= Solution
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 $u$ satisfying
$$
\boxed{u(\phi_t x)-u(x)=\int_0^t f(\phi_sx)\,ds\quad\text{for all }x,t.}
$$
Equivalently, $u$ is differentiable along the flow and $Fu=f$. The periodic obstruction is
$$
\phi_Tx=x\quad\Longrightarrow\quad\int_0^T f(\phi_sx)\,ds=0.
$$
It 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).
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