Polynomial Decay of Correlations for Nonpositively Curved Surfaces

The paper proves that for r ≥ 4 the geodesic flow on the described surface mixes at rate t^{-(a-ε)} with a=(r+2)/(r−2) by building a Poincaré map away from the neck, estimating the roof-time tail µ(τ>n)≈n^{-(a+1)} via an explicit Clairaut-based transition-time calculation, and applying renewal/transfer-operator results for flows over Young/Gibbs–Markov bases; see Theorem 1.1 and Lemma 5.3 with the induction to a nice base F=f^σ and the BBM19 machinery (yielding the t^{-(a-ε)} upper bound) . The candidate solution reproduces the same backbone: it computes T(c) by integrating 1/√(δ+|s|^r) from Clairaut’s relation to obtain the heavy-tail exponent β=a+1, and then derives the correlation rate β−1=a using operator renewal for suspension flows; the regularity and contact/nonlattice conditions are handled in the paper via BBM19’s “absence of approximate eigenfunctions,” while the candidate notes an equivalent nonlattice condition. Minor omissions in the candidate (e.g., the log-correction when passing from τ to the induced roof ϕ) do not affect the final exponent, so both arguments are essentially the same.