# Oseledets multiplicative ergodic theorem

Oseledets multiplicative ergodic theorem, or Oseledets decomposition, considerably extends the results of Furstenberg-Kesten theorem, under the same conditions.

Consider $\mu$ a probability measure, and $f:M\rightarrow M$ a measure preserving dynamical system. Consider $A:M\rightarrow GL(d,\textbf{R})$, a measurable transformation, where GL(d,R) is the space of invertible square matrices of size $d$. Consider the multiplicative cocycle $(\phi^{n}(x))_{n}$ defined by the transformation $A$, and assume $\log^{+}||A||$ and $\log^{+}||A^{-1}||$ are integrable.

Then, $\mu$ almost everywhere $x\in M$, one can find a natural number $k=k(x)$ and real numbers $\lambda_{1}(x)>\cdots>\lambda_{k}(x)$ and a filtration

 $\textbf{R}^{d}=V_{x}^{1}>\cdots>V_{x}^{k}>V_{x}^{k+1}=\{0\}$

such that, for $\mu$ almost everywhere and for all $i\in\{1,\dots,k\}$

1. 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. 2.

$\lim_{n}\frac{1}{n}\log||\phi^{n}(x)v||=\lambda_{i}(x)$ for all $v\in V_{x}^{i}\backslash V_{x}^{i+1}$;

3. 3.

$\lim_{n}\frac{1}{n}\log|\det\phi^{n}(x)|=\sum_{i=1}^{k}d_{i}(x)\lambda_{i}(x)$ 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 $\lambda_{1}=\lambda_{\max}$ and $\lambda_{k}=\lambda_{\min}$, where $\lambda_{max}$ and $\lambda_{\min}$ are as given by Furstenberg-Kesten theorem.

Title Oseledets multiplicative ergodic theorem OseledetsMultiplicativeErgodicTheorem 2014-03-26 14:21:35 2014-03-26 14:21:35 Filipe (28191) Filipe (28191) 6 Filipe (28191) Theorem msc 37H15 Oseledets decomposition Lyapunov exponent Furstenberg-Kesten Theorem