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For a volume form $\Omega$, define divergence by ${\mathcal{L}}_F\Omega =({\mathrm{div}}_{\Omega }F)\Omega$. The product rule for the Lie derivative gives ${\mathrm{div}}_{e^h\Omega }F={\mathrm{div}}_{\Omega }F+Fh$. Consequently the flow preserves a smooth positive volume precisely when its divergence in any reference volume is a smooth flow coboundary. The displayed density has the minus sign dictated by this product rule.

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.

The Livsic theorem says that for a topologically transitive Anosov flow on a closed manifold, a Hölder continuous function $f$ is a flow coboundary if and only if it has zero integral over every periodic orbit. In the integral formulation there is a Hölder continuous function $u$ satisfyingEquivalently, $u$ is differentiable along the flow and $Fu=f$. The periodic obstruction isIt is necessary because the left-hand side of the coboundary identity is zero at a period. The substantive theorem is its sufficiency. For a smooth flow and smooth $f$, the smooth regularity version gives a smooth $u$. Two such solutions differ by a constant: their difference is flow invariant and hence constant on a dense orbit and, by continuity, everywhere. This smooth version is the one needed in part (b).

Fix an orientation and a positive volume form $\Omega$. Define the divergence by the Lie derivativeA smooth function is a flow coboundary when it equals $Fu$ for some smooth $u$. Every other positive volume form is $e^h\Omega$ for a smooth $h$. The product rule for the Lie derivative givesThus changing the reference volume form changes its divergence by a flow coboundary. If volume forms of either orientation are allowed, take the logarithm of the absolute ratio on each connected component; multiplying a reference form by a constant sign leaves its divergence unchanged.

If ${\mathrm{div}}_{\Omega }F=Fu$, the preceding product rule showsConversely, if $e^h\Omega$ is invariant, then ${\mathrm{div}}_{\Omega }F=-Fh=F(-h)$ is a flow coboundary. Vanishing of the Lie derivative is equivalent to flow invariance, since ${\partial }_t{\varphi }_t^{\ast }\Omega ={\varphi }_t^{\ast }({\mathcal{L}}_F\Omega )$. HenceThe equivalence holds for every choice of reference volume form, proving the invariant volume criterion for a smooth flow. The smoothness requirement matters: a merely measurable transfer function does not give a smooth invariant volume form.