Rigidity for circle diffeomorphisms with breaks satisfying a Zygmund smoothness condition

The paper proves C^{1+ω_γ}-rigidity for break-equivalent circle diffeomorphisms under a Zygmund condition with γ>2, bounded-type rotation, and an explicit restriction on the break size c; its route is via renormalization to Möbius models with 1/n^γ accuracy, quantitative control of partition geometry, a tailored cohomological-equation argument requiring ∑ k_{n+1}Λ_n<∞, and a final modulus estimate for Dh (|Dh(x)-Dh(y)| ≤ A|log|x-y||^{-(γ/2-1)}) (Theorem 2.4, Theorem 3.2, Lemma 5.1, Lemma 7.1, inequality (52)) . The candidate solution follows the same overall renormalization–cohomology scheme but invokes Gottschalk–Hedlund for the cohomological equation. It slightly overstates two technical points: (i) a claim of uniform contraction of Möbius parameters and (ii) a ‘canonical’ Möbius model determined solely by c and combinatorics. These are not needed for the result as presented in the paper and are not established there. Nonetheless, the model’s argument reaches the same conclusion and is consistent with the paper’s quantitative bounds and hypotheses.