Lens rigidity for manifolds with hyperbolic trapped set

Abstract

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $\gamma$ with endpoints on $\partial M$, together with their endpoints $(x_{-},x_{+})\in \partial M\times \partial M$ and tangent exit vectors $(v_{-},v_{+})\in T_{x_{-}}M\times T_{x_{+}}M$. We show deformation lens rigidity for a large class of manifolds which includes all manifolds with negative curvature and strictly convex boundary, possibly with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension $2$, we prove that the set of endpoints and exit vectors of geodesics (ie. the scattering data) determines the topology and the conformal class of the surface.

1 figure, 43 pages. Minor corrections (in Prop. 5.10) and some improvements: the integrability assumption on the non-escaping mass function is now removed from Theorem 3,4 and 5

References in corpus (2)

• Pollicott-Ruelle resonances for open systems
• Invariant distributions and X-ray transform for Anosov flows

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