Livsic theorem
= Livsic theorem
{c}
{title2=$Fu=h$}
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 $u(\phi_t x)-u(x)=\int_0^t h(\phi_sx)\,ds$. 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.