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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 thebreak $c$ and satisfying a certain Zygmund type smoothness condition dependingon a parameter $\gamma>2.$ We prove that under a certain condition imposed onthe break size $c$, the diffeomorphisms $f$ and $\tilde{f}$ are$C^{1+\omega_{\gamma}}$-smoothly conjugate to each other, where$\omega_{\gamma}(\delta)=|\log \delta|^{-(\gamma/2-1)}.$

Rigidity for circle diffeomorphisms with breaks satisfying a Zygmund smoothness condition