Polynomial Convergence Rate at Infinity for the Cusp Winding Spectrum of Generalized Schottky Groups

The paper proves that for a generalized Schottky group with one parabolic generator, if s = dimH(Λc(G)) ∈ (1/2,1), then lim_{α→∞}(s − b(α)) α^x diverges for x > 1/(2−2s)−1 and vanishes for x < 1/(2−2s)−1 (Theorem 1.1), by introducing the pressure p(α,q,b)=P(q(−a1+α)−b log|f̃′|), constructing real-analytic q(α), b(α), and establishing s−b(α) = ∫_α^∞ λ(µ_t)^{-1} q(t) dt together with precise Dirichlet-series comparisons α ≍ Σ_{l≥1} e^{−l q(α)} l^{1−2b(α)} and Σ_{l≥1} e^{−l q(α)} l^{1−2b(α)} ≍ q(α)^{−2+2b(α)} via Mellin inversion, which yields the stated polynomial rate at infinity (Theorem 1.1 and the lemmas/propositions in Section 4) . The candidate solution reaches the same asymptotic by a different (but standard) thermodynamic route: introducing a two-parameter pressure P(s,β), a free energy t(β) solving P(t(β),β)=0, the Legendre relation b(α)=inf_β [t(β)+βα] with α=−t′(β), and small-β asymptotics driven by parabolic blocks, which reduce to the same Dirichlet-series behavior. While the paper works with q(α) and Mellin transform rather than t(β), the two derivations are compatible; both ultimately hinge on the parabolic-block estimates (e.g., −t log|(γ^l)′| ≍ 2t log l) and Dirichlet-series tails. The model assumes standard analyticity/Legendre duality and two-sided comparabilities for δ−t(β) and −t′(β); these are not derived in the paper’s presentation but are consistent with its estimates and with the induced system’s Gibbs/pressure framework (cf. Lemma 3.3, Lemma 4.4, Proposition 4.5, and the identity s−b(α)=∫_α^∞ λ(µ_t)^{-1}q(t)dt) .