Loop group factorization method for the magnetic and thermostatic nonabelian ray transforms

TL;DR

This paper proves the injectivity of nonabelian ray transforms on surfaces with boundary for certain magnetic and thermostatic flows, using a loop group factorization approach that extends previous methods to specific cases.

Contribution

It generalizes the loop group factorization method to cover magnetic and thermostatic flows, settling the injectivity question for simple magnetic flows in the nonabelian setting.

Findings

Proves injectivity of nonabelian ray transforms for magnetic flows.

Extends loop group factorization to the unitary group for specific Fourier degrees.

Identifies limitations of the method for higher Fourier modes.

Abstract

We study the injectivity of the matrix attenuated and nonabelian ray transforms on compact surfaces with boundary for nontrapping $\lambda$-geodesic flows and the general linear group of invertible complex matrices. We generalize the loop group factorization argument of Paternain and Salo to reduce to the setting of the unitary group when $\lambda$ has the vertical Fourier degree at most $2$. This covers the magnetic and thermostatic flows as special cases. Our article settles the general injectivity question of the nonabelian ray transform for simple magnetic flows in combination with an earlier result by Ainsworth. We stress that the injectivity question in the unitary case for simple Gaussian thermostats remains open. Furthermore, we observe that the loop group argument does not apply when $\lambda$ has higher Fourier modes.