Abstract
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.
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