Examples of $C^1$-smoothly conjugate diffeomorphisms of the circle with break that are not $C^{1+γ}$ -smoothly conjugate - Teplins’kyi, O. Yu. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not $C^{1+γ}$-smoothly conjugate for any $γ > 0$. In this way, we show that the previous result concerning the $C^1$-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.