= Periodic orbit obstruction
{title2=$\int_0^T h(\Phi_t p)\,dt=0$}
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 $T$,
$$
\int_0^T h(\Phi_t p)\,dt=u(\Phi_Tp)-u(p)=0.
$$
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.<Abelian Livšic theorems for Anosov flows by Richard Sharp>[Introduction, classical periodic-orbit theorem and equation (1.1)]{ref}
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