Exotic circles of a remarkable group of piecewise generalized (non linear) circle homeomorphisms

Let G be a subgroup of Homeo+(S1). An exotic circle of G is a subgroup of G which is conjugate to SO (2) in Homeo+(S1) but not conjugate to SO (2) in G . The existence of exotic circles shows that the subgroup G is far from being a Lie group. Let r ≥ 1 be an integer, r = +∞ or r ω. In this paper, we prove that the subgroup Pr( S1) of Homeo+(S1) consisting of piecewise class P Cr homeomorphisms of the circle has no exotic circles. However, we show that there exist exotic circles of a particular subgroup (denoted P1r (S1)) of Pr(S1) and we determine the conjugacy classes of all exotic circles in P1r (S1). In particular, for the group PL+(S1) consisting of piecewise linear homeomorphisms we give a simple proof of Minakawa's Theorems. (author)