RAPID MIXING AND SUPERPOLYNOMIAL EQUIDISTRIBUTION FOR TORUS EXTENSIONS OF HYPERBOLIC FLOWS

The paper proves rapid mixing for T^3-extensions and superpolynomial equidistribution of holonomies from an inhomogeneously Diophantine triple of closed orbits by: (i) passing to a symbolic suspension flow; (ii) Fourier-decomposing along the T^3-fibres; (iii) expressing Laplace-transformed correlations as sums of twisted transfer operators ℒ_{!−BA+i⟨m,Θ⟩}; and (iv) establishing a two-parameter Dolgopyat-type estimate (Proposition 3.5), which yields an analytic strip for the Laplace transform with width depending polynomially on |m| (Corollary 3.6), followed by integration by parts and (for equidistribution) the standard prime orbit machinery. These steps appear explicitly in the statement of the main results (Theorems 1.2 and 1.3), the symbolic reduction, and the Dolgopyat estimate and its corollary in §3–§5 of the uploaded PDF . The candidate solution mirrors this strategy, including the Fourier-mode reduction and a uniform (in b=Im s and m) Dolgopyat estimate leading to superpolynomial decay via Laplace inversion, and an equidistribution argument via twisted zeta/prime orbit methods. The only discrepancy is a side remark claiming the triple inhomogeneous Diophantine hypothesis implies Dolgopyat’s original (two-orbit) base-flow condition; setting m=0 only yields a Diophantine property for r=(ℓ1−ℓ2)/(ℓ2−ℓ3), not necessarily for ℓ1/ℓ2. This remark is not needed for the main results and is not asserted in the paper. Otherwise, the proofs are substantially the same in structure and ingredients, with compatible hypotheses and conclusions (see also the proof outline tying the symbolic model, operator estimates, and equidistribution in §3) .