On the smoothness of the conjugacy between circle maps with a break

Abstract

For any α ∈ (0, 1), c ∈ ℝ₊ \ {1} and γ > 0 and for Lebesgue almost all irrational ρ ∈ (0, 1), any two C^2+α-smooth circle diffeomorphisms with a break, with the same rotation number ρ and the same size of the breaks c, are conjugate to each other via a C¹-smooth conjugacy whose derivative is uniformly continuous with modulus of continuity ω(x) = A|log x|^−γ for some A > 0.