HOROCYCLE FLOWS ON ABELIAN COVERS OF SURFACES OF NEGATIVE CURVATURE

The paper proves (i) the existence of a normalizing time t*(x,T) with ||T − e^{h_top t*}||_∞ ≤ C T^{1−ε}, (ii) a vector CLT for the geodesic winding cycle F*, and (iii) the asymptotic ∫_0^T f∘h_s(x) ds = a(T) Φ_T(x) ∫ f dμ + O(T log log T / (log T)^{(d+1)/2}), with explicit a(T) and Φ_T(x). These statements appear as Theorem A (and related lemmas/propositions) and are proved via transfer-operator techniques on anisotropic Banach spaces, avoiding symbolic dynamics . The candidate solution establishes the same three items using a different route: symbolic coding of the geodesic flow, a multiplicative cocycle for the stable Jacobian J_t, uniform distortion along horocycles, vector-valued CLT/LLT on Zd-extensions, and Livšic theory. The normalization estimate for τ(T,t,x) and the definition/role of t*(x,T) align with the paper’s renormalization section (Lemma 3.1, Lemma 3.6, Proposition 7.3) . The CLT is derived in the paper via a spectral function z(ω) with D²z(0) = −(4π²)Σ and a standard characteristic-function argument (Remark 6.5) , which is consistent with the model’s use of Ratner/Livšic. Finally, the paper’s main asymptotic matches the model’s: the same Gaussian weight in Φ_T(x), the same a(T) = h_top^{d/2}/[(2π)^{d/2} √det Σ] · T/(log T)^{d/2}, and the same error rate . Thus, both are correct, but the proofs differ substantially in technique.