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The Anosov closing lemma turns a sufficiently close return of an Anosov flow into a nearby periodic orbit. On a compact manifold, uniform hyperbolicity ensures that an orbit segment returning close to its starting point is shadowed by a genuine closed orbit, with a small adjustment of the period and of the time parametrization.

Stable directions contract toward the shadowing orbit, while unstable directions are matched by choosing the appropriate nearby initial point. This makes periodic trajectories a tool for studying nonperiodic recurrence, for example when proving that periodic-orbit data control a cohomological equation. The time adjustment distinguishes the flow statement from its discrete-map counterpart.^[ref]

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, satisfyingif 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.^[ref]

Necessity is immediate by putting ${\Phi }_{T}p=p$ 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.^[ref] The theorem is named for A. N. Livšic.