= Flow coboundary
{title2=$h=Fu$}
An observable $h$ is a <flow coboundary> if it is the derivative of a function $u$ along a <flow map>. An integrated formulation is
$$
u(\Phi_t x)-u(x)=\int_0^t h(\Phi_sx)\,ds.
$$
When $u$ 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:
$$
\frac1T\int_0^T h(\Phi_t x)\,dt=\frac{u(\Phi_Tx)-u(x)}T\longrightarrow0.
$$
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.
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