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

摘要

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'-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'-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.