Let $A(x,n)$ be the cocycle over $f$, where $f$ is a measure-preserving transformation on a probability space $X$. Is the largest Lyapunov exponent always given by:
$\lim_{n\to +\infty} \log \|A(x,n)\|$?
Since the above limit can be bounded from above by $\int_X\log \|A\|$, can one give an example of the cocycle where the above inequality is strict? Tnx!
Hi. First, I suppose that the Lyapunov exponent is given by $$ \lim_{n\to\infty} \frac{1}{n} \int \log\|A(x,n)\| d\mu(x), $$ where $\mu$ is an appropriate ergodic measure. (You have some base dynamics for the cocycle, i.e. $A(x,n + m ) = A(T^n x, m) A(x,n)$ and $T$ is $\mu$ ergodic.
Then $$ \lim_{n\to\infty} \frac{1}{n} \log\|A(x,n)\|, $$ for almost every $x$, not for every. This follows from the subadditive ergodic theorem.
The inequality is very rarely strict. For the simplest example, consider a dynamic over a one-point space given by $A(x,n) = A^n$ for $$ A = B \begin{pmatrix} 2 & 0 \\\ 0 & \frac{1}{2} \end{pmatrix} B^{-1}. $$ It is easy to check that the Lyapunov exponent will be $2$, but using an appropriate choice of $B$, one can make $\|A\|$ arbitrarily large.
