On piece-wise class P Cr homeomorphisms of the circle which are piece-wise Cr (r ≥ 1) conjugate to irrational rotations

Let r >= 1 be a real. In this paper, we study piecewise class P C-r circle homeomorphisms with irrational rotation numbers. We give characterizations for such homeomorphisms that are piecewise C-r conjugate to C-r diffeomorphisms. As a consequence, we obtain a criterion of piecewise C-r conjugacy to diophantine rotations. This characterization extends those obtained by Liousse for the PL circle homeomorphisms and by Dzhalilov for the piecewise class P circle homeomorphisms with rotation numbers of constant type. We also show that every abelian subgroup of piecewise class P C-r circle homeomorphism which contains at least two elements with rotation numbers irrational and rationally independent, is piecewise C-r conjugate to a subgroup of C-r diffeomorphisms. An analogous to a recent result of Fayad and Khanin, is obtained concerning C-infinity (resp. C-omega) conjugacy for piecewise class P C-infinity (resp. C-omega) commuting homeomorphisms of the circle.