Quantitative Mixing Rates for Anosov Flows on Hyperbolic 3-Manifolds

We establish sharp quantitative estimates for the mixing rates of contact Anosov flows on closed hyperbolic 3-manifolds. By implementing a microlocal approach to the spectral theory of the Ruelle transfer operator, we construct a family of anisotropic Banach spaces of distributions adapted to the hyperbolic dynamics. Within this framework, we prove that the generator of the flow possesses a discrete spectrum of Ruelle resonances with a uniform spectral gap in the complex half-plane. This spectral gap implies an exponential decay of correlations for Hölder continuous observables. Our analysis relies on a rigorous adaptation of Dolgopyat's method for contact flows, exploiting the non-integrability of the temporal and strong stable/unstable foliations inherent to the contact structure. Furthermore, we discuss the implications of these results for the analytic continuation of the Ruelle zeta function and the distribution of closed geodesics.