= Kernel of the thermostat X-ray transform in degrees zero and one
{title2=$Fu=h\circ\pi+\theta_x(v)\Longrightarrow h=0,\ \theta=df$}
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 https://arxiv.org/abs/math/0601039.
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