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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.

An observable $h$ is a flow coboundary if it is the derivative of a function $u$ along a flow map. An integrated formulation isWhen $u$ is differentiable along the flow, this is $Fu=h$, where $F$ is the generating vector field. Spatial regularity must be specified separately: $u$ may be smooth or merely Hölder continuous while still being differentiable along every flow line.

The function $c(t,x)=u({\Phi }_{t}x)-u(x)$ is a cocycle over a dynamical system. Its identity follows by adding the two differences from $x$ to ${\Phi }_{t}x$ and from ${\Phi }_{t}x$ to ${\Phi }_{s+t}x$. Being a coboundary says that the accumulated observable can be recovered from endpoint values.

For a periodic orbit of period $T$, the endpoint values coincide, so its integral of $h$ is zero. This is the periodic orbit obstruction. If $u$ is bounded, then the time average of $h$ also vanishes:For a transitive Anosov flow, the Livšic theorem identifies when the periodic-orbit obstruction is sufficient. This is stronger than checking just one average or finitely many periodic trajectories.