Exponential Mixing and Decay of Correlations for Hyperbolic Flows on Polylogarithmic Surfaces

We establish exponential mixing and rapid decay of correlations for geodesic flows on a class of non-compact Riemannian surfaces of variable negative curvature, termed Polylogarithmic Surfaces. These manifolds are characterized by curvature functions K(x) that decay asymptotically as -(d(x, x0)) for > 2, inducing a weak hyperbolicity at infinity that typically suggests polynomial mixing rates. However, by constructing a specialized family of anisotropic Banach spaces adapted to the polylogarithmic scaling and employing a modified Dolgopyat-type cancellation argument for the Ruelle transfer operator, we prove that the flow retains a spectral gap. Consequently, we derive exponential decay estimates for Hölder continuous observables. This result extends the thermodynamic formalism of Anosov flows to a specific regime of infinite-volume geometrically infinite surfaces, bridging the gap between the rapid mixing of compact hyperbolic dynamics and the sub-exponential behavior observed in rank-one manifolds with heavy-tailed cusps.