On the injectivity of the X-ray transform for Anosov thermostats

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June 2009, 24(2): 471-487. doi: 10.3934/dcds.2009.24.471

On the injectivity of the X-ray transform for Anosov thermostats

Dan Jane ^1, and Gabriel P. Paternain ^1,

1.	Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom, United Kingdom

Received July 2008 Revised November 2008 Published March 2009

Abstract
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We consider Anosov thermostats on a closed surface and the X-ray transform on functions which are up to degree two in the velocities. We show that the subspace where the X-ray transform fails to be s-injective is finite dimensional. Furthermore, if the surface is negatively curved and the thermostat is pure Gaussian (i.e. no magnetic field is present), then the X-ray transform is s-injective.

Keywords: generalised thermostat, X-ray transform, Pestov identity..

Mathematics Subject Classification: Primary: 37D40, 53C65; Secondary: 45P0.

Citation: Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471

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