We study the resonant differential form at zero of the transitive Anosov flow on the $3$ manifold. We pay particular attention to the dissipative case, that is, the Anosov flow, which does not preserve absolute continuous measures. Such flows have two distinct Sinai-Ruelle-Bowen $3$ forms, $\Omega_{\text{SRB}}^{\pm}$, and cohomology class $.[\iota_{X}\Omega_{\text{SRB}}^{\pm}]$ (where $X$ is the flow’s infinitesimal generator) plays an important role in determining the space of resonant $1$ forms. When both classes disappear, we associate $\textit{helicity}$ with the flow. This is a natural extension of the classical concepts associated with null-homologous volume preservation flows. We provide a general theory involving the holocyclic invariance of resonant $1$ forms and SRB measures, and the local geometry of the map $X\mapsto.
[\iota_{X}\Omega_{\text{SRB}}^{\pm}]$ near null homologous volume-preserving flow. Now let’s look at some related example classes. Among these is the thermostat associated with the holomorphic second derivative, giving rise to the quasi-Fuchsian flow introduced by Ghys. For these flows, we explicitly compute all resonant $1$ forms at zero and denote $.[\iota_{X}\Omega_{\text{SRB}}^{\pm}]=0$, giving an explicit expression for helicity. Moreover, the general time evolution of the quasi-Fuchsian flow is semisimple, so the order of vanishing of the Luell zeta function at zero is $-\chi(M)$, the same as for the geodesic flow. indicates In contrast, when $(M,g)$ is a closed surface of negative curvature, a Gaussian thermostat driven by a (small) harmonic $1$ form has a Rouel We show that it has a zeta function. \chi(M)-1$.
