= Anosov closing lemma
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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.<Inverse Problems in Geometry and Dynamics by Will J. Merry and Gabriel P. Paternain>[Chapter 3, Anosov flows]{ref}
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