# Furstenberg-Kesten theorem

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 $A$.

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

 $\lambda_{\max}(x)=\lim_{n}\frac{1}{n}\log||\phi^{n}(x)||$

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

 $\int\lambda_{\max}d\mu=\lim_{n}\frac{1}{n}\int\log||\phi^{n}||d\mu=\inf_{n}% \frac{1}{n}\int\log||\phi^{n}||d\mu$

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

 $\lambda_{\min}(x)=\lim_{n}-\frac{1}{n}\log||\phi^{-n}(x)||$

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

 $\int\lambda_{\min}d\mu=\lim_{n}-\frac{1}{n}\int\log||\phi^{-n}||d\mu=\sup_{n}-% \frac{1}{n}\int\log||\phi^{-n}||d\mu$

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

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

and

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