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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]

Gabriel Paternain works on the interaction between differential geometry, dynamical systems and inverse problems. His notes with Will J. Merry connect the geodesic flow and Anosov flows to the X-ray transform and boundary rigidity. The guiding reconstruction problem asks how much of an interior Riemannian metric can be determined from measurements of trajectories or distances at the boundary.^[ref]

Let a nonsingular smooth flow on a closed manifold $N$ be generated by $F$. Suppose a continuous nondegenerate indefinite quadratic form $Q$ on the transverse vector bundle $TN/\mathbb{R}F$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative there. Then the flow is an Anosov flow.^[ref]

Here differentiating the form means differentiating $Q_{{\Phi }_{t}x}(D{\Phi }_{t}v)$. Compactness gives constants $M,c>0$ such that $|Q(v)|\le M\Vert v{\Vert }^2$ and $\dot{Q}(v)\ge c\Vert v{\Vert }^2$. On the positive cone,so the form increases at least exponentially. Negative cones supply the analogous backward-time estimate. Strict cone invariance produces uniformly expanding and contracting transverse bundles; the full splitting also contains the neutral flow direction.

For a geodesic flow on a compact negatively curved surface, a transverse Jacobi field has scalar coordinates satisfying $\dot{y}=z$ and $\dot{z}=-K(t)y$, with Gaussian curvature $K\le -\kappa <0$. The indefinite form $Q(y,z)=yz$ obeysfor every nonzero transverse vector. This directly verifies the criterion. The quotient by $F$ is essential: $D{\Phi }_{t}F=F\circ {\Phi }_t$ gives the neutral direction that cannot satisfy this strict transverse positivity. The estimates concern the form; constructing the invariant bundles is the additional cone argument, rather than an assertion that every positive-cone vector has monotonically growing norm.

Will J. Merry coauthored notes with Gabriel Paternain on inverse problems in geometry and dynamical systems. They develop geodesic flows, the Anosov flows and Livšic theorem as tools for reconstruction. A recurring idea is to infer information about a function or a Riemannian metric from integrals along trajectories, rather than from direct measurements at every interior point.^[ref]