= Livšic theorem
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{title2=$Fu=h$}
The <Livšic theorem> turns periodic-orbit data into a global derivative along a hyperbolic flow. Let $\Phi_t$ be a transitive <Anosov flow> on a compact <smooth manifold>, with generating <vector field> $F$, and let $h$ be a real <Hölder continuous> observable. There is a <Hölder continuous> function $u$, differentiable along flow lines, satisfying
$$
u(\Phi_t x)-u(x)=\int_0^t h(\Phi_sx)\,ds
$$
if and only if $\int_0^T h(\Phi_t p)\,dt=0$ for every <periodic orbit> $p$ of period $T$. The solution preserves the Hölder exponent.<Abelian Livšic theorems for Anosov flows by Richard Sharp>[Introduction, equation (1.1) and its following paragraph: the classical all-periodic-orbit theorem]{ref}
Necessity is immediate by putting $\Phi_Tp=p$ in the identity. Two solutions differ by a continuous invariant function. Transitivity supplies a <dense orbit>; their difference is constant on that orbit, and continuity makes it constant everywhere. Thus the solution is unique up to an additive constant.
The harder sufficiency argument defines $u(\Phi_t x_0)$ by integrating $h$ along a dense orbit. The <Anosov closing lemma> converts a sufficiently close return into a nearby periodic trajectory. Its vanishing integral and exponential shadowing bound the discrepancy between the endpoint values in terms of their distance. This gives Hölder control on the dense orbit and permits extension to the whole manifold.
When both the flow and $h$ are smooth, a separate regularity theorem gives smooth $u$. This does not follow merely from the preceding continuity argument.<Canonical perturbation theory of Anosov systems, and regularity results for the Livsic cohomology equation by Rafael de la Llave and others (1985)>[Printed p. 92, Theorem 1 and proof sketch; p. 93 explicitly states the flow analogue]{ref} The theorem is named for <A. N. Livšic>.
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