Rapid mixing and superpolynomial equidistribution for torus extensions of hyperbolic flows

In this paper, we study mixing rates for ${\mathbb{T}}^d$-extensions of hyperbolic flows. Given three closed orbits with their holonomies, we can relate them to a point in ${\mathbb{R}}^{d+1}$. We prove that the extension flow enjoys rapid mixing, if the associated point is an inhomogeneously Diophantine number. Under the same assumption, we also obtain the superpolynomial equidistribution, namely, a superpolynomial error term in the equidistribution of the holonomy around closed orbits. Lastly, we apply these results to a class of three-dimensional frame flows.