On piecewise smoothness of conjugacy of class P circle homeomorphisms to diffeomorphisms and rotations

We give a characterization of piecewise C ¹ class P-homeomorphism f of the circle with irrational rotation number and finitely many break points which is piecewise C ¹ conjugate to a C ¹-diffeomorphism. The following properties are equivalent:

?(i)?f is conjugate to a C ¹-diffeomorphism of the circle by a piecewise C ¹-homeomorphism. (ii)?The product of jumps of f in the break points contained in a same orbit is trivial. (iii)?f is conjugate to a C ¹-diffeomorphism of the circle by a piecewise linear (PL)-homeomorphism or a piecewise quadratic homeomorphism.

For a PL-homeomorphism f having the property (ii): f is conjugate to a rotation either by a PL-homeomorphism or by a piecewise analytic homeomorphism.