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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.

A periodic orbit obstruction is a necessary compatibility condition for solving a cohomological equation. If an dynamical observable is a flow coboundary, $h=Xu$, where $X$ differentiates along the flow, then on a periodic orbit of period $T$,A nonzero integral therefore rules out a single-valued transfer function $u$. Under suitable transitive Anosov flow and Hölder regularity hypotheses, the Livšic theorem provides a converse. Without those dynamical and regularity assumptions, vanishing periodic integrals alone is not a general existence theorem.^[ref]