# 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

Although I'm hardly qualified, let me try to answer three of your questions.

Question 1

In the simple case I'm going to talk about, I don't think it matters which norm you pick. This is because if $V$ is a finite-dimensional real vector space, then $\operatorname{End}(V)$ is too, and any two norms $\|\cdot\|$ and $\|\cdot\|'$ on a finite-dimensional real vector space are equivalent in the sense that $\log \|\cdot\| - \log \|\cdot\|'$ is bounded.

Question 3

As Akhil Mathew said, the noncommutative analogue of Birkhoff's ergodic theorem is Oseledec's multiplicative ergodic theorem. There are several different versions of it. Here's a sketch of a version for autonomous discrete-time dynamical systems, based on the notes by Quas mentioned below.

Say $X$ is a probability space equipped with an ergodic map $\sigma \colon X \to X$. I like to imagine a marble that starts at some point $x \in X$ and then rolls through a mess of pipes and lifts and kickers and stuff and finally ends up at some other point $\sigma(x) \in X$, and then it keeps doing that over and over.

Say the marble carries a "charge" taking values in a finite-dimensional real vector space $V$. Maybe the marble is spinning, for example, and the charge is its angular momentum vector. When the marble goes through the machine, its charge might be different when it comes back. What happens to the charge depends on how the marble goes through the machine, which is determined by where the marble starts. A marble that starts at $x \in X$ with charge $v \in V$ will end up at $\sigma(x)$ with charge $A(x)\,v$, where $A$ is a measurable function from $X$ to $\operatorname{End}(V)$.

As the marble keeps rolling through the machine, passing through the points $x_1, x_2, x_3, \ldots$, it passes through with charges

\begin{align*} v_1 \\ v_2 & = A(x_1)\,v_1 \\ v_3 & = A(x_2)\,A(x_1)\,v_1 \\ \vdots \end{align*}

What happens to the charge $v_n$ in the long term, as $n \to \infty$? To find out, pick norms on $V$ and $\operatorname{End}(V)$—I don't think it matters which ones, as I said before. If $\log \|A\|$ is an integrable function on $X$, the multiplicative ergodic theorem gives a nice answer, in the form of...

• A sequence of nested subspaces $$V = V_1(x) \supset V_2(x) \supset \ldots \supset V_m(x) \supset V_{m + 1}(x) = 0,$$ defined at almost every $x \in X$, called the Lyapunov filtration.

• A sequence of numbers $\lambda_1 > \lambda_2 > \ldots > \lambda_m$, called the Lyapunov exponents.

such that...

• If the marble's charge starts in $V_j$, it stays in $V_j$. In other words, $A(x)$ sends $V_j(x)$ into $V_j(\sigma(x))$.

• In the long term, charges in $V_j \smallsetminus V_{j+1}$ grow exponentially with exponent $\lambda_j$. In other words, $$\lim_{n \to \infty} \frac{1}{n} \log \|v_n\| = \lambda_j$$ whenever $v_1 \in V_j(x_1)$. Notice that this presupposes that $x_1$ is one of the points where the Lyapunov filtration is defined.

By the way...

• The subspace $V_j(x)$ has the same dimension at every $x \in X$ where it's defined, and it varies measurably with respect to $x$.

Question 4

The lecture notes

are really nice. You might also try the notes

especially if you're interested in the more geometric scenario where you follow the marble all the way through the machine, instead of just looking at "snapshots." In this case, the marble machine is a manifold, and the charge takes values in a flat vector bundle over it.

The bibliography of Kelliher's notes is very descriptive, and mentions several references with detailed and not-so-detailed proofs. I like the look of the proof in

• "A proof of Oseledec's multiplicative ergodic theorem," by M.S. Raghunathan,

although I haven't read it closely.

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