Anosov closing lemma

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]