The following linear ODE on $\Bbb{C}$

$\dot{z} = (a + i b)z$

has solutions $z(t) = e^{(a+ib)t} z(0)$. Hence the real part $a$ captures expansion rate and the imaginary part $b$ captures rotation rate.

It seems to me that this notion of expansion rate is generalized to a linear flow $\Phi^t$ on a normed vector bundle $\pi:V\to B$ through the notion of Lyapunov exponents $\lambda(v)$ of $v \in V$, where

$$\lambda(v) := \limsup_{t\to\infty}\frac{1}{t}\log |\Phi^tv|.$$

Question: Is it also possible to generalize the "rotation rate" to linear flows on vector bundles? Any explanation/references would be much appreciated.


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