Sur les homéomorphismes du cercle de classe P Cr par morceaux (r1) qui sont conjugués Cr par morceaux aux rotations irrationnelles

— Let r > 1 be a real. In this paper, we study piecewise class P Cr circle homeomorphisms with irrational rotation numbers. We give characterizations for such homeomorphisms that are piecewise Cr conjugate to Cr diffeomorphisms. As a consequence, we obtain a criterion of piecewise Cr 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 Cr circle homeomorphism which contains at least two elements with rotation numbers irrational and rationally independent, is piecewise Cr conjugate to a subgroup of Cr diffeomorphisms. An analogous to a recent result of Fayad and Khanin, is obtained concerning C∞ (resp. Cω) conjugacy for piecewise class P C∞ (resp. Cω) commuting homeomorphisms of the circle. Mots-clés : homéomorphisme de classe P Cr par morceaux, condition de Hölder, nombre de rotation, conjugaison, point de coupure, point singulier, saut, mesure invariante, mesure équivalente, mesure singulière. Classification math. : 37C15, 37E10. 756 Abdelhamid ADOUANI & Habib MARZOUGUI