Spectral rigidity and invariant distributions on Anosov surfaces

By Gabriel P. Paternain, Mikko Salo, Gunther Uhlmann. 2012

59 citations recorded by the source corpus.

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 $f$.

Subfields: Astronomy and Astrophysics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b4c95d7cb1e80b082348c3062d2850c4e96a4d6a

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In Astronomy and Astrophysics

In Mathematical Physics

Citations in the corpus

Invariant distributions and X-ray transform for Anosov flows (2014, unknown): “The properties of our operator Π allows to solve the problem encountered in Remark 9.4 of [ PSU1 ], which was why they had to assume m ≤ 2.”
Invariant distributions, Beurling transforms and tensor tomography in higher dimensions (2014, methodology): “pherical harmonic expansions in the vertical variable, which generalize Fourier expansions in the angular variable for dimM= 2. Section 4 considers the notion of α-controlled manifolds (introduced in [PSU14c] for surfaces), and contains certain useful estimates. In Section 5we consider the Beurling ”