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Scattering rigidity for Hamiltonian systems with an application to Finsler geometry

Nikolas Eptaminitakis, Plamen Stefanov

Abstract

We study scattering rigidity for Hamiltonian systems on $T^*M\setminus 0$, where $M$ is a manifold with boundary equipped with a positively homogeneous Hamiltonian function $H(x,ξ)$. We show that $H$ can be uniquely determined by the scattering relation up to a canonical transformation fixing the boundary (in a suitable sense) for positive energy levels $H=E>0$. We define the travel times $T(x,y)$ between boundary points, and show that their linearization leads to an X-ray transform over Hamiltonian curves, which we invert. When $E=0$, scattering rigidity can be formulated in terms of a diffeomorphism of the zero energy surfaces which preserves the boundary and respects the orbits of the Hamiltonian flows there, as well as the restricted symplectic form. The travel times are replaced by a defining function of pairs of boundary points which can be connected by a locally unique zero bicharacteristic. Its linearization leads to the "Hamiltonian light ray transform" which we invert modulo a gauge as well. As an application of this phase space approach, we prove semiglobal lens rigidity of non-trapping Finsler manifolds. The group of the gauge transformations consists of certain canonical transformations composed with Legendre transforms.

Scattering rigidity for Hamiltonian systems with an application to Finsler geometry

Abstract

We study scattering rigidity for Hamiltonian systems on , where is a manifold with boundary equipped with a positively homogeneous Hamiltonian function . We show that can be uniquely determined by the scattering relation up to a canonical transformation fixing the boundary (in a suitable sense) for positive energy levels . We define the travel times between boundary points, and show that their linearization leads to an X-ray transform over Hamiltonian curves, which we invert. When , scattering rigidity can be formulated in terms of a diffeomorphism of the zero energy surfaces which preserves the boundary and respects the orbits of the Hamiltonian flows there, as well as the restricted symplectic form. The travel times are replaced by a defining function of pairs of boundary points which can be connected by a locally unique zero bicharacteristic. Its linearization leads to the "Hamiltonian light ray transform" which we invert modulo a gauge as well. As an application of this phase space approach, we prove semiglobal lens rigidity of non-trapping Finsler manifolds. The group of the gauge transformations consists of certain canonical transformations composed with Legendre transforms.
Paper Structure (21 sections, 15 theorems, 91 equations, 3 figures)

This paper contains 21 sections, 15 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Positive energy levels. The scattering relation and the travel times
  3. Preliminaries
  4. The scattering relation and the travel times
  5. Rigidity results
  6. Linearization of the travel times
  7. Travel times T(x,y)
  8. A variational formula
  9. Linearization of T(x,y)
  10. T(x,y) as a generating function of S
  11. The X-ray transform
  12. Zero energy level: Scattering rigidity for operators of real principal type
  13. Setup
  14. Scattering rigidity
  15. Linearization
  16. ...and 6 more sections

Key Result

Lemma 2.1

If $\gamma(t) = (x(t),\xi(t))$ is a Hamiltonian curve at energy level $E$, then for every $\lambda \not=0$, $\tilde{\gamma}(t)=\mathcal{M}_\lambda \circ \gamma(\lambda t)=(x(\lambda t), \lambda\xi(\lambda t))$ is a Hamiltonian curve at energy level $\lambda^2E$. In terms of the flow map, $\Phi^t\cir

Figures (3)

  • Figure 1: The scattering relation $S$. The covectors $\xi$, $\xi'$, $\eta$ and $\eta'$ are not plotted. The domain between the dotted lines represents $\Gamma$, see \ref{['eq:Gamma']} (projected to the base).
  • Figure 2: The slowness surface $|\xi|_x=1$ (in 2D) in dual coordinates associated with boundary coordinates $(x',x^n)$ at $U$, where locally $x^n$ is a boundary defining function. Condition \ref{['A1']} means that the vertical line dropped from the vertex of $\xi$ is transversal to the surface there. It does not exclude existence of other $\xi$'s pointing into $M$ with the same restriction $\xi'$. For the $\xi$ pictured above, $\mathrm{d} x_n H_\xi =H_{\xi_{n}}<0$, indicating that $\xi$ is outgoing.
  • Figure 3: The canonical relation $\kappa$. The codirections are not plotted.

Theorems & Definitions (48)

  • Lemma 2.1
  • proof
  • Definition 2.1
  • Theorem 2.1: semi-global rigidity for Hamiltonian systems at a positive energy level
  • Remark 2.1
  • proof : Proof of Theorem \ref{['thm_H']}
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • ...and 38 more