Cocycle over a dynamical system ($c(s+t,x)=c(s,{\Phi }_{t}x)+c(t,x)$)

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.