Geodesic
On the smoothness of the conjugacy between circle maps with a~break
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Order and chaos in dynamical systems, Tome 297 (2017), pp. 224-231

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For any $\alpha\in(0,1)$, $c\in\mathbb R_+\setminus\{1\}$ and $\gamma>0$ and for Lebesgue almost all irrational $\rho\in(0,1)$, any two $C^{2+\alpha}$-smooth circle diffeomorphisms with a break, with the same rotation number $\rho$ and the same size of the breaks $c$, are conjugate to each other via a $C^1$-smooth conjugacy whose derivative is uniformly continuous with modulus of continuity $\omega(x)=A|{\log x}|^{-\gamma}$ for some $A>0$. 
@article{TM_2017_297_a11,
     author = {Konstantin Khanin and Sa\v{s}a Koci\'c},
     title = {On the smoothness of the conjugacy between circle maps with a~break},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {224--231},
     publisher = {mathdoc},
     volume = {297},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_297_a11/}
}
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Konstantin Khanin; Saša Kocić. On the smoothness of the conjugacy between circle maps with a~break. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Order and chaos in dynamical systems, Tome 297 (2017), pp. 224-231. http://geodesic.mathdoc.fr/item/TM_2017_297_a11/