Holography of Geodesic Flows, Harmonizing Metrics, and Billiards' Dynamics (original) (raw)
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Spaces of pseudo-Riemannian geodesics and pseudo-Euclidean billiards
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Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. We discuss the geometry of these structures in detail, as well as introduce and study pseudo-Euclidean billiards. In particular, we prove pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.
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We consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non-perturbed ones. One condition says that the "Hamilton-Jacobi equation" (introduced in this work) for the conjugation problem should admit a solution as a formal power series (not necessarily convergent) in the perturbation parameter. The alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem. The relation with the similar problems arising in the KAM theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed. As a byproduct of our analysis we obtain some results on the Livscic, Guillemin, Kazhdan equation and on the Fourier series for the SL(2, R) group. We also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property (with respect to the geodesic flow and to the invariant volume measure) which is exponential with a universal exponent, independent on the particular function, equal to the curvature of the surface divided by 2. This result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow: in this case we find that the decay rate is the inverse of the time ("up to logarithms").
Causal holography in application to the inverse scattering problems
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For a given smooth compact manifold M , we introduce an open class G(M) of Riemannian metrics, which we call metrics of the gradient type. For such metrics g, the geodesic flow v g on the spherical tangent bundle SM → M admits a Lyapunov function (so the v g-flow is traversing). It turns out, that metrics of the gradient type are exactly the non-trapping metrics. For every g ∈ G(M), the geodesic scattering along the boundary ∂M can be expressed in terms of the scattering map C v g : ∂ + 1 (SM) → ∂ − 1 (SM). It acts from a domain ∂ + 1 (SM) in the boundary ∂(SM) to the complementary domain ∂ − 1 (SM), both domains being diffeomorphic. We prove that, for a boundary generic metric g ∈ G(M), the map C v g allows for a reconstruction of SM and of the geodesic foliation F (v g) on it, up to a homeomorphism (often a diffeomorphism). Also, for such g, the knowledge of the scattering map C v g makes it possible to recover the homology of M , the Gromov simplicial semi-norm on it, and the fundamental group of M. Additionally, C v g allows to reconstruct the naturally stratified topological type of the space of geodesics on M. We aim to understand the constraints on (M, g), under which the scattering data allow for a reconstruction of M and the metric g on it, up to a natural action of the diffeomorphism group Diff(M, ∂M). In particular, we consider a closed Riemannian n-manifold (N, g) which is locally symmetric and of negative sectional curvature. Let M is obtained from N by removing an n-domain U , such that the metric g| M is boundary generic, of the gradient type, and the homomorphism π 1 (U) → π 1 (N) of the fundamental groups is trivial. Then we prove that the scattering map C v g| M makes it possible to recover N and the metric g on it.
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