Let f be a class P-homeomorphism of the circle. We prove that there exists a piecewise analytic homeomorphism that conjugate f to a one-class P with prescribed break points lying on pairwise distinct orbits. As a consequence, we give a sharp estimate for the smoothness of a conjugation of class P-homeomorphism f of the circle satisfying the (D)-property (i.e. the product of f-jumps in the break points contained in a same orbit is trivial), to diffeomorphism. When f does not satisfy the (D)-property the conjugating homeomorphism is never a class P and even more it is not absolutely continuous function when the total product of f-jumps in all the break points is non-trivial.
"A sharp smoothness of the conjugation of class P-homeomorphisms to diffeomorphisms." Kodai Math. J. 39 (2) 425 - 438, June 2016. https://doi.org/10.2996/kmj/1467830148