Exotic Circles in Groups of Piecewise Smooth Circle Homeomorphisms

In this paper, we deal with the existence of exotic circles in some groups of circle homeomorphisms: this means the existence of subgroups of a given group G that are topologically conjugate to the group SO(2) consisting of circle rotations but not conjugate to SO(2) in G. The existence of exotic circle shows that the group G is far from being a Lie group. Let r >= 1 be an integer, we prove on one hand, that the group P-r(S-1) consisting of piece-wise C-r class P circle homeomorphisms has no exotic circle, on the other hand, its subgroup P-1(r)(S-1) consisting of elements for which the product of all jumps at the break points is trivial has exotic circles, and we give a classification of them.