# Relationship between Multiplicative Ergodic Theorems

One version of Oseledets' Multiplicative Ergodic Theorem states that if $\sigma$ is an ergodic measure-preserving transformation of a space $(\Omega,\mathbb P)$ and if $A\colon\Omega\to GL(d,\mathbb R)$ is such that $\log\|A(\omega)\|$ and $\log\|(A(\omega))^{-1}\|$ are integrable, then there exist $k\le d$, $\infty>\lambda_1>\ldots>\lambda_k>-\infty$, $d_1,\ldots,d_k$ such that $d_1+\ldots+d_k=d$ and measurable maps $V_i\colon\Omega\to \mathcal G(d,d_i)$ (the Grassmannian of $d_i$-dimensional subspaces of $\mathbb R^d$) such that:

1. $A(\omega)V_i(\omega)=V_i(\sigma(\omega))$ a.s. (equivariance);
2. $V_1(\omega)\oplus \ldots\oplus V_k(\omega)=\mathbb R^d$ a.s. (direct sum decomposition);
3. For almost every $\omega$ and all $v\in V_i(\omega)\setminus\{0\}$, $\frac 1n\log\|A(\sigma^{n-1}\omega) \cdots A(\omega)v\|\to \lambda_i$ and $\frac 1n\log \| A(\sigma^{-n}\omega)^{-1}\cdots A(\sigma^{-1}\omega)^{-1}v\|\to-\lambda_i$.
4. $\sum_{i=1}^k d_i\lambda_i=\int \log|\det A(\omega)|\,d\mathbb P(\omega)$.

Karlsson and Margulis (in a 1999 paper in CMP) proved a generalization of this theorem, where instead of a matrix cocycle, one has a cocycle mapping into the semi-contractions of a Hadamard metric space $(X,d)$: $a\colon \Omega\to SC(X)$ (a semi-contraction being a map of Lipschitz constant at most 1). They prove that given such a cocycle, and a measurable map $b\colon \Omega\to X$, $\lim_{n\to\infty}\frac 1nd(a(\omega)\circ a(\sigma\omega)\circ a(\sigma^{n-1}\omega)(b(x)),b(x))$ exists a.s; and $a(\omega)\circ a(\sigma\omega)\circ a(\sigma^{n-1}\omega)(b(x))$ shadows a geodesic at a sub-linear distance.

They then explain how the Oseledets theorem is a corollary of their theorem. Unfortunately, that part of the paper is extremely brief and assumes an understanding of the symmetric space $GL(d,\mathbb R)/O(d)$ (they don't directly describe the metric, but combining some statements in the paper, it appears from the paper that $d(A,B)$ is the Euclidean norm of the logarithms of the singular values of $A^{-1}B$ - I don't know how to prove this is a metric on $GL(d,\mathbb R)/O(d)$ for example; still less how to prove that it satisfies the required non-positive curvature conditions).

I'm looking for either a reference carrying out this deduction slowly, explaining explaining the necessary background; or failing that a reference giving enough background to make it feasible to understand the Karlsson-Margulis deduction.
• A classical reference for symmetric space of non-positive curvature is Eberlein's book "Geometry of Nonpositively Curved Manifolds", but it takes some persistence to get to a good understanding of it (and it might contain logical loops, if my memory serves me well). – Benoît Kloeckner May 5 '16 at 11:38

## 3 Answers

The equivalence of "sublinear shadowing" property and the Oseledec theorem (or, rather, of what is called "Lyapunov regularity") is due to Kaimanovich MR0947327 (89m:22006), and is explained there in detail (actually, this is a purely geometric property which has nothing to do with random products). By the way, your formulation of the Oseledec theorem is incomplete: one should add that (4) the sum of the Lyapunov exponents (taken with their multiplicities) should be equal to the exponent of the determinant (the expectation of $\log\det A(\omega)$).

• after reading Kaimanovich, "Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups" you could also read "Noncommutative Ergodic Theorems" by Karlsson-Ledrappier and "Subadditive and mutiplicative ergodic theorems" by Karlsson-Gouezel. – Uri Bader May 5 '16 at 6:48
• As a complement to the answer of RW and the comment of Uri Bader, you might want to take a look at the lecture notes (math.uchicago.edu/~sfilip/MET_lectures.pdf) of Simion Filip. – Matheus May 5 '16 at 7:00

It seems to me that the OP is asking specifically what the metric on the symmetric space is, and why it is non-positively curved. These facts are discussed in Bhatia's paper (2003), where he refers to Lang's 1999 differential geometry book and Ballmann/Gromov/Schroeder for proofs (Bhatia's paper is relevant to the question as well).

The question has already been answered, so let me just complement what has been said.

The book which I find most accessible is Lang's Fundamentals of Diff. Geom. (it was recommended to me by Karlsson himself). Another book that discusses SL(d,R)/O(d) specifically is Bridson & Haefliger's Metric Spaces of Non-positive curvature.

Finally, the details of the relation between Oseledets and Karlsson-Margulis have been worked out by my former MSc student Cong Zhou. Here's a link for his dissertation, which unfortunately is written in Portuguese.