 Kernel of the thermostat X-ray transform in degrees zero and one

For a smooth Anosov flow generated by a surface thermostat on a closed oriented Riemannian surface, a smooth base function plus a smooth one-form is a smooth flow coboundary precisely when the base function is zero and the one-form is an exact differential form. This is the nontrivial degree-zero/one kernel theorem. It does not require negative curvature or preservation of Liouville volume. The converse follows directly from $F(f\circ \pi )=df(v)$. A primary proof of the kernel theorem is arxiv.org/abs/math/0601039.