Furstenberg-Kesten theorem

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 $A$.

If ${\log }^{+}||A||$ is integrable, where ${\log }^{+}||A||=\max \{\log ||A||,0\}$, then:

$${\lambda }_{\max }(x)=\underset{n}{lim}\frac{1}{n}\log ||{\varphi }^n(x)||$$

exists almost everywhere, and ${\lambda }_{\max }^{+}$ is integrable and

$$\int {\lambda }_{\max }𝑑\mu =\underset{n}{lim}\frac{1}{n}\int \log ||{\varphi }^n||d\mu =\underset{n}{inf}\frac{1}{n}\int \log ||{\varphi }^n||d\mu$$

If ${\log }^{+}||A^{-1}||$ is integrable, then:

$${\lambda }_{\min }(x)=\underset{n}{lim}-\frac{1}{n}\log ||{\varphi }^{-n}(x)||$$

exists almost everywhere, and ${\lambda }_{\min }^{+}$ is integrable and

$$\int {\lambda }_{\min }𝑑\mu =\underset{n}{lim}-\frac{1}{n}\int \log ||{\varphi }^{-n}||d\mu =\underset{n}{sup}-\frac{1}{n}\int \log ||{\varphi }^{-n}||d\mu$$

Furthermore, both ${\lambda }_{\min }$ and ${\lambda }_{\max }$ are invariant for the tranformation $f$, that is, ${\lambda }_{\min }\circ f(x)={\lambda }_{\min }(x)$ and ${\lambda }_{\max }\circ f(x)={\lambda }_{\max }(x)$, for $\mu$ almost everywhere.

This theorem is a direct consequence of Kingman’s subadditive ergodic theorem, by observing that both

$$\log ||{\varphi }^n(x)||$$

and

$$\log ||{\varphi }^{-n}(x)||$$

are subadditive sequences.

The results in this theorem are strongly improved by Oseledet’s multiplicative ergodic theorem, or Oseledet’s decomposition.