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

Dan Jane; Gabriel P. Paternain

doi:10.3934/dcds.2009.24.471

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2009, Volume 24, Issue 2: 471-487. Doi: 10.3934/dcds.2009.24.471

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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

Received: June 30, 2008

Revised: October 31, 2008

Published: May 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D40, 53C65; Secondary: 45P05.

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