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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.

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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 $$\lim_{n\to\infty} \int \log\|A(x,n)\|,$$ 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.

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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 $$\lim_{n\to\infty} \int \log\|A(x,n)\|,$$ 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.