Scattering rigidity for Hamiltonian systems with an application to Finsler geometry

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.