '11 at 16:38