 Flow coboundary

A smooth function $h$ is a flow coboundary if $h=Fu$ for some smooth function $u$, where $F$ is the flow generator. Equivalently $u({\varphi }_{t}x)-u(x)={\int }_0^{t}h({\varphi }_{s}x)ds$. Thus its integral over every periodic orbit is zero. For a transitive Anosov flow, the Livsic theorem supplies the converse and smooth regularity.