Oseledets multiplicative ergodic theorem
Oseledets multiplicative ergodic theorem, or Oseledets decomposition, considerably extends the results of FurstenbergKesten theorem, under the same conditions.
Consider $\mu $ a probability measure^{}, and $f:M\to M$ a measure preserving dynamical system^{}. Consider $A:M\to GL(d,\text{\mathbf{R}})$, a measurable transformation, where GL(d,R) is the space of invertible square matrices of size $d$. Consider the multiplicative cocycle ${({\varphi}^{n}(x))}_{n}$ defined by the transformation $A$, and assume ${\mathrm{log}}^{+}A$ and ${\mathrm{log}}^{+}{A}^{1}$ are integrable.
Then, $\mu $ almost everywhere $x\in M$, one can find a natural number^{} $k=k(x)$ and real numbers ${\lambda}_{1}(x)>\mathrm{\cdots}>{\lambda}_{k}(x)$ and a filtration^{}
$${\text{\mathbf{R}}}^{d}={V}_{x}^{1}>\mathrm{\cdots}>{V}_{x}^{k}>{V}_{x}^{k+1}=\{0\}$$ 
such that, for $\mu $ almost everywhere and for all $i\in \{1,\mathrm{\dots},k\}$

1.
$k(f(x))=k(x)$ and ${\lambda}_{i}(f(x))={\lambda}_{i}(x)$ and $A(x)\cdot {V}_{x}^{i}={V}_{f(x)}^{i}$;

2.
${lim}_{n}\frac{1}{n}\mathrm{log}{\varphi}^{n}(x)v={\lambda}_{i}(x)$ for all $v\in {V}_{x}^{i}\backslash {V}_{x}^{i+1}$;

3.
${lim}_{n}\frac{1}{n}\mathrm{log}det{\varphi}^{n}(x)={\sum}_{i=1}^{k}{d}_{i}(x){\lambda}_{i}(x)$ where ${d}_{i}(x)=dim{V}_{x}^{i}dim{V}_{x}^{i+1}$
Furthermore, the numbers ${k}_{i}(x)$ and the subspaces ${V}_{x}^{i}$ depend measurably on the point $x$.
The numbers ${\lambda}_{i}(x)$ are called Lyapunov exponents of $A$ relatively to $f$ at the point $x$. Each number ${d}_{i}(x)$ is called the multiplicity of the Lyapunov exponent ${\lambda}_{i}(x)$. We also have that ${\lambda}_{1}={\lambda}_{\mathrm{max}}$ and ${\lambda}_{k}={\lambda}_{\mathrm{min}}$, where ${\lambda}_{max}$ and ${\lambda}_{\mathrm{min}}$ are as given by FurstenbergKesten theorem.
Title  Oseledets multiplicative ergodic theorem 

Canonical name  OseledetsMultiplicativeErgodicTheorem 
Date of creation  20140326 14:21:35 
Last modified on  20140326 14:21:35 
Owner  Filipe (28191) 
Last modified by  Filipe (28191) 
Numerical id  6 
Author  Filipe (28191) 
Entry type  Theorem 
Classification  msc 37H15 
Synonym  Oseledets decomposition 
Related topic  Lyapunov exponent 
Related topic  FurstenbergKesten Theorem 