Periodic orbit obstruction (${\int }_0^{T}h({\Phi }_{t}p)dt=0$)

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 $T$,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.^[ref]