Smooth invariant function of an Anosov flow
= Smooth invariant function of an Anosov flow
{title2=$Fh=0\Longrightarrow dh=0$}
Invariance gives $dh(v_s)=dh(D\phi_t v_s)$ for a stable direction. Boundedness of $dh$ and exponential contraction make this zero; negative time does the same for unstable directions. Also $dh(F)=0$. The Anosov splitting spans the tangent bundle, so $dh=0$ and the function is constant on each connected component. This proof does not need an ergodicity hypothesis.