Rapid mixing and superpolynomial equidistribution for torus extensions of hyperbolic flows

In this paper, we study error estimates on the equidistribution of holonomies around closed orbits and mixing rates for Tᵈ-extensions of hyperbolic flows. Given three closed orbits with their holonomies, we can relate them to a point in Rᵈ⁺¹. We prove that the extension flow enjoys rapid mixing as well as superpolynomial equidistribution, if the associated point is an inhomogeneously Diophantine number. Additionally, we also provide examples of frame flows that meet this criterion.