LOG HÖLDER CONTINUITY OF THE ROTATION NUMBER

The uploaded paper’s Theorem 2.5 states and proves that for a one-parameter family of fiber circle maps over an ergodic base, with a uniform parameter derivative bound and integrable log-Lipschitz bound M_ω, the rotation number ρ(a) is log-Hölder in the parameter a; see the precise hypotheses and statement in Theorem 2.5 and the beginning of its proof . The proof hinges on a clean one-step comparison (Lemma 2.8) for the j-shifted distances x'_n − x_n − j, yielding d_{n,j} := δ + max(0, x'_n − x_n − j) that obeys d_{n,j} ≤ M_{σ^{n-1}ω} d_{n-1,j} , and the lap-time construction n_j together with the block lower bound log(1/(2δ)) ≤ ∑_{ℓ=n_j+1}^{n_{j+1}} log M_{σ^{ℓ-1}ω} (Lemma 2.10) . Birkhoff’s ergodic theorem is then applied to the logarithms to conclude the claimed log-Hölder modulus and constant R , with the final estimate written explicitly in (8) and the closing lines of the proof . By contrast, the candidate solution replaces the paper’s j-indexed distance d_{n,j} with r_k = dist_Z(y_k−x_k) and claims that at a penultimate step before creating a new “lap,” a necessary condition is M_k r_k + Cδ ≥ 1 (later adjusted to ≥ 1/2). This claim is not justified: r_k is the distance to the nearest integer and does not directly control the increment needed to cross the next integer; lap creation can occur with r_{k+1} small, and r_{k+1} ≤ M_k r_k + Cδ does not imply any lower bound needed for crossing. Moreover, the argument assumes “resets” r_{t_{i-1}} = 0 at successive lap boundaries to start block estimates, but the candidate’s definition of lap times t_i does not ensure r_{t_{i-1}} = 0. The paper’s proof avoids both issues by working with the asymmetric quantities d_{n,j} and using Lemma 2.8 and Corollary 2.9 to obtain the block inequality rigorously . Therefore, the paper’s argument is correct, while the candidate solution contains a critical gap at the lap-creation step and an unproven reset assumption.