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 $f:M\to M$ a measure preserving dynamical system. Consider $A:M\to GL(d,\text{𝐑})$, 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 ${\log }^{+}||A||$ 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 ${\lambda }_1(x)>\mathrm{\cdots }>{\lambda }_k(x)$ and a filtration

$${\text{𝐑}}^d=V_x^1>\mathrm{\cdots }>V_x^k>V_x^{k+1}=\{0\}$$

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. ${lim}_n\frac{1}{n}\log ||{\varphi }^n(x)v||={\lambda }_i(x)$ for all $v\in V_x^i\backslash \backslash V_x^{i+1}$;

3. ${lim}_n\frac{1}{n}\log |det{\varphi }^n(x)|={\sum }_{i=1}^k{d}_i(x){\lambda }_i(x)$ 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 ${\lambda }_1={\lambda }_{\max }$ and ${\lambda }_k={\lambda }_{\min }$, where ${\lambda }_{max}$ and ${\lambda }_{\min }$ are as given by Furstenberg-Kesten theorem.