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We prove that every two circle diffeomorphisms with break that have the same size of break (i.e., the ratio of the left and right derivatives at the break point) and the same irrational revolution number from a certain class that contains non-Diophantine numbers are C-smoothly conjugate. [ABSTRACT FROM AUTHOR]

We prove that a C2+ α-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class D δ, 0≤ δ< α≤1, α− δ≠1, is C1+ α− δ-smoothly conjugate to a rigid rotation. This is the first sharp result on the smoothness of the conjugacy. We also derive the most precise version of Denjoy’s inequality for such diffeomorphisms. [ABSTRACT FROM AUTHOR]