Spectral rigidity and invariant distributions on Anosov surfaces

TL;DR

This paper proves spectral rigidity and injectivity of the geodesic ray transform on Anosov surfaces, establishing new results in inverse geometry and invariant distributions related to geodesic flows.

Contribution

It demonstrates spectral rigidity for all Anosov surfaces and proves injectivity and surjectivity of the geodesic ray transform on solenoidal tensors, advancing inverse problems in geometric analysis.

Findings

Spectral rigidity holds for any Anosov surface.

Injectivity of the geodesic ray transform on solenoidal 2-tensors is established.

Existence of invariant distributions with prescribed projections on Anosov surfaces.

Abstract

This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal tensors. The surjectivity results are of independent interest and imply the existence of many geometric invariant distributions on the unit sphere bundle. In particular, we show that on any Anosov surface $(M,g)$, given a smooth function $f$ on $M$ there is a distribution in the Sobolev space $H^{-1}(SM)$ that is invariant under the geodesic flow and whose projection to $M$ is the given function $f$.