  We consider Anosov flows on closed 3-manifolds preserving a volume form
$\Omega$. Following Dyatlov and Zworski (2017) we study spaces of invariant
distributions with values in the bundle of exterior forms whose wavefront set
is contained in the dual of the unstable bundle. Our first result computes the
dimension of these spaces in terms of the first Betti number of the manifold,
the cohomology class $[\iota_{X}\Omega]$ (where $X$ is the infinitesimal
generator of the flow) and the helicity. These dimensions coincide with the
Pollicott-Ruelle resonance multiplicities under the assumption of
$\textit{semisimplicity}$. We prove various results regarding semisimplicity on
1-forms, including an example showing that it may fail for time changes of
hyperbolic geodesic flows. We also study non null-homologous deformations of
contact Anosov flows and we show that there is always a splitting
Pollicott-Ruelle resonance on 1-forms and that semisimplicity persists in this
instance. These results have consequences for the order of vanishing at zero of
the Ruelle zeta function. Finally our analysis also incorporates a flat unitary
twist in both, the resonant spaces and the Ruelle zeta function.
