Rigidity for circle diffeomorphisms with breaks satisfying a Zygmund smoothness condition

Let $f$ and $\tilde{f}$ be two circle diffeomorphisms with a break point, with the same irrational rotation number of bounded type, the same size of the break $c$ and satisfying a certain Zygmund type smoothness condition depending on a parameter $γ>2.$ We prove that under a certain condition imposed on the break size $c$, the diffeomorphisms $f$ and $\tilde{f}$ are $C^{1+ω_γ}$-smoothly conjugate to each other, where $ω_γ(δ)=|\log δ|^{-(γ/2-1)}.$

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