On the range of the attenuated magnetic ray transform for connections and Higgs fields
Gareth Ainsworth Yernat M. Assylbekov
Inverse Problems & Imaging 2015, 9(2): 317-335 doi: 10.3934/ipi.2015.9.317
For a two-dimensional simple magnetic system, we study the attenuated magnetic ray transform $I_{A,\Phi}$, with attenuation given by a unitary connection $A$ and a skew-Hermitian Higgs field $\Phi$. We give a description for the range of $I_{A,\Phi}$ acting on $\mathbb{C}^n$-valued tensor fields.
keywords: inverse problems Ray transforms tensor tomography magnetic geodesics.
The attenuated magnetic ray transform on surfaces
Gareth Ainsworth
Inverse Problems & Imaging 2013, 7(1): 27-46 doi: 10.3934/ipi.2013.7.27
It has been shown in [10] that on a simple, compact Riemannian 2-manifold the attenuated geodesic ray transform, with attenuation given by a connection and Higgs field, is injective on functions and 1-forms modulo the natural obstruction. Furthermore, the scattering relation determines the connection and Higgs field modulo a gauge transformation. We extend the results obtained therein to the case of magnetic geodesics. In addition, we provide an application to tensor tomography in the magnetic setting, along the lines of [11].
keywords: Ray transforms tensor tomography. inverse problems
The magnetic ray transform on Anosov surfaces
Gareth Ainsworth
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1801-1816 doi: 10.3934/dcds.2015.35.1801
Assume (M,g,$\Omega$) is a closed, oriented Riemannian surface equipped with an Anosov magnetic flow. We establish certain results on the surjectivity of the adjoint of the magnetic ray transform, and use these to prove the injectivity of the magnetic ray transform on sums of tensors of degree at most two. In the final section of the paper we give an application to the entropy production of magnetic flows perturbed by symmetric 2-tensors.
keywords: X-ray transform Anosov surface geodesics.

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