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

TL;DR

This paper investigates the injectivity properties of the X-ray transform on Anosov thermostats on closed surfaces, showing finite-dimensional failure cases and establishing injectivity under negative curvature and pure Gaussian conditions.

Contribution

It proves that the non-injective subspace of the X-ray transform is finite dimensional and confirms injectivity for negatively curved surfaces with pure Gaussian thermostats.

Findings

Finite dimensional subspace where injectivity fails.

Injectivity holds for negatively curved surfaces with pure Gaussian thermostats.

Results extend understanding of X-ray transform in dynamical systems.

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.