= Contact rigidity of a zero-flux Anosov magnetic flow
On a connected closed oriented surface, a magnetic Anosov flow with $\int f\Omega_a=0$ preserves a smooth contact form only when $f=0$. To see the obstruction, take $d\eta=f\Omega_a$ and $\sigma=\alpha-\pi^*\eta$. An invariant contact form can be normalized to evaluate to one on $F$; its differential is a constant multiple of $\iota_F\mu$, since smooth invariant functions are constant. Averaging closed one-forms forces that multiple to be $-1$, so the contact form differs from $\sigma$ by a closed form. Its fibre-average decomposition gives $Fh=(\eta-\xi)(v)-af$. The degree-zero/one kernel theorem gives $af=0$ and $d\eta=d\xi=aK\Omega_a$, forcing $f=0$ whether $a$ vanishes or not. For $f=0$, the canonical form $\alpha$ is invariant and contact.
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