 Livsic theorem

For a topologically transitive Anosov flow, a Hölder function has a Hölder flow coboundary precisely when its integral around every periodic orbit vanishes. The coboundary identity is $u({\varphi }_{t}x)-u(x)={\int }_0^{t}h({\varphi }_{s}x)ds$. For smooth flow and smooth $h$, the smooth regularity theorem gives smooth $u$. Necessity follows by setting $t$ equal to a period. The solution is unique up to a constant by density of an orbit.