CONJUGACY OF PIECEWISE C1-HOMEOMORPHISMS OF CLASS P OF THE CIRCLE

Abstract. We give a characterization of piecewise C1-homeomorphism of class P of the circle with irrational rotation number and finitely many break points which are piecewise differentiably conjugate to C1-diffeomorphims. The following properties are equivalent: i) f is conjugate to a C1-diffeomorphism of the circle by a piecewise C1-homeomorphism of class P. ii) The number of break points of fn is bounded by some constant that doesn’t depend on n. iii) The product of jumps of f in the break points contained in a same orbit is equal 1. iv) f is conjugate to a C1-diffeomorphism of the circle by a piecewise quadratic homeomorphism of class P. This characterization extend Liousse’s Theorem for PL homeomor-phisms of the circle ([3])