= Cocycle over a dynamical system
{title2=$c(s+t,x)=c(s,\Phi_t x)+c(t,x)$}
An additive <cocycle over a dynamical system> records a quantity accumulated along a <flow>. It satisfies $c(0,x)=0$ and $c(s+t,x)=c(t,x)+c(s,\Phi_t x)$: accumulation over two intervals equals accumulation over the first interval plus accumulation over the second interval from the evolved state.
For an <dynamical observable> $h$, the integral $c(t,x)=\int_0^t h(\Phi_u x)\,du$ is an example, as splitting the integral at $t$ proves the identity. A <coboundary> has the special form $c(t,x)=u(\Phi_t x)-u(x)$. Matrix-valued multiplicative versions replace addition by an ordered product; the order matters when the matrices do not commute.
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